"""
Sage class for PARI's GEN type
See the ``PariInstance`` class for documentation and examples.
AUTHORS:
- William Stein (2006-03-01): updated to work with PARI 2.2.12-beta
- William Stein (2006-03-06): added newtonpoly
- Justin Walker: contributed some of the function definitions
- Gonzalo Tornaria: improvements to conversions; much better error
handling.
- Robert Bradshaw, Jeroen Demeyer, William Stein (2010-08-15):
Upgrade to PARI 2.4.3 (#9343)
- Jeroen Demeyer (2011-11-12): rewrite various conversion routines
(#11611, #11854, #11952)
- Peter Bruin (2013-11-17): move PariInstance to a separate file
(#15185)
"""
import math
import types
import operator
import sage.structure.element
from sage.structure.element cimport ModuleElement, RingElement, Element
from sage.misc.randstate cimport randstate, current_randstate
from sage.misc.misc_c import is_64_bit
include 'pari_err.pxi'
include 'sage/ext/stdsage.pxi'
include 'sage/ext/python.pxi'
include 'sage/ext/interrupt.pxi'
cimport cython
cdef extern from "misc.h":
int factorint_withproof_sage(GEN* ans, GEN x, GEN cutoff)
int gcmp_sage(GEN x, GEN y)
cdef extern from "mpz_pylong.h":
cdef int mpz_set_pylong(mpz_t dst, src) except -1
Integer = None
import pari_instance
from pari_instance cimport PariInstance, prec_bits_to_words
cdef PariInstance P = pari_instance.pari
@cython.final
cdef class gen(sage.structure.element.RingElement):
"""
Python extension class that models the PARI GEN type.
"""
def __init__(self):
raise RuntimeError("PARI objects cannot be instantiated directly; use pari(x) to convert x to PARI")
def __dealloc__(self):
if self.b:
sage_free(<void*> self.b)
def __repr__(self):
cdef char *c
pari_catch_sig_on()
sig_block()
c = GENtostr(self.g)
sig_unblock()
pari_catch_sig_off()
s = str(c)
pari_free(c)
return s
def __hash__(self):
"""
Return the hash of self, computed using PARI's hash_GEN().
TESTS::
sage: type(pari('1 + 2.0*I').__hash__())
<type 'int'>
"""
cdef long h
pari_catch_sig_on()
h = hash_GEN(self.g)
pari_catch_sig_off()
return h
def _testclass(self):
import test
T = test.testclass()
T._init(self)
return T
def list(self):
"""
Convert self to a list of PARI gens.
EXAMPLES:
A PARI vector becomes a Sage list::
sage: L = pari("vector(10,i,i^2)").list()
sage: L
[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]
sage: type(L)
<type 'list'>
sage: type(L[0])
<type 'sage.libs.pari.gen.gen'>
For polynomials, list() behaves as for ordinary Sage polynomials::
sage: pol = pari("x^3 + 5/3*x"); pol.list()
[0, 5/3, 0, 1]
For power series or laurent series, we get all coefficients starting
from the lowest degree term. This includes trailing zeros::
sage: R.<x> = LaurentSeriesRing(QQ)
sage: s = x^2 + O(x^8)
sage: s.list()
[1]
sage: pari(s).list()
[1, 0, 0, 0, 0, 0]
sage: s = x^-2 + O(x^0)
sage: s.list()
[1]
sage: pari(s).list()
[1, 0]
For matrices, we get a list of columns::
sage: M = matrix(ZZ,3,2,[1,4,2,5,3,6]); M
[1 4]
[2 5]
[3 6]
sage: pari(M).list()
[[1, 2, 3]~, [4, 5, 6]~]
For "scalar" types, we get a 1-element list containing ``self``::
sage: pari("42").list()
[42]
"""
if typ(self.g) == t_POL:
return list(self.Vecrev())
return list(self.Vec())
def __reduce__(self):
"""
EXAMPLES::
sage: f = pari('x^3 - 3')
sage: loads(dumps(f)) == f
True
"""
s = str(self)
import sage.libs.pari.gen_py
return sage.libs.pari.gen_py.pari, (s,)
cpdef ModuleElement _add_(self, ModuleElement right):
pari_catch_sig_on()
return P.new_gen(gadd(self.g, (<gen>right).g))
cpdef ModuleElement _sub_(self, ModuleElement right):
pari_catch_sig_on()
return P.new_gen(gsub(self.g, (<gen> right).g))
cpdef RingElement _mul_(self, RingElement right):
pari_catch_sig_on()
return P.new_gen(gmul(self.g, (<gen>right).g))
cpdef RingElement _div_(self, RingElement right):
pari_catch_sig_on()
return P.new_gen(gdiv(self.g, (<gen>right).g))
def _add_one(gen self):
"""
Return self + 1.
OUTPUT: pari gen
EXAMPLES::
sage: n = pari(5)
sage: n._add_one()
6
sage: n = pari('x^3')
sage: n._add_one()
x^3 + 1
"""
pari_catch_sig_on()
return P.new_gen(gaddsg(1, self.g))
def __mod__(self, other):
cdef gen selfgen
cdef gen othergen
if isinstance(other, gen) and isinstance(self, gen):
selfgen = self
othergen = other
pari_catch_sig_on()
return P.new_gen(gmod(selfgen.g, othergen.g))
return sage.structure.element.bin_op(self, other, operator.mod)
def __pow__(gen self, n, m):
cdef gen t0 = objtogen(n)
pari_catch_sig_on()
return P.new_gen(gpow(self.g, t0.g, prec_bits_to_words(0)))
def __neg__(gen self):
pari_catch_sig_on()
return P.new_gen(gneg(self.g))
def __xor__(gen self, n):
raise RuntimeError, "Use ** for exponentiation, not '^', which means xor\n"+\
"in Python, and has the wrong precedence."
def __rshift__(gen self, long n):
pari_catch_sig_on()
return P.new_gen(gshift(self.g, -n))
def __lshift__(gen self, long n):
pari_catch_sig_on()
return P.new_gen(gshift(self.g, n))
def __invert__(gen self):
pari_catch_sig_on()
return P.new_gen(ginv(self.g))
def getattr(gen self, attr):
"""
Return the PARI attribute with the given name.
EXAMPLES::
sage: K = pari("nfinit(x^2 - x - 1)")
sage: K.getattr("pol")
x^2 - x - 1
sage: K.getattr("disc")
5
sage: K.getattr("reg")
Traceback (most recent call last):
...
PariError: _.reg: incorrect type in reg
sage: K.getattr("zzz")
Traceback (most recent call last):
...
PariError: no function named "_.zzz"
"""
cdef str s = "_." + attr
cdef char *t = PyString_AsString(s)
pari_catch_sig_on()
return P.new_gen(closure_callgen1(strtofunction(t), self.g))
def mod(self):
"""
Given an INTMOD or POLMOD ``Mod(a,m)``, return the modulus `m`.
EXAMPLES::
sage: pari(4).Mod(5).mod()
5
sage: pari("Mod(x, x*y)").mod()
y*x
sage: pari("[Mod(4,5)]").mod()
Traceback (most recent call last):
...
TypeError: Not an INTMOD or POLMOD in mod()
"""
if typ(self.g) != t_INTMOD and typ(self.g) != t_POLMOD:
raise TypeError("Not an INTMOD or POLMOD in mod()")
pari_catch_sig_on()
return P.new_gen(gel(self.g, 1))
def nf_get_pol(self):
"""
Returns the defining polynomial of this number field.
INPUT:
- ``self`` -- A PARI number field being the output of ``nfinit()``,
``bnfinit()`` or ``bnrinit()``.
EXAMPLES::
sage: K.<a> = NumberField(x^4 - 4*x^2 + 1)
sage: pari(K).nf_get_pol()
y^4 - 4*y^2 + 1
sage: bnr = pari("K = bnfinit(x^4 - 4*x^2 + 1); bnrinit(K, 2*x)")
sage: bnr.nf_get_pol()
x^4 - 4*x^2 + 1
For relative extensions, this returns the absolute polynomial,
not the relative one::
sage: L.<b> = K.extension(x^2 - 5)
sage: pari(L).nf_get_pol() # Absolute polynomial
y^8 - 28*y^6 + 208*y^4 - 408*y^2 + 36
sage: L.pari_rnf().nf_get_pol()
x^8 - 28*x^6 + 208*x^4 - 408*x^2 + 36
TESTS::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^4 - 4*x^2 + 1)
sage: K.pari_nf().nf_get_pol()
y^4 - 4*y^2 + 1
sage: K.pari_bnf().nf_get_pol()
y^4 - 4*y^2 + 1
An error is raised for invalid input::
sage: pari("[0]").nf_get_pol()
Traceback (most recent call last):
...
PariError: incorrect type in pol
"""
pari_catch_sig_on()
return P.new_gen(member_pol(self.g))
def nf_get_diff(self):
"""
Returns the different of this number field as a PARI ideal.
INPUT:
- ``self`` -- A PARI number field being the output of ``nfinit()``,
``bnfinit()`` or ``bnrinit()``.
EXAMPLES::
sage: K.<a> = NumberField(x^4 - 4*x^2 + 1)
sage: pari(K).nf_get_diff()
[12, 0, 0, 0; 0, 12, 8, 0; 0, 0, 4, 0; 0, 0, 0, 4]
"""
pari_catch_sig_on()
return P.new_gen(member_diff(self.g))
def nf_get_sign(self):
"""
Returns a Python list ``[r1, r2]``, where ``r1`` and ``r2`` are
Python ints representing the number of real embeddings and pairs
of complex embeddings of this number field, respectively.
INPUT:
- ``self`` -- A PARI number field being the output of ``nfinit()``,
``bnfinit()`` or ``bnrinit()``.
EXAMPLES::
sage: K.<a> = NumberField(x^4 - 4*x^2 + 1)
sage: s = K.pari_nf().nf_get_sign(); s
[4, 0]
sage: type(s); type(s[0])
<type 'list'>
<type 'int'>
sage: CyclotomicField(15).pari_nf().nf_get_sign()
[0, 4]
"""
cdef long r1
cdef long r2
cdef GEN sign
pari_catch_sig_on()
sign = member_sign(self.g)
r1 = itos(gel(sign, 1))
r2 = itos(gel(sign, 2))
pari_catch_sig_off()
return [r1, r2]
def nf_get_zk(self):
"""
Returns a vector with a `\ZZ`-basis for the ring of integers of
this number field. The first element is always `1`.
INPUT:
- ``self`` -- A PARI number field being the output of ``nfinit()``,
``bnfinit()`` or ``bnrinit()``.
EXAMPLES::
sage: K.<a> = NumberField(x^4 - 4*x^2 + 1)
sage: pari(K).nf_get_zk()
[1, y, y^3 - 4*y, y^2 - 2]
"""
pari_catch_sig_on()
return P.new_gen(member_zk(self.g))
def bnf_get_no(self):
"""
Returns the class number of ``self``, a "big number field" (``bnf``).
EXAMPLES::
sage: K.<a> = QuadraticField(-65)
sage: K.pari_bnf().bnf_get_no()
8
"""
pari_catch_sig_on()
return P.new_gen(bnf_get_no(self.g))
def bnf_get_cyc(self):
"""
Returns the structure of the class group of this number field as
a vector of SNF invariants.
NOTE: ``self`` must be a "big number field" (``bnf``).
EXAMPLES::
sage: K.<a> = QuadraticField(-65)
sage: K.pari_bnf().bnf_get_cyc()
[4, 2]
"""
pari_catch_sig_on()
return P.new_gen(bnf_get_cyc(self.g))
def bnf_get_gen(self):
"""
Returns a vector of generators of the class group of this
number field.
NOTE: ``self`` must be a "big number field" (``bnf``).
EXAMPLES::
sage: K.<a> = QuadraticField(-65)
sage: G = K.pari_bnf().bnf_get_gen(); G
[[3, 2; 0, 1], [2, 1; 0, 1]]
sage: map(lambda J: K.ideal(J), G)
[Fractional ideal (3, a + 2), Fractional ideal (2, a + 1)]
"""
pari_catch_sig_on()
return P.new_gen(bnf_get_gen(self.g))
def bnf_get_reg(self):
"""
Returns the regulator of this number field.
NOTE: ``self`` must be a "big number field" (``bnf``).
EXAMPLES::
sage: K.<a> = NumberField(x^4 - 4*x^2 + 1)
sage: K.pari_bnf().bnf_get_reg()
2.66089858019037...
"""
pari_catch_sig_on()
return P.new_gen(bnf_get_reg(self.g))
def pr_get_p(self):
"""
Returns the prime of `\ZZ` lying below this prime ideal.
NOTE: ``self`` must be a PARI prime ideal (as returned by
``idealfactor`` for example).
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: F = pari(K).idealfactor(K.ideal(5)); F
[[5, [-2, 1]~, 1, 1, [2, 1]~], 1; [5, [2, 1]~, 1, 1, [-2, 1]~], 1]
sage: F[0,0].pr_get_p()
5
"""
pari_catch_sig_on()
return P.new_gen(pr_get_p(self.g))
def pr_get_e(self):
"""
Returns the ramification index (over `\QQ`) of this prime ideal.
NOTE: ``self`` must be a PARI prime ideal (as returned by
``idealfactor`` for example).
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: pari(K).idealfactor(K.ideal(2))[0,0].pr_get_e()
2
sage: pari(K).idealfactor(K.ideal(3))[0,0].pr_get_e()
1
sage: pari(K).idealfactor(K.ideal(5))[0,0].pr_get_e()
1
"""
cdef long e
pari_catch_sig_on()
e = pr_get_e(self.g)
pari_catch_sig_off()
return e
def pr_get_f(self):
"""
Returns the residue class degree (over `\QQ`) of this prime ideal.
NOTE: ``self`` must be a PARI prime ideal (as returned by
``idealfactor`` for example).
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: pari(K).idealfactor(K.ideal(2))[0,0].pr_get_f()
1
sage: pari(K).idealfactor(K.ideal(3))[0,0].pr_get_f()
2
sage: pari(K).idealfactor(K.ideal(5))[0,0].pr_get_f()
1
"""
cdef long f
pari_catch_sig_on()
f = pr_get_f(self.g)
pari_catch_sig_off()
return f
def pr_get_gen(self):
"""
Returns the second generator of this PARI prime ideal, where the
first generator is ``self.pr_get_p()``.
NOTE: ``self`` must be a PARI prime ideal (as returned by
``idealfactor`` for example).
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: g = pari(K).idealfactor(K.ideal(2))[0,0].pr_get_gen(); g; K(g)
[1, 1]~
i + 1
sage: g = pari(K).idealfactor(K.ideal(3))[0,0].pr_get_gen(); g; K(g)
[3, 0]~
3
sage: g = pari(K).idealfactor(K.ideal(5))[0,0].pr_get_gen(); g; K(g)
[-2, 1]~
i - 2
"""
pari_catch_sig_on()
return P.new_gen(pr_get_gen(self.g))
def bid_get_cyc(self):
"""
Returns the structure of the group `(O_K/I)^*`, where `I` is the
ideal represented by ``self``.
NOTE: ``self`` must be a "big ideal" (``bid``) as returned by
``idealstar`` for example.
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: J = pari(K).idealstar(K.ideal(4*i + 2))
sage: J.bid_get_cyc()
[4, 2]
"""
pari_catch_sig_on()
return P.new_gen(bid_get_cyc(self.g))
def bid_get_gen(self):
"""
Returns a vector of generators of the group `(O_K/I)^*`, where
`I` is the ideal represented by ``self``.
NOTE: ``self`` must be a "big ideal" (``bid``) with generators,
as returned by ``idealstar`` with ``flag`` = 2.
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: J = pari(K).idealstar(K.ideal(4*i + 2), 2)
sage: J.bid_get_gen()
[7, [-2, -1]~]
We get an exception if we do not supply ``flag = 2`` to
``idealstar``::
sage: J = pari(K).idealstar(K.ideal(3))
sage: J.bid_get_gen()
Traceback (most recent call last):
...
PariError: missing bid generators. Use idealstar(,,2)
"""
pari_catch_sig_on()
return P.new_gen(bid_get_gen(self.g))
def __getitem__(gen self, n):
"""
Return the nth entry of self. The indexing is 0-based, like in
Python. Note that this is *different* than the default behavior
of the PARI/GP interpreter.
EXAMPLES::
sage: p = pari('1 + 2*x + 3*x^2')
sage: p[0]
1
sage: p[2]
3
sage: p[100]
0
sage: p[-1]
0
sage: q = pari('x^2 + 3*x^3 + O(x^6)')
sage: q[3]
3
sage: q[5]
0
sage: q[6]
Traceback (most recent call last):
...
IndexError: index out of bounds
sage: m = pari('[1,2;3,4]')
sage: m[0]
[1, 3]~
sage: m[1,0]
3
sage: l = pari('List([1,2,3])')
sage: l[1]
2
sage: s = pari('"hello, world!"')
sage: s[0]
'h'
sage: s[4]
'o'
sage: s[12]
'!'
sage: s[13]
Traceback (most recent call last):
...
IndexError: index out of bounds
sage: v = pari('[1,2,3]')
sage: v[0]
1
sage: c = pari('Col([1,2,3])')
sage: c[1]
2
sage: sv = pari('Vecsmall([1,2,3])')
sage: sv[2]
3
sage: type(sv[2])
<type 'int'>
sage: tuple(pari(3/5))
(3, 5)
sage: tuple(pari('1 + 5*I'))
(1, 5)
sage: tuple(pari('Qfb(1, 2, 3)'))
(1, 2, 3)
sage: pari(57)[0]
Traceback (most recent call last):
...
TypeError: unindexable object
sage: m = pari("[[1,2;3,4],5]") ; m[0][1,0]
3
sage: v = pari(xrange(20))
sage: v[2:5]
[2, 3, 4]
sage: v[:]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: v[10:]
[10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: v[:5]
[0, 1, 2, 3, 4]
sage: v
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: v[-1]
Traceback (most recent call last):
...
IndexError: index out of bounds
sage: v[:-3]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
sage: v[5:]
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: pari([])[::]
[]
"""
cdef int pari_type
pari_type = typ(self.g)
if isinstance(n, tuple):
if pari_type != t_MAT:
raise TypeError, "self must be of pari type t_MAT"
if len(n) != 2:
raise IndexError, "index must be an integer or a 2-tuple (i,j)"
i = int(n[0])
j = int(n[1])
if i < 0 or i >= glength(<GEN>(self.g[1])):
raise IndexError, "row index out of bounds"
if j < 0 or j >= glength(self.g):
raise IndexError, "column index out of bounds"
ind = (i,j)
if self.refers_to is not None and ind in self.refers_to:
return self.refers_to[ind]
else:
val = P.new_ref(gmael(self.g, j+1, i+1), self)
if self.refers_to is None:
self.refers_to = {ind: val}
else:
self.refers_to[ind] = val
return val
elif isinstance(n, slice):
l = glength(self.g)
start,stop,step = n.indices(l)
inds = xrange(start,stop,step)
k = len(inds)
if k==0:
return P.vector(0)
if pari_type == t_VEC:
if step==1:
return self.vecextract('"'+str(start+1)+".."+str(stop)+'"')
if step==-1:
return self.vecextract('"'+str(start+1)+".."+str(stop+2)+'"')
v = P.vector(k)
for i,j in enumerate(inds):
v[i] = self[j]
return v
if pari_type == t_POL:
return self.polcoeff(n)
elif pari_type == t_SER:
bound = valp(self.g) + lg(self.g) - 2
if n >= bound:
raise IndexError, "index out of bounds"
return self.polcoeff(n)
elif pari_type in (t_INT, t_REAL, t_PADIC, t_QUAD):
raise TypeError, "unindexable object"
elif n < 0 or n >= glength(self.g):
raise IndexError, "index out of bounds"
elif pari_type == t_VEC or pari_type == t_MAT:
if self.refers_to is not None and n in self.refers_to:
return self.refers_to[n]
else:
val = P.new_ref(gel(self.g, n+1), self)
if self.refers_to is None:
self.refers_to = {n: val}
else:
self.refers_to[n] = val
return val
elif pari_type == t_VECSMALL:
return self.g[n+1]
elif pari_type == t_STR:
return chr( (<char *>(self.g+1))[n] )
elif pari_type == t_LIST:
return self.component(n+1)
elif pari_type in (t_INTMOD, t_POLMOD):
raise TypeError, "unindexable object"
else:
return P.new_ref(gel(self.g,n+1), self)
def __setitem__(gen self, n, y):
r"""
Set the nth entry to a reference to y.
- The indexing is 0-based, like everywhere else in Python, but
*unlike* in PARI/GP.
- Assignment sets the nth entry to a reference to y, assuming y is
an object of type gen. This is the same as in Python, but
*different* than what happens in the gp interpreter, where
assignment makes a copy of y.
- Because setting creates references it is *possible* to make
circular references, unlike in GP. Do *not* do this (see the
example below). If you need circular references, work at the Python
level (where they work well), not the PARI object level.
EXAMPLES::
sage: v = pari(range(10))
sage: v
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: v[0] = 10
sage: w = pari([5,8,-20])
sage: v
[10, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: v[1] = w
sage: v
[10, [5, 8, -20], 2, 3, 4, 5, 6, 7, 8, 9]
sage: w[0] = -30
sage: v
[10, [-30, 8, -20], 2, 3, 4, 5, 6, 7, 8, 9]
sage: t = v[1]; t[1] = 10 ; v
[10, [-30, 10, -20], 2, 3, 4, 5, 6, 7, 8, 9]
sage: v[1][0] = 54321 ; v
[10, [54321, 10, -20], 2, 3, 4, 5, 6, 7, 8, 9]
sage: w
[54321, 10, -20]
sage: v = pari([[[[0,1],2],3],4]) ; v[0][0][0][1] = 12 ; v
[[[[0, 12], 2], 3], 4]
sage: m = pari(matrix(2,2,range(4))) ; l = pari([5,6]) ; n = pari(matrix(2,2,[7,8,9,0])) ; m[1,0] = l ; l[1] = n ; m[1,0][1][1,1] = 1111 ; m
[0, 1; [5, [7, 8; 9, 1111]], 3]
sage: m = pari("[[1,2;3,4],5,6]") ; m[0][1,1] = 11 ; m
[[1, 2; 3, 11], 5, 6]
Finally, we create a circular reference::
sage: v = pari([0])
sage: w = pari([v])
sage: v
[0]
sage: w
[[0]]
sage: v[0] = w
Now there is a circular reference. Accessing v[0] will crash Sage.
::
sage: s=pari.vector(2,[0,0])
sage: s[:1]
[0]
sage: s[:1]=[1]
sage: s
[1, 0]
sage: type(s[0])
<type 'sage.libs.pari.gen.gen'>
sage: s = pari(range(20)) ; s
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: s[0:10:2] = range(50,55) ; s
[50, 1, 51, 3, 52, 5, 53, 7, 54, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
sage: s[10:20:3] = range(100,150) ; s
[50, 1, 51, 3, 52, 5, 53, 7, 54, 9, 100, 11, 12, 101, 14, 15, 102, 17, 18, 103]
TESTS::
sage: v = pari(xrange(10)) ; v
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: v[:] = [20..29]
sage: v
[20, 21, 22, 23, 24, 25, 26, 27, 28, 29]
sage: type(v[0])
<type 'sage.libs.pari.gen.gen'>
"""
cdef int i, j
cdef gen x = objtogen(y)
cdef long l
cdef Py_ssize_t ii, jj, step
pari_catch_sig_on()
try:
if isinstance(n, tuple):
if typ(self.g) != t_MAT:
raise TypeError, "cannot index PARI type %s by tuple"%typ(self.g)
if len(n) != 2:
raise ValueError, "matrix index must be of the form [row, column]"
i = int(n[0])
j = int(n[1])
ind = (i,j)
if i < 0 or i >= glength(<GEN>(self.g[1])):
raise IndexError, "row i(=%s) must be between 0 and %s"%(i,self.nrows()-1)
if j < 0 or j >= glength(self.g):
raise IndexError, "column j(=%s) must be between 0 and %s"%(j,self.ncols()-1)
if self.refers_to is None:
self.refers_to = {ind: x}
else:
self.refers_to[ind] = x
(<GEN>(self.g)[j+1])[i+1] = <long>(x.g)
return
elif isinstance(n, slice):
l = glength(self.g)
inds = xrange(*n.indices(l))
k = len(inds)
if k > len(y):
raise ValueError, "attempt to assign sequence of size %s to slice of size %s"%(len(y), k)
for i,j in enumerate(inds):
self[j] = y[i]
return
i = int(n)
if i < 0 or i >= glength(self.g):
raise IndexError, "index (%s) must be between 0 and %s"%(i,glength(self.g)-1)
if self.refers_to is None:
self.refers_to = {i: x}
else:
self.refers_to[i] = x
if typ(self.g) == t_POL:
i = i + 1
(self.g)[i+1] = <long>(x.g)
return
finally:
pari_catch_sig_off()
def __len__(gen self):
return glength(self.g)
def __richcmp__(left, right, int op):
return (<Element>left)._richcmp(right, op)
cdef int _cmp_c_impl(left, Element right) except -2:
"""
Comparisons
First uses PARI's cmp routine; if it decides the objects are not
comparable, it then compares the underlying strings (since in
Python all objects are supposed to be comparable).
EXAMPLES::
sage: a = pari(5)
sage: b = 10
sage: a < b
True
sage: a <= b
True
sage: a <= 5
True
sage: a > b
False
sage: a >= b
False
sage: a >= pari(10)
False
sage: a == 5
True
sage: a is 5
False
::
sage: pari(2.5) > None
True
sage: pari(3) == pari(3)
True
sage: pari('x^2 + 1') == pari('I-1')
False
sage: pari(I) == pari(I)
True
"""
return gcmp_sage(left.g, (<gen>right).g)
def __copy__(gen self):
pari_catch_sig_on()
return P.new_gen(gcopy(self.g))
def list_str(gen self):
"""
Return str that might correctly evaluate to a Python-list.
"""
s = str(self)
if s[:4] == "Mat(":
s = "[" + s[4:-1] + "]"
s = s.replace("~","")
if s.find(";") != -1:
s = s.replace(";","], [")
s = "[" + s + "]"
return eval(s)
else:
return eval(s)
def __hex__(gen self):
"""
Return the hexadecimal digits of self in lower case.
EXAMPLES::
sage: print hex(pari(0))
0
sage: print hex(pari(15))
f
sage: print hex(pari(16))
10
sage: print hex(pari(16938402384092843092843098243))
36bb1e3929d1a8fe2802f083
sage: print hex(long(16938402384092843092843098243))
0x36bb1e3929d1a8fe2802f083L
sage: print hex(pari(-16938402384092843092843098243))
-36bb1e3929d1a8fe2802f083
"""
cdef GEN x
cdef long lx, *xp
cdef long w
cdef char *s, *sp
cdef char *hexdigits
hexdigits = "0123456789abcdef"
cdef int i, j
cdef int size
x = self.g
if typ(x) != t_INT:
raise TypeError, "gen must be of PARI type t_INT"
if not signe(x):
return "0"
lx = lgefint(x)-2
size = lx*2*sizeof(long)
s = <char *>sage_malloc(size+2)
sp = s + size+1
sp[0] = 0
xp = int_LSW(x)
for i from 0 <= i < lx:
w = xp[0]
for j from 0 <= j < 2*sizeof(long):
sp = sp-1
sp[0] = hexdigits[w & 15]
w = w>>4
xp = int_nextW(xp)
while sp[0] == c'0':
sp = sp+1
if signe(x) < 0:
sp = sp-1
sp[0] = c'-'
k = <object>sp
sage_free(s)
return k
def __int__(gen self):
"""
Convert ``self`` to a Python integer.
If the number is too large to fit into a Pyhon ``int``, a
Python ``long`` is returned instead.
EXAMPLES::
sage: int(pari(0))
0
sage: int(pari(10))
10
sage: int(pari(-10))
-10
sage: int(pari(123456789012345678901234567890))
123456789012345678901234567890L
sage: int(pari(-123456789012345678901234567890))
-123456789012345678901234567890L
sage: int(pari(2^31-1))
2147483647
sage: int(pari(-2^31))
-2147483648
sage: int(pari("Pol(10)"))
10
sage: int(pari("Mod(2, 7)"))
2
sage: int(pari(RealField(63)(2^63-1)))
9223372036854775807L # 32-bit
9223372036854775807 # 64-bit
sage: int(pari(RealField(63)(2^63+2)))
9223372036854775810L
"""
global Integer
if Integer is None:
import sage.rings.integer
Integer = sage.rings.integer.Integer
return int(Integer(self))
def python_list_small(gen self):
"""
Return a Python list of the PARI gens. This object must be of type
t_VECSMALL, and the resulting list contains python 'int's.
EXAMPLES::
sage: v=pari([1,2,3,10,102,10]).Vecsmall()
sage: w = v.python_list_small()
sage: w
[1, 2, 3, 10, 102, 10]
sage: type(w[0])
<type 'int'>
"""
cdef long n, m
if typ(self.g) != t_VECSMALL:
raise TypeError, "Object (=%s) must be of type t_VECSMALL."%self
V = []
m = glength(self.g)
for n from 0 <= n < m:
V.append(self.g[n+1])
return V
def python_list(gen self):
"""
Return a Python list of the PARI gens. This object must be of type
t_VEC.
INPUT: None
OUTPUT:
- ``list`` - Python list whose elements are the
elements of the input gen.
EXAMPLES::
sage: v=pari([1,2,3,10,102,10])
sage: w = v.python_list()
sage: w
[1, 2, 3, 10, 102, 10]
sage: type(w[0])
<type 'sage.libs.pari.gen.gen'>
sage: pari("[1,2,3]").python_list()
[1, 2, 3]
"""
cdef long n, m
cdef gen t
if typ(self.g) != t_VEC:
raise TypeError, "Object (=%s) must be of type t_VEC."%self
m = glength(self.g)
V = []
for n from 0 <= n < m:
V.append(self.__getitem__(n))
return V
def python(self, locals=None):
"""
Return Python eval of self.
Note: is self is a real (type t_REAL) the result will be a
RealField element of the equivalent precision; if self is a complex
(type t_COMPLEX) the result will be a ComplexField element of
precision the minimum precision of the real and imaginary parts.
EXAMPLES::
sage: pari('389/17').python()
389/17
sage: f = pari('(2/3)*x^3 + x - 5/7 + y'); f
2/3*x^3 + x + (y - 5/7)
sage: var('x,y')
(x, y)
sage: f.python({'x':x, 'y':y})
2/3*x^3 + x + y - 5/7
You can also use .sage, which is a psynonym::
sage: f.sage({'x':x, 'y':y})
2/3*x^3 + x + y - 5/7
"""
import sage.libs.pari.gen_py
return sage.libs.pari.gen_py.python(self, locals=locals)
sage = python
_sage_ = _eval_ = python
def __long__(gen self):
"""
Convert ``self`` to a Python ``long``.
EXAMPLES::
sage: long(pari(0))
0L
sage: long(pari(10))
10L
sage: long(pari(-10))
-10L
sage: long(pari(123456789012345678901234567890))
123456789012345678901234567890L
sage: long(pari(-123456789012345678901234567890))
-123456789012345678901234567890L
sage: long(pari(2^31-1))
2147483647L
sage: long(pari(-2^31))
-2147483648L
sage: long(pari("Pol(10)"))
10L
sage: long(pari("Mod(2, 7)"))
2L
"""
global Integer
if Integer is None:
import sage.rings.integer
Integer = sage.rings.integer.Integer
return long(Integer(self))
def __float__(gen self):
"""
Return Python float.
"""
cdef double d
pari_catch_sig_on()
d = gtodouble(self.g)
pari_catch_sig_off()
return d
def __complex__(self):
r"""
Return ``self`` as a Python ``complex``
value.
EXAMPLES::
sage: g = pari(-1.0)^(1/5); g
0.809016994374947 + 0.587785252292473*I
sage: g.__complex__()
(0.8090169943749475+0.5877852522924731j)
sage: complex(g)
(0.8090169943749475+0.5877852522924731j)
::
sage: g = pari(Integers(5)(3)); g
Mod(3, 5)
sage: complex(g)
Traceback (most recent call last):
...
PariError: incorrect type in greal/gimag
"""
cdef double re, im
pari_catch_sig_on()
re = gtodouble(greal(self.g))
im = gtodouble(gimag(self.g))
pari_catch_sig_off()
return complex(re, im)
def __nonzero__(self):
"""
EXAMPLES::
sage: pari('1').__nonzero__()
True
sage: pari('x').__nonzero__()
True
sage: bool(pari(0))
False
sage: a = pari('Mod(0,3)')
sage: a.__nonzero__()
False
"""
return not gequal0(self.g)
def gequal(gen a, b):
r"""
Check whether `a` and `b` are equal using PARI's ``gequal``.
EXAMPLES::
sage: a = pari(1); b = pari(1.0); c = pari('"some_string"')
sage: a.gequal(a)
True
sage: b.gequal(b)
True
sage: c.gequal(c)
True
sage: a.gequal(b)
True
sage: a.gequal(c)
False
WARNING: this relation is not transitive::
sage: a = pari('[0]'); b = pari(0); c = pari('[0,0]')
sage: a.gequal(b)
True
sage: b.gequal(c)
True
sage: a.gequal(c)
False
"""
cdef gen t0 = objtogen(b)
pari_catch_sig_on()
cdef int ret = gequal(a.g, t0.g)
pari_catch_sig_off()
return ret != 0
def gequal0(gen a):
r"""
Check whether `a` is equal to zero.
EXAMPLES::
sage: pari(0).gequal0()
True
sage: pari(1).gequal0()
False
sage: pari(1e-100).gequal0()
False
sage: pari("0.0 + 0.0*I").gequal0()
True
sage: pari(GF(3^20,'t')(0)).gequal0()
True
"""
pari_catch_sig_on()
cdef int ret = gequal0(a.g)
pari_catch_sig_off()
return ret != 0
def gequal_long(gen a, long b):
r"""
Check whether `a` is equal to the ``long int`` `b` using PARI's ``gequalsg``.
EXAMPLES::
sage: a = pari(1); b = pari(2.0); c = pari('3*matid(3)')
sage: a.gequal_long(1)
True
sage: a.gequal_long(-1)
False
sage: a.gequal_long(0)
False
sage: b.gequal_long(2)
True
sage: b.gequal_long(-2)
False
sage: c.gequal_long(3)
True
sage: c.gequal_long(-3)
False
"""
pari_catch_sig_on()
cdef int ret = gequalsg(b, a.g)
pari_catch_sig_off()
return ret != 0
def isprime(gen self, long flag=0):
"""
isprime(x, flag=0): Returns True if x is a PROVEN prime number, and
False otherwise.
INPUT:
- ``flag`` - int 0 (default): use a combination of
algorithms. 1: certify primality using the Pocklington-Lehmer Test.
2: certify primality using the APRCL test.
OUTPUT:
- ``bool`` - True or False
EXAMPLES::
sage: pari(9).isprime()
False
sage: pari(17).isprime()
True
sage: n = pari(561) # smallest Carmichael number
sage: n.isprime() # not just a pseudo-primality test!
False
sage: n.isprime(1)
False
sage: n.isprime(2)
False
"""
pari_catch_sig_on()
cdef long t = signe(gisprime(self.g, flag))
P.clear_stack()
return t != 0
def qfbhclassno(gen n):
r"""
Computes the Hurwitz-Kronecker class number of `n`.
INPUT:
- `n` (gen) -- a non-negative integer
.. note::
If `n` is large (more than `5*10^5`), the result is
conditional upon GRH.
EXAMPLES:
The Hurwitx class number is 0 is n is congruent to 1 or 2 modulo 4::
sage: pari(-10007).qfbhclassno()
0
sage: pari(-2).qfbhclassno()
0
It is -1/12 for n=0::
sage: pari(0).qfbhclassno()
-1/12
Otherwise it is the number of classes of positive definite
binary quadratic forms with discriminant `-n`, weighted by
`1/m` where `m` is the number of automorphisms of the form::
sage: pari(4).qfbhclassno()
1/2
sage: pari(3).qfbhclassno()
1/3
sage: pari(23).qfbhclassno()
3
"""
pari_catch_sig_on()
return P.new_gen(hclassno(n.g))
def ispseudoprime(gen self, long flag=0):
"""
ispseudoprime(x, flag=0): Returns True if x is a pseudo-prime
number, and False otherwise.
INPUT:
- ``flag`` - int 0 (default): checks whether x is a
Baillie-Pomerance-Selfridge-Wagstaff pseudo prime (strong
Rabin-Miller pseudo prime for base 2, followed by strong Lucas test
for the sequence (P,-1), P smallest positive integer such that
`P^2 - 4` is not a square mod x). 0: checks whether x is a
strong Miller-Rabin pseudo prime for flag randomly chosen bases
(with end-matching to catch square roots of -1).
OUTPUT:
- ``bool`` - True or False
EXAMPLES::
sage: pari(9).ispseudoprime()
False
sage: pari(17).ispseudoprime()
True
sage: n = pari(561) # smallest Carmichael number
sage: n.ispseudoprime(2)
False
"""
pari_catch_sig_on()
cdef long t = ispseudoprime(self.g, flag)
pari_catch_sig_off()
return t != 0
def ispower(gen self, k=None):
r"""
Determine whether or not self is a perfect k-th power. If k is not
specified, find the largest k so that self is a k-th power.
INPUT:
- ``k`` - int (optional)
OUTPUT:
- ``power`` - int, what power it is
- ``g`` - what it is a power of
EXAMPLES::
sage: pari(9).ispower()
(2, 3)
sage: pari(17).ispower()
(1, 17)
sage: pari(17).ispower(2)
(False, None)
sage: pari(17).ispower(1)
(1, 17)
sage: pari(2).ispower()
(1, 2)
"""
cdef int n
cdef GEN x
cdef gen t0
if k is None:
pari_catch_sig_on()
n = gisanypower(self.g, &x)
if n == 0:
pari_catch_sig_off()
return 1, self
else:
return n, P.new_gen(x)
else:
t0 = objtogen(k)
pari_catch_sig_on()
n = ispower(self.g, t0.g, &x)
if n == 0:
pari_catch_sig_off()
return False, None
else:
return k, P.new_gen(x)
def divrem(gen x, y, var=-1):
"""
divrem(x, y, v): Euclidean division of x by y giving as a
2-dimensional column vector the quotient and the remainder, with
respect to v (to main variable if v is omitted).
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(divrem(x.g, t0.g, P.get_var(var)))
def lex(gen x, y):
"""
lex(x,y): Compare x and y lexicographically (1 if xy, 0 if x==y, -1
if xy)
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
r = lexcmp(x.g, t0.g)
pari_catch_sig_off()
return r
def max(gen x, y):
"""
max(x,y): Return the maximum of x and y.
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gmax(x.g, t0.g))
def min(gen x, y):
"""
min(x,y): Return the minimum of x and y.
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gmin(x.g, t0.g))
def shift(gen x, long n):
"""
shift(x,n): shift x left n bits if n=0, right -n bits if n0.
"""
pari_catch_sig_on()
return P.new_gen(gshift(x.g, n))
def shiftmul(gen x, long n):
"""
shiftmul(x,n): Return the product of x by `2^n`.
"""
pari_catch_sig_on()
return P.new_gen(gmul2n(x.g, n))
def moebius(gen x):
"""
moebius(x): Moebius function of x.
"""
pari_catch_sig_on()
return P.new_gen(gmoebius(x.g))
def sign(gen x):
"""
Return the sign of x, where x is of type integer, real or
fraction.
EXAMPLES::
sage: pari(pi).sign()
1
sage: pari(0).sign()
0
sage: pari(-1/2).sign()
-1
PARI throws an error if you attempt to take the sign of a
complex number::
sage: pari(I).sign()
Traceback (most recent call last):
...
PariError: incorrect type in gsigne
"""
pari_catch_sig_on()
r = gsigne(x.g)
pari_catch_sig_off()
return r
def vecmax(gen x):
"""
vecmax(x): Return the maximum of the elements of the vector/matrix
x.
"""
pari_catch_sig_on()
return P.new_gen(vecmax(x.g))
def vecmin(gen x):
"""
vecmin(x): Return the maximum of the elements of the vector/matrix
x.
"""
pari_catch_sig_on()
return P.new_gen(vecmin(x.g))
def Col(gen x, long n = 0):
"""
Transform the object `x` into a column vector with minimal size `|n|`.
INPUT:
- ``x`` -- gen
- ``n`` -- Make the column vector of minimal length `|n|`. If `n > 0`,
append zeros; if `n < 0`, prepend zeros.
OUTPUT:
A PARI column vector (type ``t_COL``)
EXAMPLES::
sage: pari(1.5).Col()
[1.50000000000000]~
sage: pari([1,2,3,4]).Col()
[1, 2, 3, 4]~
sage: pari('[1,2; 3,4]').Col()
[[1, 2], [3, 4]]~
sage: pari('"Sage"').Col()
["S", "a", "g", "e"]~
sage: pari('x + 3*x^3').Col()
[3, 0, 1, 0]~
sage: pari('x + 3*x^3 + O(x^5)').Col()
[1, 0, 3, 0]~
We demonstate the `n` argument::
sage: pari([1,2,3,4]).Col(2)
[1, 2, 3, 4]~
sage: pari([1,2,3,4]).Col(-2)
[1, 2, 3, 4]~
sage: pari([1,2,3,4]).Col(6)
[1, 2, 3, 4, 0, 0]~
sage: pari([1,2,3,4]).Col(-6)
[0, 0, 1, 2, 3, 4]~
See also :meth:`Vec` (create a row vector) for more examples
and :meth:`Colrev` (create a column in reversed order).
"""
pari_catch_sig_on()
return P.new_gen(_Vec_append(gtocol(x.g), gen_0, n))
def Colrev(gen x, long n = 0):
"""
Transform the object `x` into a column vector with minimal size `|n|`.
The order of the resulting vector is reversed compared to :meth:`Col`.
INPUT:
- ``x`` -- gen
- ``n`` -- Make the vector of minimal length `|n|`. If `n > 0`,
prepend zeros; if `n < 0`, append zeros.
OUTPUT:
A PARI column vector (type ``t_COL``)
EXAMPLES::
sage: pari(1.5).Colrev()
[1.50000000000000]~
sage: pari([1,2,3,4]).Colrev()
[4, 3, 2, 1]~
sage: pari('[1,2; 3,4]').Colrev()
[[3, 4], [1, 2]]~
sage: pari('x + 3*x^3').Colrev()
[0, 1, 0, 3]~
We demonstate the `n` argument::
sage: pari([1,2,3,4]).Colrev(2)
[4, 3, 2, 1]~
sage: pari([1,2,3,4]).Colrev(-2)
[4, 3, 2, 1]~
sage: pari([1,2,3,4]).Colrev(6)
[0, 0, 4, 3, 2, 1]~
sage: pari([1,2,3,4]).Colrev(-6)
[4, 3, 2, 1, 0, 0]~
"""
pari_catch_sig_on()
cdef GEN v = _Vec_append(gtocol(x.g), gen_0, n)
cdef GEN L = v + 1
cdef GEN R = v + (lg(v)-1)
cdef long t
while (L < R):
t = L[0]
L[0] = R[0]
R[0] = t
L += 1
R -= 1
return P.new_gen(v)
def List(gen x):
"""
List(x): transforms the PARI vector or list x into a list.
EXAMPLES::
sage: v = pari([1,2,3])
sage: v
[1, 2, 3]
sage: v.type()
't_VEC'
sage: w = v.List()
sage: w
List([1, 2, 3])
sage: w.type()
't_LIST'
"""
pari_catch_sig_on()
return P.new_gen(gtolist(x.g))
def Mat(gen x):
"""
Mat(x): Returns the matrix defined by x.
- If x is already a matrix, a copy of x is created and returned.
- If x is not a vector or a matrix, this function returns a 1x1
matrix.
- If x is a row (resp. column) vector, this functions returns
a 1-row (resp. 1-column) matrix, *unless* all elements are
column (resp. row) vectors of the same length, in which case
the vectors are concatenated sideways and the associated big
matrix is returned.
INPUT:
- ``x`` - gen
OUTPUT:
- ``gen`` - a PARI matrix
EXAMPLES::
sage: x = pari(5)
sage: x.type()
't_INT'
sage: y = x.Mat()
sage: y
Mat(5)
sage: y.type()
't_MAT'
sage: x = pari('[1,2;3,4]')
sage: x.type()
't_MAT'
sage: x = pari('[1,2,3,4]')
sage: x.type()
't_VEC'
sage: y = x.Mat()
sage: y
Mat([1, 2, 3, 4])
sage: y.type()
't_MAT'
::
sage: v = pari('[1,2;3,4]').Vec(); v
[[1, 3]~, [2, 4]~]
sage: v.Mat()
[1, 2; 3, 4]
sage: v = pari('[1,2;3,4]').Col(); v
[[1, 2], [3, 4]]~
sage: v.Mat()
[1, 2; 3, 4]
"""
pari_catch_sig_on()
return P.new_gen(gtomat(x.g))
def Mod(gen x, y):
"""
Mod(x, y): Returns the object x modulo y, denoted Mod(x, y).
The input y must be a an integer or a polynomial:
- If y is an INTEGER, x must also be an integer, a rational
number, or a p-adic number compatible with the modulus y.
- If y is a POLYNOMIAL, x must be a scalar (which is not a
polmod), a polynomial, a rational function, or a power
series.
.. warning::
This function is not the same as ``x % y`` which is an
integer or a polynomial.
INPUT:
- ``x`` - gen
- ``y`` - integer or polynomial
OUTPUT:
- ``gen`` - intmod or polmod
EXAMPLES::
sage: z = pari(3)
sage: x = z.Mod(pari(7))
sage: x
Mod(3, 7)
sage: x^2
Mod(2, 7)
sage: x^100
Mod(4, 7)
sage: x.type()
't_INTMOD'
::
sage: f = pari("x^2 + x + 1")
sage: g = pari("x")
sage: a = g.Mod(f)
sage: a
Mod(x, x^2 + x + 1)
sage: a*a
Mod(-x - 1, x^2 + x + 1)
sage: a.type()
't_POLMOD'
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gmodulo(x.g, t0.g))
def Pol(self, v=-1):
"""
Pol(x, v): convert x into a polynomial with main variable v and
return the result.
- If x is a scalar, returns a constant polynomial.
- If x is a power series, the effect is identical to
``truncate``, i.e. it chops off the `O(X^k)`.
- If x is a vector, this function creates the polynomial whose
coefficients are given in x, with x[0] being the leading
coefficient (which can be zero).
.. warning::
This is *not* a substitution function. It will not
transform an object containing variables of higher priority
than v::
sage: pari('x+y').Pol('y')
Traceback (most recent call last):
...
PariError: variable must have higher priority in gtopoly
INPUT:
- ``x`` - gen
- ``v`` - (optional) which variable, defaults to 'x'
OUTPUT:
- ``gen`` - a polynomial
EXAMPLES::
sage: v = pari("[1,2,3,4]")
sage: f = v.Pol()
sage: f
x^3 + 2*x^2 + 3*x + 4
sage: f*f
x^6 + 4*x^5 + 10*x^4 + 20*x^3 + 25*x^2 + 24*x + 16
::
sage: v = pari("[1,2;3,4]")
sage: v.Pol()
[1, 3]~*x + [2, 4]~
"""
pari_catch_sig_on()
return P.new_gen(gtopoly(self.g, P.get_var(v)))
def Polrev(self, v=-1):
"""
Polrev(x, v): Convert x into a polynomial with main variable v and
return the result. This is the reverse of Pol if x is a vector,
otherwise it is identical to Pol. By "reverse" we mean that the
coefficients are reversed.
INPUT:
- ``x`` - gen
OUTPUT:
- ``gen`` - a polynomial
EXAMPLES::
sage: v = pari("[1,2,3,4]")
sage: f = v.Polrev()
sage: f
4*x^3 + 3*x^2 + 2*x + 1
sage: v.Pol()
x^3 + 2*x^2 + 3*x + 4
sage: v.Polrev('y')
4*y^3 + 3*y^2 + 2*y + 1
Note that Polrev does *not* reverse the coefficients of a
polynomial! ::
sage: f
4*x^3 + 3*x^2 + 2*x + 1
sage: f.Polrev()
4*x^3 + 3*x^2 + 2*x + 1
sage: v = pari("[1,2;3,4]")
sage: v.Polrev()
[2, 4]~*x + [1, 3]~
"""
pari_catch_sig_on()
return P.new_gen(gtopolyrev(self.g, P.get_var(v)))
def Qfb(gen a, b, c, D=0, unsigned long precision=0):
"""
Qfb(a,b,c,D=0.): Returns the binary quadratic form
.. math::
ax^2 + bxy + cy^2.
The optional D is 0 by default and initializes Shank's distance if
`b^2 - 4ac > 0`. The discriminant of the quadratic form must not
be a perfect square.
.. note::
Negative definite forms are not implemented, so use their
positive definite counterparts instead. (I.e., if f is a
negative definite quadratic form, then -f is positive
definite.)
INPUT:
- ``a`` - gen
- ``b`` - gen
- ``c`` - gen
- ``D`` - gen (optional, defaults to 0)
OUTPUT:
- ``gen`` - binary quadratic form
EXAMPLES::
sage: pari(3).Qfb(7, 1)
Qfb(3, 7, 1, 0.E-19)
sage: pari(3).Qfb(7, 2) # discriminant is 25
Traceback (most recent call last):
...
PariError: square discriminant in Qfb
"""
cdef gen t0 = objtogen(b)
cdef gen t1 = objtogen(c)
cdef gen t2 = objtogen(D)
pari_catch_sig_on()
return P.new_gen(Qfb0(a.g, t0.g, t1.g, t2.g, prec_bits_to_words(precision)))
def Ser(gen x, v=-1, long seriesprecision=16):
"""
Ser(x,v=x): Create a power series from x with main variable v and
return the result.
- If x is a scalar, this gives a constant power series with
precision given by the default series precision, as returned
by get_series_precision().
- If x is a polynomial, the precision is the greatest of
get_series_precision() and the degree of the polynomial.
- If x is a vector, the precision is similarly given, and the
coefficients of the vector are understood to be the
coefficients of the power series starting from the constant
term (i.e. the reverse of the function Pol).
.. warning::
This is *not* a substitution function. It will not
transform an object containing variables of higher priority
than v.
INPUT:
- ``x`` - gen
- ``v`` - PARI variable (default: x)
OUTPUT:
- ``gen`` - PARI object of PARI type t_SER
EXAMPLES::
sage: pari(2).Ser()
2 + O(x^16)
sage: x = pari([1,2,3,4,5])
sage: x.Ser()
1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + O(x^5)
sage: f = x.Ser('v'); print f
1 + 2*v + 3*v^2 + 4*v^3 + 5*v^4 + O(v^5)
sage: pari(1)/f
1 - 2*v + v^2 + O(v^5)
sage: pari('x^5').Ser(seriesprecision=20)
x^5 + O(x^25)
sage: pari('1/x').Ser(seriesprecision=1)
x^-1 + O(x^0)
"""
pari_catch_sig_on()
return P.new_gen(gtoser(x.g, P.get_var(v), seriesprecision))
def Set(gen x):
"""
Set(x): convert x into a set, i.e. a row vector of strings in
increasing lexicographic order.
INPUT:
- ``x`` - gen
OUTPUT:
- ``gen`` - a vector of strings in increasing
lexicographic order.
EXAMPLES::
sage: pari([1,5,2]).Set()
["1", "2", "5"]
sage: pari([]).Set() # the empty set
[]
sage: pari([1,1,-1,-1,3,3]).Set()
["-1", "1", "3"]
sage: pari(1).Set()
["1"]
sage: pari('1/(x*y)').Set()
["1/(y*x)"]
sage: pari('["bc","ab","bc"]').Set()
["\"ab\"", "\"bc\""]
"""
pari_catch_sig_on()
return P.new_gen(gtoset(x.g))
def Str(self):
"""
Str(self): Return the print representation of self as a PARI
object.
INPUT:
- ``self`` - gen
OUTPUT:
- ``gen`` - a PARI gen of type t_STR, i.e., a PARI
string
EXAMPLES::
sage: pari([1,2,['abc',1]]).Str()
"[1, 2, [abc, 1]]"
sage: pari([1,1, 1.54]).Str()
"[1, 1, 1.54000000000000]"
sage: pari(1).Str() # 1 is automatically converted to string rep
"1"
sage: x = pari('x') # PARI variable "x"
sage: x.Str() # is converted to string rep.
"x"
sage: x.Str().type()
't_STR'
"""
cdef char* c
pari_catch_sig_on()
sig_block()
c = GENtostr(self.g)
sig_unblock()
v = P.new_gen(strtoGENstr(c))
pari_free(c)
return v
def Strchr(gen x):
"""
Strchr(x): converts x to a string, translating each integer into a
character (in ASCII).
.. note::
:meth:`.Vecsmall` is (essentially) the inverse to :meth:`.Strchr`.
INPUT:
- ``x`` - PARI vector of integers
OUTPUT:
- ``gen`` - a PARI string
EXAMPLES::
sage: pari([65,66,123]).Strchr()
"AB{"
sage: pari('"Sage"').Vecsmall() # pari('"Sage"') --> PARI t_STR
Vecsmall([83, 97, 103, 101])
sage: _.Strchr()
"Sage"
sage: pari([83, 97, 103, 101]).Strchr()
"Sage"
"""
pari_catch_sig_on()
return P.new_gen(Strchr(x.g))
def Strexpand(gen x):
"""
Strexpand(x): Concatenate the entries of the vector x into a single
string, performing tilde expansion.
.. note::
I have no clue what the point of this function is. - William
"""
if typ(x.g) != t_VEC:
raise TypeError, "x must be of type t_VEC."
pari_catch_sig_on()
return P.new_gen(Strexpand(x.g))
def Strtex(gen x):
r"""
Strtex(x): Translates the vector x of PARI gens to TeX format and
returns the resulting concatenated strings as a PARI t_STR.
INPUT:
- ``x`` - gen
OUTPUT:
- ``gen`` - PARI t_STR (string)
EXAMPLES::
sage: v=pari('x^2')
sage: v.Strtex()
"x^2"
sage: v=pari(['1/x^2','x'])
sage: v.Strtex()
"\\frac{1}{x^2}x"
sage: v=pari(['1 + 1/x + 1/(y+1)','x-1'])
sage: v.Strtex()
"\\frac{ \\left(y\n + 2\\right) x\n + \\left(y\n + 1\\right) }{ \\left(y\n + 1\\right) x}x\n - 1"
"""
if typ(x.g) != t_VEC:
x = P.vector(1, [x])
pari_catch_sig_on()
return P.new_gen(Strtex(x.g))
def printtex(gen x):
return x.Strtex()
def Vec(gen x, long n = 0):
"""
Transform the object `x` into a vector with minimal size `|n|`.
INPUT:
- ``x`` -- gen
- ``n`` -- Make the vector of minimal length `|n|`. If `n > 0`,
append zeros; if `n < 0`, prepend zeros.
OUTPUT:
A PARI vector (type ``t_VEC``)
EXAMPLES::
sage: pari(1).Vec()
[1]
sage: pari('x^3').Vec()
[1, 0, 0, 0]
sage: pari('x^3 + 3*x - 2').Vec()
[1, 0, 3, -2]
sage: pari([1,2,3]).Vec()
[1, 2, 3]
sage: pari('[1, 2; 3, 4]').Vec()
[[1, 3]~, [2, 4]~]
sage: pari('"Sage"').Vec()
["S", "a", "g", "e"]
sage: pari('2*x^2 + 3*x^3 + O(x^5)').Vec()
[2, 3, 0]
sage: pari('2*x^-2 + 3*x^3 + O(x^5)').Vec()
[2, 0, 0, 0, 0, 3, 0]
Note the different term ordering for polynomials and series::
sage: pari('1 + x + 3*x^3 + O(x^5)').Vec()
[1, 1, 0, 3, 0]
sage: pari('1 + x + 3*x^3').Vec()
[3, 0, 1, 1]
We demonstate the `n` argument::
sage: pari([1,2,3,4]).Vec(2)
[1, 2, 3, 4]
sage: pari([1,2,3,4]).Vec(-2)
[1, 2, 3, 4]
sage: pari([1,2,3,4]).Vec(6)
[1, 2, 3, 4, 0, 0]
sage: pari([1,2,3,4]).Vec(-6)
[0, 0, 1, 2, 3, 4]
See also :meth:`Col` (create a column vector) and :meth:`Vecrev`
(create a vector in reversed order).
"""
pari_catch_sig_on()
return P.new_gen(_Vec_append(gtovec(x.g), gen_0, n))
def Vecrev(gen x, long n = 0):
"""
Transform the object `x` into a vector with minimal size `|n|`.
The order of the resulting vector is reversed compared to :meth:`Vec`.
INPUT:
- ``x`` -- gen
- ``n`` -- Make the vector of minimal length `|n|`. If `n > 0`,
prepend zeros; if `n < 0`, append zeros.
OUTPUT:
A PARI vector (type ``t_VEC``)
EXAMPLES::
sage: pari(1).Vecrev()
[1]
sage: pari('x^3').Vecrev()
[0, 0, 0, 1]
sage: pari('x^3 + 3*x - 2').Vecrev()
[-2, 3, 0, 1]
sage: pari([1, 2, 3]).Vecrev()
[3, 2, 1]
sage: pari('Col([1, 2, 3])').Vecrev()
[3, 2, 1]
sage: pari('[1, 2; 3, 4]').Vecrev()
[[2, 4]~, [1, 3]~]
sage: pari('"Sage"').Vecrev()
["e", "g", "a", "S"]
We demonstate the `n` argument::
sage: pari([1,2,3,4]).Vecrev(2)
[4, 3, 2, 1]
sage: pari([1,2,3,4]).Vecrev(-2)
[4, 3, 2, 1]
sage: pari([1,2,3,4]).Vecrev(6)
[0, 0, 4, 3, 2, 1]
sage: pari([1,2,3,4]).Vecrev(-6)
[4, 3, 2, 1, 0, 0]
"""
pari_catch_sig_on()
return P.new_gen(_Vec_append(gtovecrev(x.g), gen_0, -n))
def Vecsmall(gen x, long n = 0):
"""
Transform the object `x` into a ``t_VECSMALL`` with minimal size `|n|`.
INPUT:
- ``x`` -- gen
- ``n`` -- Make the vector of minimal length `|n|`. If `n > 0`,
append zeros; if `n < 0`, prepend zeros.
OUTPUT:
A PARI vector of small integers (type ``t_VECSMALL``)
EXAMPLES::
sage: pari([1,2,3]).Vecsmall()
Vecsmall([1, 2, 3])
sage: pari('"Sage"').Vecsmall()
Vecsmall([83, 97, 103, 101])
sage: pari(1234).Vecsmall()
Vecsmall([1234])
sage: pari('x^2 + 2*x + 3').Vecsmall()
Traceback (most recent call last):
...
PariError: incorrect type in vectosmall
We demonstate the `n` argument::
sage: pari([1,2,3]).Vecsmall(2)
Vecsmall([1, 2, 3])
sage: pari([1,2,3]).Vecsmall(-2)
Vecsmall([1, 2, 3])
sage: pari([1,2,3]).Vecsmall(6)
Vecsmall([1, 2, 3, 0, 0, 0])
sage: pari([1,2,3]).Vecsmall(-6)
Vecsmall([0, 0, 0, 1, 2, 3])
"""
pari_catch_sig_on()
return P.new_gen(_Vec_append(gtovecsmall(x.g), <GEN>0, n))
def binary(gen x):
"""
binary(x): gives the vector formed by the binary digits of abs(x),
where x is of type t_INT.
INPUT:
- ``x`` - gen of type t_INT
OUTPUT:
- ``gen`` - of type t_VEC
EXAMPLES::
sage: pari(0).binary()
[0]
sage: pari(-5).binary()
[1, 0, 1]
sage: pari(5).binary()
[1, 0, 1]
sage: pari(2005).binary()
[1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1]
::
sage: pari('"2"').binary()
Traceback (most recent call last):
...
TypeError: x (="2") must be of type t_INT, but is of type t_STR.
"""
if typ(x.g) != t_INT:
raise TypeError, "x (=%s) must be of type t_INT, but is of type %s."%(x,x.type())
pari_catch_sig_on()
return P.new_gen(binaire(x.g))
def bitand(gen x, y):
"""
bitand(x,y): Bitwise and of two integers x and y. Negative numbers
behave as if modulo some large power of 2.
INPUT:
- ``x`` - gen (of type t_INT)
- ``y`` - coercible to gen (of type t_INT)
OUTPUT:
- ``gen`` - of type type t_INT
EXAMPLES::
sage: pari(8).bitand(4)
0
sage: pari(8).bitand(8)
8
sage: pari(6).binary()
[1, 1, 0]
sage: pari(7).binary()
[1, 1, 1]
sage: pari(6).bitand(7)
6
sage: pari(19).bitand(-1)
19
sage: pari(-1).bitand(-1)
-1
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gbitand(x.g, t0.g))
def bitneg(gen x, long n=-1):
r"""
bitneg(x,n=-1): Bitwise negation of the integer x truncated to n
bits. n=-1 (the default) represents an infinite sequence of the bit
1. Negative numbers behave as if modulo some large power of 2.
With n=-1, this function returns -n-1. With n = 0, it returns a
number a such that `a\cong -n-1 \pmod{2^n}`.
INPUT:
- ``x`` - gen (t_INT)
- ``n`` - long, default = -1
OUTPUT:
- ``gen`` - t_INT
EXAMPLES::
sage: pari(10).bitneg()
-11
sage: pari(1).bitneg()
-2
sage: pari(-2).bitneg()
1
sage: pari(-1).bitneg()
0
sage: pari(569).bitneg()
-570
sage: pari(569).bitneg(10)
454
sage: 454 % 2^10
454
sage: -570 % 2^10
454
"""
pari_catch_sig_on()
return P.new_gen(gbitneg(x.g,n))
def bitnegimply(gen x, y):
"""
bitnegimply(x,y): Bitwise negated imply of two integers x and y, in
other words, x BITAND BITNEG(y). Negative numbers behave as if
modulo big power of 2.
INPUT:
- ``x`` - gen (of type t_INT)
- ``y`` - coercible to gen (of type t_INT)
OUTPUT:
- ``gen`` - of type type t_INT
EXAMPLES::
sage: pari(14).bitnegimply(0)
14
sage: pari(8).bitnegimply(8)
0
sage: pari(8+4).bitnegimply(8)
4
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gbitnegimply(x.g, t0.g))
def bitor(gen x, y):
"""
bitor(x,y): Bitwise or of two integers x and y. Negative numbers
behave as if modulo big power of 2.
INPUT:
- ``x`` - gen (of type t_INT)
- ``y`` - coercible to gen (of type t_INT)
OUTPUT:
- ``gen`` - of type type t_INT
EXAMPLES::
sage: pari(14).bitor(0)
14
sage: pari(8).bitor(4)
12
sage: pari(12).bitor(1)
13
sage: pari(13).bitor(1)
13
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gbitor(x.g, t0.g))
def bittest(gen x, long n):
"""
bittest(x, long n): Returns bit number n (coefficient of
`2^n` in binary) of the integer x. Negative numbers behave
as if modulo a big power of 2.
INPUT:
- ``x`` - gen (pari integer)
OUTPUT:
- ``bool`` - a Python bool
EXAMPLES::
sage: x = pari(6)
sage: x.bittest(0)
False
sage: x.bittest(1)
True
sage: x.bittest(2)
True
sage: x.bittest(3)
False
sage: pari(-3).bittest(0)
True
sage: pari(-3).bittest(1)
False
sage: [pari(-3).bittest(n) for n in range(10)]
[True, False, True, True, True, True, True, True, True, True]
"""
pari_catch_sig_on()
cdef long b = bittest(x.g, n)
pari_catch_sig_off()
return b != 0
def bitxor(gen x, y):
"""
bitxor(x,y): Bitwise exclusive or of two integers x and y. Negative
numbers behave as if modulo big power of 2.
INPUT:
- ``x`` - gen (of type t_INT)
- ``y`` - coercible to gen (of type t_INT)
OUTPUT:
- ``gen`` - of type type t_INT
EXAMPLES::
sage: pari(6).bitxor(4)
2
sage: pari(0).bitxor(4)
4
sage: pari(6).bitxor(0)
6
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gbitxor(x.g, t0.g))
def ceil(gen x):
"""
For real x: return the smallest integer = x. For rational
functions: the quotient of numerator by denominator. For lists:
apply componentwise.
INPUT:
- ``x`` - gen
OUTPUT:
- ``gen`` - depends on type of x
EXAMPLES::
sage: pari(1.4).ceil()
2
sage: pari(-1.4).ceil()
-1
sage: pari(3/4).ceil()
1
sage: pari(x).ceil()
x
sage: pari((x^2+x+1)/x).ceil()
x + 1
This may be unexpected: but it is correct, treating the argument as
a rational function in RR(x).
::
sage: pari(x^2+5*x+2.5).ceil()
x^2 + 5*x + 2.50000000000000
"""
pari_catch_sig_on()
return P.new_gen(gceil(x.g))
def centerlift(gen x, v=-1):
"""
centerlift(x,v): Centered lift of x. This function returns exactly
the same thing as lift, except if x is an integer mod.
INPUT:
- ``x`` - gen
- ``v`` - var (default: x)
OUTPUT: gen
EXAMPLES::
sage: x = pari(-2).Mod(5)
sage: x.centerlift()
-2
sage: x.lift()
3
sage: f = pari('x-1').Mod('x^2 + 1')
sage: f.centerlift()
x - 1
sage: f.lift()
x - 1
sage: f = pari('x-y').Mod('x^2+1')
sage: f
Mod(x - y, x^2 + 1)
sage: f.centerlift('x')
x - y
sage: f.centerlift('y')
Mod(x - y, x^2 + 1)
"""
pari_catch_sig_on()
return P.new_gen(centerlift0(x.g, P.get_var(v)))
def component(gen x, long n):
"""
component(x, long n): Return n'th component of the internal
representation of x. This function is 1-based instead of 0-based.
.. note::
For vectors or matrices, it is simpler to use x[n-1]. For
list objects such as is output by nfinit, it is easier to
use member functions.
INPUT:
- ``x`` - gen
- ``n`` - C long (coercible to)
OUTPUT: gen
EXAMPLES::
sage: pari([0,1,2,3,4]).component(1)
0
sage: pari([0,1,2,3,4]).component(2)
1
sage: pari([0,1,2,3,4]).component(4)
3
sage: pari('x^3 + 2').component(1)
2
sage: pari('x^3 + 2').component(2)
0
sage: pari('x^3 + 2').component(4)
1
::
sage: pari('x').component(0)
Traceback (most recent call last):
...
PariError: nonexistent component
"""
pari_catch_sig_on()
return P.new_gen(compo(x.g, n))
def conj(gen x):
"""
conj(x): Return the algebraic conjugate of x.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari('x+1').conj()
x + 1
sage: pari('x+I').conj()
x - I
sage: pari('1/(2*x+3*I)').conj()
1/(2*x - 3*I)
sage: pari([1,2,'2-I','Mod(x,x^2+1)', 'Mod(x,x^2-2)']).conj()
[1, 2, 2 + I, Mod(-x, x^2 + 1), Mod(-x, x^2 - 2)]
sage: pari('Mod(x,x^2-2)').conj()
Mod(-x, x^2 - 2)
sage: pari('Mod(x,x^3-3)').conj()
Traceback (most recent call last):
...
PariError: incorrect type in gconj
"""
pari_catch_sig_on()
return P.new_gen(gconj(x.g))
def conjvec(gen x, unsigned long precision=0):
"""
conjvec(x): Returns the vector of all conjugates of the algebraic
number x. An algebraic number is a polynomial over Q modulo an
irreducible polynomial.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari('Mod(1+x,x^2-2)').conjvec()
[-0.414213562373095, 2.41421356237310]~
sage: pari('Mod(x,x^3-3)').conjvec()
[1.44224957030741, -0.721124785153704 + 1.24902476648341*I, -0.721124785153704 - 1.24902476648341*I]~
sage: pari('Mod(1+x,x^2-2)').conjvec(precision=192)[0].sage()
-0.414213562373095048801688724209698078569671875376948073177
"""
pari_catch_sig_on()
return P.new_gen(conjvec(x.g, prec_bits_to_words(precision)))
def denominator(gen x):
"""
denominator(x): Return the denominator of x. When x is a vector,
this is the least common multiple of the denominators of the
components of x.
what about poly? INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari('5/9').denominator()
9
sage: pari('(x+1)/(x-2)').denominator()
x - 2
sage: pari('2/3 + 5/8*x + 7/3*x^2 + 1/5*y').denominator()
1
sage: pari('2/3*x').denominator()
1
sage: pari('[2/3, 5/8, 7/3, 1/5]').denominator()
120
"""
pari_catch_sig_on()
return P.new_gen(denom(x.g))
def floor(gen x):
"""
For real x: return the largest integer = x. For rational functions:
the quotient of numerator by denominator. For lists: apply
componentwise.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari(5/9).floor()
0
sage: pari(11/9).floor()
1
sage: pari(1.17).floor()
1
sage: pari([1.5,2.3,4.99]).floor()
[1, 2, 4]
sage: pari([[1.1,2.2],[3.3,4.4]]).floor()
[[1, 2], [3, 4]]
sage: pari(x).floor()
x
sage: pari((x^2+x+1)/x).floor()
x + 1
sage: pari(x^2+5*x+2.5).floor()
x^2 + 5*x + 2.50000000000000
::
sage: pari('"hello world"').floor()
Traceback (most recent call last):
...
PariError: incorrect type in gfloor
"""
pari_catch_sig_on()
return P.new_gen(gfloor(x.g))
def frac(gen x):
"""
frac(x): Return the fractional part of x, which is x - floor(x).
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari(1.75).frac()
0.750000000000000
sage: pari(sqrt(2)).frac()
0.414213562373095
sage: pari('sqrt(-2)').frac()
Traceback (most recent call last):
...
PariError: incorrect type in gfloor
"""
pari_catch_sig_on()
return P.new_gen(gfrac(x.g))
def imag(gen x):
"""
imag(x): Return the imaginary part of x. This function also works
component-wise.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari('1+2*I').imag()
2
sage: pari(sqrt(-2)).imag()
1.41421356237310
sage: pari('x+I').imag()
1
sage: pari('x+2*I').imag()
2
sage: pari('(1+I)*x^2+2*I').imag()
x^2 + 2
sage: pari('[1,2,3] + [4*I,5,6]').imag()
[4, 0, 0]
"""
pari_catch_sig_on()
return P.new_gen(gimag(x.g))
def length(self):
"""
"""
return glength(self.g)
def lift(gen x, v=-1):
"""
lift(x,v): Returns the lift of an element of Z/nZ to Z or R[x]/(P)
to R[x] for a type R if v is omitted. If v is given, lift only
polymods with main variable v. If v does not occur in x, lift only
intmods.
INPUT:
- ``x`` - gen
- ``v`` - (optional) variable
OUTPUT: gen
EXAMPLES::
sage: x = pari("x")
sage: a = x.Mod('x^3 + 17*x + 3')
sage: a
Mod(x, x^3 + 17*x + 3)
sage: b = a^4; b
Mod(-17*x^2 - 3*x, x^3 + 17*x + 3)
sage: b.lift()
-17*x^2 - 3*x
??? more examples
"""
pari_catch_sig_on()
if v == -1:
return P.new_gen(lift(x.g))
return P.new_gen(lift0(x.g, P.get_var(v)))
def numbpart(gen x):
"""
numbpart(x): returns the number of partitions of x.
EXAMPLES::
sage: pari(20).numbpart()
627
sage: pari(100).numbpart()
190569292
"""
pari_catch_sig_on()
return P.new_gen(numbpart(x.g))
def numerator(gen x):
"""
numerator(x): Returns the numerator of x.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES:
"""
pari_catch_sig_on()
return P.new_gen(numer(x.g))
def numtoperm(gen k, long n):
"""
numtoperm(k, n): Return the permutation number k (mod n!) of n
letters, where n is an integer.
INPUT:
- ``k`` - gen, integer
- ``n`` - int
OUTPUT:
- ``gen`` - vector (permutation of 1,...,n)
EXAMPLES:
"""
pari_catch_sig_on()
return P.new_gen(numtoperm(n, k.g))
def padicprec(gen x, p):
"""
padicprec(x,p): Return the absolute p-adic precision of the object
x.
INPUT:
- ``x`` - gen
OUTPUT: int
EXAMPLES::
sage: K = Qp(11,5)
sage: x = K(11^-10 + 5*11^-7 + 11^-6)
sage: y = pari(x)
sage: y.padicprec(11)
-5
sage: y.padicprec(17)
Traceback (most recent call last):
...
PariError: not the same prime in padicprec
This works for polynomials too::
sage: R.<t> = PolynomialRing(Zp(3))
sage: pol = R([O(3^4), O(3^6), O(3^5)])
sage: pari(pol).padicprec(3)
4
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
cdef long prec = padicprec(x.g, t0.g)
pari_catch_sig_off()
return prec
def padicprime(gen x):
"""
The uniformizer of the p-adic ring this element lies in, as a t_INT.
INPUT:
- ``x`` - gen, of type t_PADIC
OUTPUT:
- ``p`` - gen, of type t_INT
EXAMPLES::
sage: K = Qp(11,5)
sage: x = K(11^-10 + 5*11^-7 + 11^-6)
sage: y = pari(x)
sage: y.padicprime()
11
sage: y.padicprime().type()
't_INT'
"""
pari_catch_sig_on()
return P.new_gen(gel(x.g, 2))
def permtonum(gen x):
"""
permtonum(x): Return the ordinal (between 1 and n!) of permutation
vector x. ??? Huh ??? say more. what is a perm vector. 0 to n-1 or
1-n.
INPUT:
- ``x`` - gen (vector of integers)
OUTPUT:
- ``gen`` - integer
EXAMPLES:
"""
if typ(x.g) != t_VEC:
raise TypeError, "x (=%s) must be of type t_VEC, but is of type %s."%(x,x.type())
pari_catch_sig_on()
return P.new_gen(permtonum(x.g))
def precision(gen x, long n=-1):
"""
precision(x,n): Change the precision of x to be n, where n is a
C-integer). If n is omitted, output the real precision of x.
INPUT:
- ``x`` - gen
- ``n`` - (optional) int
OUTPUT: nothing or gen if n is omitted
EXAMPLES:
"""
if n <= -1:
return precision(x.g)
pari_catch_sig_on()
return P.new_gen(precision0(x.g, n))
def random(gen N):
r"""
``random(N=2^31)``: Return a pseudo-random integer
between 0 and `N-1`.
INPUT:
-``N`` - gen, integer
OUTPUT:
- ``gen`` - integer
EXAMPLES:
"""
if typ(N.g) != t_INT:
raise TypeError, "x (=%s) must be of type t_INT, but is of type %s."%(N,N.type())
pari_catch_sig_on()
return P.new_gen(genrand(N.g))
def real(gen x):
"""
real(x): Return the real part of x.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES:
"""
pari_catch_sig_on()
return P.new_gen(greal(x.g))
def round(gen x, estimate=False):
"""
round(x,estimate=False): If x is a real number, returns x rounded
to the nearest integer (rounding up). If the optional argument
estimate is True, also returns the binary exponent e of the
difference between the original and the rounded value (the
"fractional part") (this is the integer ceiling of log_2(error)).
When x is a general PARI object, this function returns the result
of rounding every coefficient at every level of PARI object. Note
that this is different than what the truncate function does (see
the example below).
One use of round is to get exact results after a long approximate
computation, when theory tells you that the coefficients must be
integers.
INPUT:
- ``x`` - gen
- ``estimate`` - (optional) bool, False by default
OUTPUT:
- if estimate is False, return a single gen.
- if estimate is True, return rounded version of x and error
estimate in bits, both as gens.
EXAMPLES::
sage: pari('1.5').round()
2
sage: pari('1.5').round(True)
(2, -1)
sage: pari('1.5 + 2.1*I').round()
2 + 2*I
sage: pari('1.0001').round(True)
(1, -14)
sage: pari('(2.4*x^2 - 1.7)/x').round()
(2*x^2 - 2)/x
sage: pari('(2.4*x^2 - 1.7)/x').truncate()
2.40000000000000*x
"""
cdef int n
cdef long e
cdef gen y
pari_catch_sig_on()
if not estimate:
return P.new_gen(ground(x.g))
y = P.new_gen(grndtoi(x.g, &e))
return y, e
def simplify(gen x):
"""
simplify(x): Simplify the object x as much as possible, and return
the result.
A complex or quadratic number whose imaginary part is an exact 0
(i.e., not an approximate one such as O(3) or 0.E-28) is converted
to its real part, and a a polynomial of degree 0 is converted to
its constant term. Simplification occurs recursively.
This function is useful before using arithmetic functions, which
expect integer arguments:
EXAMPLES::
sage: y = pari('y')
sage: x = pari('9') + y - y
sage: x
9
sage: x.type()
't_POL'
sage: x.factor()
matrix(0,2)
sage: pari('9').factor()
Mat([3, 2])
sage: x.simplify()
9
sage: x.simplify().factor()
Mat([3, 2])
sage: x = pari('1.5 + 0*I')
sage: x.type()
't_REAL'
sage: x.simplify()
1.50000000000000
sage: y = x.simplify()
sage: y.type()
't_REAL'
"""
pari_catch_sig_on()
return P.new_gen(simplify(x.g))
def sizeword(gen x):
"""
Return the total number of machine words occupied by the
complete tree of the object x. A machine word is 32 or
64 bits, depending on the computer.
INPUT:
- ``x`` - gen
OUTPUT: int (a Python int)
EXAMPLES::
sage: pari('0').sizeword()
2
sage: pari('1').sizeword()
3
sage: pari('1000000').sizeword()
3
sage: pari('10^100').sizeword()
13 # 32-bit
8 # 64-bit
sage: pari(RDF(1.0)).sizeword()
4 # 32-bit
3 # 64-bit
sage: pari('x').sizeword()
9
sage: pari('x^20').sizeword()
66
sage: pari('[x, I]').sizeword()
20
"""
return gsizeword(x.g)
def sizebyte(gen x):
"""
Return the total number of bytes occupied by the complete tree
of the object x. Note that this number depends on whether the
computer is 32-bit or 64-bit.
INPUT:
- ``x`` - gen
OUTPUT: int (a Python int)
EXAMPLE::
sage: pari('1').sizebyte()
12 # 32-bit
24 # 64-bit
"""
return gsizebyte(x.g)
def sizedigit(gen x):
"""
sizedigit(x): Return a quick estimate for the maximal number of
decimal digits before the decimal point of any component of x.
INPUT:
- ``x`` - gen
OUTPUT:
- ``int`` - Python integer
EXAMPLES::
sage: x = pari('10^100')
sage: x.Str().length()
101
sage: x.sizedigit()
101
Note that digits after the decimal point are ignored.
::
sage: x = pari('1.234')
sage: x
1.23400000000000
sage: x.sizedigit()
1
The estimate can be one too big::
sage: pari('7234.1').sizedigit()
4
sage: pari('9234.1').sizedigit()
5
"""
return sizedigit(x.g)
def truncate(gen x, estimate=False):
"""
truncate(x,estimate=False): Return the truncation of x. If estimate
is True, also return the number of error bits.
When x is in the real numbers, this means that the part after the
decimal point is chopped away, e is the binary exponent of the
difference between the original and truncated value (the
"fractional part"). If x is a rational function, the result is the
integer part (Euclidean quotient of numerator by denominator) and
if requested the error estimate is 0.
When truncate is applied to a power series (in X), it transforms it
into a polynomial or a rational function with denominator a power
of X, by chopping away the `O(X^k)`. Similarly, when
applied to a p-adic number, it transforms it into an integer or a
rational number by chopping away the `O(p^k)`.
INPUT:
- ``x`` - gen
- ``estimate`` - (optional) bool, which is False by
default
OUTPUT:
- if estimate is False, return a single gen.
- if estimate is True, return rounded version of x and error
estimate in bits, both as gens.
EXAMPLES::
sage: pari('(x^2+1)/x').round()
(x^2 + 1)/x
sage: pari('(x^2+1)/x').truncate()
x
sage: pari('1.043').truncate()
1
sage: pari('1.043').truncate(True)
(1, -5)
sage: pari('1.6').truncate()
1
sage: pari('1.6').round()
2
sage: pari('1/3 + 2 + 3^2 + O(3^3)').truncate()
34/3
sage: pari('sin(x+O(x^10))').truncate()
1/362880*x^9 - 1/5040*x^7 + 1/120*x^5 - 1/6*x^3 + x
sage: pari('sin(x+O(x^10))').round() # each coefficient has abs < 1
x + O(x^10)
"""
cdef long e
cdef gen y
pari_catch_sig_on()
if not estimate:
return P.new_gen(gtrunc(x.g))
y = P.new_gen(gcvtoi(x.g, &e))
return y, e
def valuation(gen x, p):
"""
valuation(x,p): Return the valuation of x with respect to p.
The valuation is the highest exponent of p dividing x.
- If p is an integer, x must be an integer, an intmod whose
modulus is divisible by p, a rational number, a p-adic
number, or a polynomial or power series in which case the
valuation is the minimum of the valuations of the
coefficients.
- If p is a polynomial, x must be a polynomial or a rational
function. If p is a monomial then x may also be a power
series.
- If x is a vector, complex or quadratic number, then the
valuation is the minimum of the component valuations.
- If x = 0, the result is `2^31-1` on 32-bit machines or
`2^63-1` on 64-bit machines if x is an exact
object. If x is a p-adic number or power series, the result
is the exponent of the zero.
INPUT:
- ``x`` - gen
- ``p`` - coercible to gen
OUTPUT:
- ``gen`` - integer
EXAMPLES::
sage: pari(9).valuation(3)
2
sage: pari(9).valuation(9)
1
sage: x = pari(9).Mod(27); x.valuation(3)
2
sage: pari('5/3').valuation(3)
-1
sage: pari('9 + 3*x + 15*x^2').valuation(3)
1
sage: pari([9,3,15]).valuation(3)
1
sage: pari('9 + 3*x + 15*x^2 + O(x^5)').valuation(3)
1
::
sage: pari('x^2*(x+1)^3').valuation(pari('x+1'))
3
sage: pari('x + O(x^5)').valuation('x')
1
sage: pari('2*x^2 + O(x^5)').valuation('x')
2
::
sage: pari(0).valuation(3)
2147483647 # 32-bit
9223372036854775807 # 64-bit
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
v = ggval(x.g, t0.g)
pari_catch_sig_off()
return v
def _valp(gen x):
"""
Return the valuation of x where x is a p-adic number (t_PADIC)
or a laurent series (t_SER). If x is a different type, this
will give a bogus number.
EXAMPLES::
sage: pari('1/x^2 + O(x^10)')._valp()
-2
sage: pari('O(x^10)')._valp()
10
sage: pari('(1145234796 + O(3^10))/771966234')._valp()
-2
sage: pari('O(2^10)')._valp()
10
sage: pari('x')._valp() # random
-35184372088832
"""
return valp(x.g)
def variable(gen x):
"""
variable(x): Return the main variable of the object x, or p if x is
a p-adic number.
This function raises a TypeError exception on scalars, i.e., on
objects with no variable associated to them.
INPUT:
- ``x`` - gen
OUTPUT: gen
EXAMPLES::
sage: pari('x^2 + x -2').variable()
x
sage: pari('1+2^3 + O(2^5)').variable()
2
sage: pari('x+y0').variable()
x
sage: pari('y0+z0').variable()
y0
"""
pari_catch_sig_on()
return P.new_gen(gpolvar(x.g))
def abs(gen x, unsigned long precision=0):
"""
Returns the absolute value of x (its modulus, if x is complex).
Rational functions are not allowed. Contrary to most transcendental
functions, an exact argument is not converted to a real number
before applying abs and an exact result is returned if possible.
EXAMPLES::
sage: x = pari("-27.1")
sage: x.abs()
27.1000000000000
sage: pari('1 + I').abs(precision=128).sage()
1.4142135623730950488016887242096980786
If x is a polynomial, returns -x if the leading coefficient is real
and negative else returns x. For a power series, the constant
coefficient is considered instead.
EXAMPLES::
sage: pari('x-1.2*x^2').abs()
1.20000000000000*x^2 - x
sage: pari('-2 + t + O(t^2)').abs()
2 - t + O(t^2)
"""
pari_catch_sig_on()
return P.new_gen(gabs(x.g, prec_bits_to_words(precision)))
def acos(gen x, unsigned long precision=0):
r"""
The principal branch of `\cos^{-1}(x)`, so that
`\RR e(\mathrm{acos}(x))` belongs to `[0,Pi]`. If `x`
is real and `|x| > 1`, then `\mathrm{acos}(x)` is complex.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(0.5).acos()
1.04719755119660
sage: pari(1/2).acos()
1.04719755119660
sage: pari(1.1).acos()
0.443568254385115*I
sage: C.<i> = ComplexField()
sage: pari(1.1+i).acos()
0.849343054245252 - 1.09770986682533*I
"""
pari_catch_sig_on()
return P.new_gen(gacos(x.g, prec_bits_to_words(precision)))
def acosh(gen x, unsigned long precision=0):
r"""
The principal branch of `\cosh^{-1}(x)`, so that
`\Im(\mathrm{acosh}(x))` belongs to `[0,Pi]`. If
`x` is real and `x < 1`, then
`\mathrm{acosh}(x)` is complex.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).acosh()
1.31695789692482
sage: pari(0).acosh()
1.57079632679490*I
sage: C.<i> = ComplexField()
sage: pari(i).acosh()
0.881373587019543 + 1.57079632679490*I
"""
pari_catch_sig_on()
return P.new_gen(gach(x.g, prec_bits_to_words(precision)))
def agm(gen x, y, unsigned long precision=0):
r"""
The arithmetic-geometric mean of x and y. In the case of complex or
negative numbers, the principal square root is always chosen.
p-adic or power series arguments are also allowed. Note that a
p-adic AGM exists only if x/y is congruent to 1 modulo p (modulo 16
for p=2). x and y cannot both be vectors or matrices.
If any of `x` or `y` is an exact argument, it is
first converted to a real or complex number using the optional
parameter precision (in bits). If the arguments are inexact (e.g.
real), the smallest of their two precisions is used in the
computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).agm(2)
2.00000000000000
sage: pari(0).agm(1)
0
sage: pari(1).agm(2)
1.45679103104691
sage: C.<i> = ComplexField()
sage: pari(1+i).agm(-3)
-0.964731722290876 + 1.15700282952632*I
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(agm(x.g, t0.g, prec_bits_to_words(precision)))
def arg(gen x, unsigned long precision=0):
r"""
arg(x): argument of x,such that `-\pi < \arg(x) \leq \pi`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: C.<i> = ComplexField()
sage: pari(2+i).arg()
0.463647609000806
"""
pari_catch_sig_on()
return P.new_gen(garg(x.g, prec_bits_to_words(precision)))
def asin(gen x, unsigned long precision=0):
r"""
The principal branch of `\sin^{-1}(x)`, so that
`\RR e(\mathrm{asin}(x))` belongs to `[-\pi/2,\pi/2]`. If
`x` is real and `|x| > 1` then `\mathrm{asin}(x)`
is complex.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(pari(0.5).sin()).asin()
0.500000000000000
sage: pari(2).asin()
1.57079632679490 - 1.31695789692482*I
"""
pari_catch_sig_on()
return P.new_gen(gasin(x.g, prec_bits_to_words(precision)))
def asinh(gen x, unsigned long precision=0):
r"""
The principal branch of `\sinh^{-1}(x)`, so that
`\Im(\mathrm{asinh}(x))` belongs to `[-\pi/2,\pi/2]`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).asinh()
1.44363547517881
sage: C.<i> = ComplexField()
sage: pari(2+i).asinh()
1.52857091948100 + 0.427078586392476*I
"""
pari_catch_sig_on()
return P.new_gen(gash(x.g, prec_bits_to_words(precision)))
def atan(gen x, unsigned long precision=0):
r"""
The principal branch of `\tan^{-1}(x)`, so that
`\RR e(\mathrm{atan}(x))` belongs to `]-\pi/2, \pi/2[`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).atan()
0.785398163397448
sage: C.<i> = ComplexField()
sage: pari(1.5+i).atan()
1.10714871779409 + 0.255412811882995*I
"""
pari_catch_sig_on()
return P.new_gen(gatan(x.g, prec_bits_to_words(precision)))
def atanh(gen x, unsigned long precision=0):
r"""
The principal branch of `\tanh^{-1}(x)`, so that
`\Im(\mathrm{atanh}(x))` belongs to `]-\pi/2,\pi/2]`. If
`x` is real and `|x| > 1` then `\mathrm{atanh}(x)`
is complex.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(0).atanh()
0.E-19
sage: pari(2).atanh()
0.549306144334055 - 1.57079632679490*I
"""
pari_catch_sig_on()
return P.new_gen(gath(x.g, prec_bits_to_words(precision)))
def bernfrac(gen x):
r"""
The Bernoulli number `B_x`, where `B_0 = 1`,
`B_1 = -1/2`, `B_2 = 1/6,\ldots,` expressed as a
rational number. The argument `x` should be of type
integer.
EXAMPLES::
sage: pari(18).bernfrac()
43867/798
sage: [pari(n).bernfrac() for n in range(10)]
[1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0]
"""
pari_catch_sig_on()
return P.new_gen(bernfrac(x))
def bernreal(gen x, unsigned long precision=0):
r"""
The Bernoulli number `B_x`, as for the function bernfrac,
but `B_x` is returned as a real number (with the current
precision).
EXAMPLES::
sage: pari(18).bernreal()
54.9711779448622
sage: pari(18).bernreal(precision=192).sage()
54.9711779448621553884711779448621553884711779448621553885
"""
pari_catch_sig_on()
return P.new_gen(bernreal(x, prec_bits_to_words(precision)))
def bernvec(gen x):
r"""
Creates a vector containing, as rational numbers, the Bernoulli
numbers `B_0, B_2,\ldots, B_{2x}`. This routine is
obsolete. Use bernfrac instead each time you need a Bernoulli
number in exact form.
Note: this routine is implemented using repeated independent calls
to bernfrac, which is faster than the standard recursion in exact
arithmetic.
EXAMPLES::
sage: pari(8).bernvec()
[1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510]
sage: [pari(2*n).bernfrac() for n in range(9)]
[1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510]
"""
pari_catch_sig_on()
return P.new_gen(bernvec(x))
def besselh1(gen nu, x, unsigned long precision=0):
r"""
The `H^1`-Bessel function of index `\nu` and
argument `x`.
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(2).besselh1(3)
0.486091260585891 - 0.160400393484924*I
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(hbessel1(nu.g, t0.g, prec_bits_to_words(precision)))
def besselh2(gen nu, x, unsigned long precision=0):
r"""
The `H^2`-Bessel function of index `\nu` and
argument `x`.
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(2).besselh2(3)
0.486091260585891 + 0.160400393484924*I
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(hbessel2(nu.g, t0.g, prec_bits_to_words(precision)))
def besselj(gen nu, x, unsigned long precision=0):
r"""
Bessel J function (Bessel function of the first kind), with index
`\nu` and argument `x`. If `x` converts to
a power series, the initial factor
`(x/2)^{\nu}/\Gamma(\nu+1)` is omitted (since it cannot be
represented in PARI when `\nu` is not integral).
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(2).besselj(3)
0.486091260585891
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(jbessel(nu.g, t0.g, prec_bits_to_words(precision)))
def besseljh(gen nu, x, unsigned long precision=0):
"""
J-Bessel function of half integral index (Spherical Bessel
function of the first kind). More precisely, besseljh(n,x) computes
`J_{n+1/2}(x)` where n must an integer, and x is any
complex value. In the current implementation (PARI, version
2.2.11), this function is not very accurate when `x` is
small.
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(2).besseljh(3)
0.4127100324 # 32-bit
0.412710032209716 # 64-bit
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(jbesselh(nu.g, t0.g, prec_bits_to_words(precision)))
def besseli(gen nu, x, unsigned long precision=0):
r"""
Bessel I function (Bessel function of the second kind), with index
`\nu` and argument `x`. If `x` converts to
a power series, the initial factor
`(x/2)^{\nu}/\Gamma(\nu+1)` is omitted (since it cannot be
represented in PARI when `\nu` is not integral).
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(2).besseli(3)
2.24521244092995
sage: C.<i> = ComplexField()
sage: pari(2).besseli(3+i)
1.12539407613913 + 2.08313822670661*I
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(ibessel(nu.g, t0.g, prec_bits_to_words(precision)))
def besselk(gen nu, x, long flag=0, unsigned long precision=0):
"""
nu.besselk(x, flag=0): K-Bessel function (modified Bessel function
of the second kind) of index nu, which can be complex, and argument
x.
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
INPUT:
- ``nu`` - a complex number
- ``x`` - real number (positive or negative)
- ``flag`` - default: 0 or 1: use hyperu (hyperu is
much slower for small x, and doesn't work for negative x).
EXAMPLES::
sage: C.<i> = ComplexField()
sage: pari(2+i).besselk(3)
0.0455907718407551 + 0.0289192946582081*I
::
sage: pari(2+i).besselk(-3)
-4.34870874986752 - 5.38744882697109*I
::
sage: pari(2+i).besselk(300, flag=1)
3.74224603319728 E-132 + 2.49071062641525 E-134*I
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(kbessel(nu.g, t0.g, prec_bits_to_words(precision)))
def besseln(gen nu, x, unsigned long precision=0):
"""
nu.besseln(x): Bessel N function (Spherical Bessel function of the
second kind) of index nu and argument x.
If `nu` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: C.<i> = ComplexField()
sage: pari(2+i).besseln(3)
-0.280775566958244 - 0.486708533223726*I
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(nbessel(nu.g, t0.g, prec_bits_to_words(precision)))
def cos(gen x, unsigned long precision=0):
"""
The cosine function.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1.5).cos()
0.0707372016677029
sage: C.<i> = ComplexField()
sage: pari(1+i).cos()
0.833730025131149 - 0.988897705762865*I
sage: pari('x+O(x^8)').cos()
1 - 1/2*x^2 + 1/24*x^4 - 1/720*x^6 + 1/40320*x^8 + O(x^9)
"""
pari_catch_sig_on()
return P.new_gen(gcos(x.g, prec_bits_to_words(precision)))
def cosh(gen x, unsigned long precision=0):
"""
The hyperbolic cosine function.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1.5).cosh()
2.35240961524325
sage: C.<i> = ComplexField()
sage: pari(1+i).cosh()
0.833730025131149 + 0.988897705762865*I
sage: pari('x+O(x^8)').cosh()
1 + 1/2*x^2 + 1/24*x^4 + 1/720*x^6 + O(x^8)
"""
pari_catch_sig_on()
return P.new_gen(gch(x.g, prec_bits_to_words(precision)))
def cotan(gen x, unsigned long precision=0):
"""
The cotangent of x.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(5).cotan()
-0.295812915532746
Computing the cotangent of `\pi` doesn't raise an error,
but instead just returns a very large (positive or negative)
number.
::
sage: x = RR(pi)
sage: pari(x).cotan() # random
-8.17674825 E15
"""
pari_catch_sig_on()
return P.new_gen(gcotan(x.g, prec_bits_to_words(precision)))
def dilog(gen x, unsigned long precision=0):
r"""
The principal branch of the dilogarithm of `x`, i.e. the
analytic continuation of the power series
`\log_2(x) = \sum_{n>=1} x^n/n^2`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).dilog()
1.64493406684823
sage: C.<i> = ComplexField()
sage: pari(1+i).dilog()
0.616850275068085 + 1.46036211675312*I
"""
pari_catch_sig_on()
return P.new_gen(dilog(x.g, prec_bits_to_words(precision)))
def eint1(gen x, long n=0, unsigned long precision=0):
r"""
x.eint1(n): exponential integral E1(x):
.. math::
\int_{x}^{\infty} \frac{e^{-t}}{t} dt
If n is present, output the vector [eint1(x), eint1(2\*x), ...,
eint1(n\*x)]. This is faster than repeatedly calling eint1(i\*x).
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
REFERENCE:
- See page 262, Prop 5.6.12, of Cohen's book "A Course in
Computational Algebraic Number Theory".
EXAMPLES:
"""
pari_catch_sig_on()
if n <= 0:
return P.new_gen(eint1(x.g, prec_bits_to_words(precision)))
else:
return P.new_gen(veceint1(x.g, stoi(n), prec_bits_to_words(precision)))
def erfc(gen x, unsigned long precision=0):
r"""
Return the complementary error function:
.. math::
(2/\sqrt{\pi}) \int_{x}^{\infty} e^{-t^2} dt.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).erfc()
0.157299207050285
"""
pari_catch_sig_on()
return P.new_gen(gerfc(x.g, prec_bits_to_words(precision)))
def eta(gen x, long flag=0, unsigned long precision=0):
r"""
x.eta(flag=0): if flag=0, `\eta` function without the
`q^{1/24}`; otherwise `\eta` of the complex number
`x` in the upper half plane intelligently computed using
`\mathrm{SL}(2,\ZZ)` transformations.
DETAILS: This functions computes the following. If the input
`x` is a complex number with positive imaginary part, the
result is `\prod_{n=1}^{\infty} (q-1^n)`, where
`q=e^{2 i \pi x}`. If `x` is a power series
(or can be converted to a power series) with positive valuation,
the result is `\prod_{n=1}^{\infty} (1-x^n)`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: C.<i> = ComplexField()
sage: pari(i).eta()
0.998129069925959
"""
pari_catch_sig_on()
if flag == 1:
return P.new_gen(trueeta(x.g, prec_bits_to_words(precision)))
return P.new_gen(eta(x.g, prec_bits_to_words(precision)))
def exp(gen self, unsigned long precision=0):
"""
x.exp(): exponential of x.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(0).exp()
1.00000000000000
sage: pari(1).exp()
2.71828182845905
sage: pari('x+O(x^8)').exp()
1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 + O(x^8)
"""
pari_catch_sig_on()
return P.new_gen(gexp(self.g, prec_bits_to_words(precision)))
def gamma(gen s, unsigned long precision=0):
"""
s.gamma(precision): Gamma function at s.
If `s` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `s` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).gamma()
1.00000000000000
sage: pari(5).gamma()
24.0000000000000
sage: C.<i> = ComplexField()
sage: pari(1+i).gamma()
0.498015668118356 - 0.154949828301811*I
TESTS::
sage: pari(-1).gamma()
Traceback (most recent call last):
...
PariError: non-positive integer argument in ggamma
"""
pari_catch_sig_on()
return P.new_gen(ggamma(s.g, prec_bits_to_words(precision)))
def gammah(gen s, unsigned long precision=0):
"""
s.gammah(): Gamma function evaluated at the argument x+1/2.
If `s` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `s` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).gammah()
1.32934038817914
sage: pari(5).gammah()
52.3427777845535
sage: C.<i> = ComplexField()
sage: pari(1+i).gammah()
0.575315188063452 + 0.0882106775440939*I
"""
pari_catch_sig_on()
return P.new_gen(ggamd(s.g, prec_bits_to_words(precision)))
def hyperu(gen a, b, x, unsigned long precision=0):
r"""
a.hyperu(b,x): U-confluent hypergeometric function.
If `a`, `b`, or `x` is an exact argument,
it is first converted to a real or complex number using the
optional parameter precision (in bits). If the arguments are
inexact (e.g. real), the smallest of their precisions is used in
the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).hyperu(2,3)
0.333333333333333
"""
cdef gen t0 = objtogen(b)
cdef gen t1 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(hyperu(a.g, t0.g, t1.g, prec_bits_to_words(precision)))
def incgam(gen s, x, y=None, unsigned long precision=0):
r"""
s.incgam(x, y, precision): incomplete gamma function. y is optional
and is the precomputed value of gamma(s).
If `s` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `s` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: C.<i> = ComplexField()
sage: pari(1+i).incgam(3-i)
-0.0458297859919946 + 0.0433696818726677*I
"""
cdef gen t0 = objtogen(x)
cdef gen t1
if y is None:
pari_catch_sig_on()
return P.new_gen(incgam(s.g, t0.g, prec_bits_to_words(precision)))
else:
t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(incgam0(s.g, t0.g, t1.g, prec_bits_to_words(precision)))
def incgamc(gen s, x, unsigned long precision=0):
r"""
s.incgamc(x): complementary incomplete gamma function.
The arguments `x` and `s` are complex numbers such
that `s` is not a pole of `\Gamma` and
`|x|/(|s|+1)` is not much larger than `1`
(otherwise, the convergence is very slow). The function returns the
value of the integral
`\int_{0}^{x} e^{-t} t^{s-1} dt.`
If `s` or `x` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(1).incgamc(2)
0.864664716763387
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(incgamc(s.g, t0.g, prec_bits_to_words(precision)))
def log(gen x, unsigned long precision=0):
r"""
x.log(): natural logarithm of x.
This function returns the principal branch of the natural logarithm
of `x`, i.e., the branch such that
`\Im(\log(x)) \in ]-\pi, \pi].` The result is
complex (with imaginary part equal to `\pi`) if
`x\in \RR` and `x<0`. In general, the algorithm uses
the formula
.. math::
\log(x) \simeq \frac{\pi}{2{\rm agm}(1,4/s)} - m\log(2),
if `s=x 2^m` is large enough. (The result is exact to
`B` bits provided that `s>2^{B/2}`.) At low
accuracies, this function computes `\log` using the series
expansion near `1`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
Note that `p`-adic arguments can also be given as input,
with the convention that `\log(p)=0`. Hence, in particular,
`\exp(\log(x))/x` is not in general equal to `1`
but instead to a `(p-1)`-st root of unity (or
`\pm 1` if `p=2`) times a power of `p`.
EXAMPLES::
sage: pari(5).log()
1.60943791243410
sage: C.<i> = ComplexField()
sage: pari(i).log()
0.E-19 + 1.57079632679490*I
"""
pari_catch_sig_on()
return P.new_gen(glog(x.g, prec_bits_to_words(precision)))
def lngamma(gen x, unsigned long precision=0):
r"""
This method is deprecated, please use :meth:`.log_gamma` instead.
See the :meth:`.log_gamma` method for documentation and examples.
EXAMPLES::
sage: pari(100).lngamma()
doctest:...: DeprecationWarning: The method lngamma() is deprecated. Use log_gamma() instead.
See http://trac.sagemath.org/6992 for details.
359.134205369575
"""
from sage.misc.superseded import deprecation
deprecation(6992, "The method lngamma() is deprecated. Use log_gamma() instead.")
return x.log_gamma(precision)
def log_gamma(gen x, unsigned long precision=0):
r"""
Logarithm of the gamma function of x.
This function returns the principal branch of the logarithm of the
gamma function of `x`. The function
`\log(\Gamma(x))` is analytic on the complex plane with
non-positive integers removed. This function can have much larger
inputs than `\Gamma` itself.
The `p`-adic analogue of this function is unfortunately not
implemented.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(100).log_gamma()
359.134205369575
"""
pari_catch_sig_on()
return P.new_gen(glngamma(x.g, prec_bits_to_words(precision)))
def polylog(gen x, long m, long flag=0, unsigned long precision=0):
"""
x.polylog(m,flag=0): m-th polylogarithm of x. flag is optional, and
can be 0: default, 1: D_m -modified m-th polylog of x, 2:
D_m-modified m-th polylog of x, 3: P_m-modified m-th polylog of
x.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
TODO: Add more explanation, copied from the PARI manual.
EXAMPLES::
sage: pari(10).polylog(3)
5.64181141475134 - 8.32820207698027*I
sage: pari(10).polylog(3,0)
5.64181141475134 - 8.32820207698027*I
sage: pari(10).polylog(3,1)
0.523778453502411
sage: pari(10).polylog(3,2)
-0.400459056163451
"""
pari_catch_sig_on()
return P.new_gen(polylog0(m, x.g, flag, prec_bits_to_words(precision)))
def psi(gen x, unsigned long precision=0):
r"""
x.psi(): psi-function at x.
Return the `\psi`-function of `x`, i.e., the
logarithmic derivative `\Gamma'(x)/\Gamma(x)`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).psi()
-0.577215664901533
"""
pari_catch_sig_on()
return P.new_gen(gpsi(x.g, prec_bits_to_words(precision)))
def sin(gen x, unsigned long precision=0):
"""
x.sin(): The sine of x.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).sin()
0.841470984807897
sage: C.<i> = ComplexField()
sage: pari(1+i).sin()
1.29845758141598 + 0.634963914784736*I
"""
pari_catch_sig_on()
return P.new_gen(gsin(x.g, prec_bits_to_words(precision)))
def sinh(gen x, unsigned long precision=0):
"""
The hyperbolic sine function.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(0).sinh()
0.E-19
sage: C.<i> = ComplexField()
sage: pari(1+i).sinh()
0.634963914784736 + 1.29845758141598*I
"""
pari_catch_sig_on()
return P.new_gen(gsh(x.g, prec_bits_to_words(precision)))
def sqr(gen x):
"""
x.sqr(): square of x. Faster than, and most of the time (but not
always - see the examples) identical to x\*x.
EXAMPLES::
sage: pari(2).sqr()
4
For `2`-adic numbers, x.sqr() may not be identical to x\*x
(squaring a `2`-adic number increases its precision)::
sage: pari("1+O(2^5)").sqr()
1 + O(2^6)
sage: pari("1+O(2^5)")*pari("1+O(2^5)")
1 + O(2^5)
However::
sage: x = pari("1+O(2^5)"); x*x
1 + O(2^6)
"""
pari_catch_sig_on()
return P.new_gen(gsqr(x.g))
def sqrt(gen x, unsigned long precision=0):
"""
x.sqrt(precision): The square root of x.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).sqrt()
1.41421356237310
"""
pari_catch_sig_on()
return P.new_gen(gsqrt(x.g, prec_bits_to_words(precision)))
def sqrtn(gen x, n, unsigned long precision=0):
r"""
x.sqrtn(n): return the principal branch of the n-th root of x,
i.e., the one such that
`\arg(\sqrt(x)) \in ]-\pi/n, \pi/n]`. Also returns a second
argument which is a suitable root of unity allowing one to recover
all the other roots. If it was not possible to find such a number,
then this second return value is 0. If the argument is present and
no square root exists, return 0 instead of raising an error.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
.. note::
intmods (modulo a prime) and `p`-adic numbers are
allowed as arguments.
INPUT:
- ``x`` - gen
- ``n`` - integer
OUTPUT:
- ``gen`` - principal n-th root of x
- ``gen`` - root of unity z that gives the other
roots
EXAMPLES::
sage: s, z = pari(2).sqrtn(5)
sage: z
0.309016994374947 + 0.951056516295154*I
sage: s
1.14869835499704
sage: s^5
2.00000000000000
sage: z^5
1.00000000000000 + 5.42101086 E-19*I # 32-bit
1.00000000000000 + 5.96311194867027 E-19*I # 64-bit
sage: (s*z)^5
2.00000000000000 + 1.409462824 E-18*I # 32-bit
2.00000000000000 + 9.21571846612679 E-19*I # 64-bit
"""
cdef GEN zetan
cdef gen t0 = objtogen(n)
pari_catch_sig_on()
ans = P.new_gen_noclear(gsqrtn(x.g, t0.g, &zetan, prec_bits_to_words(precision)))
return ans, P.new_gen(zetan)
def tan(gen x, unsigned long precision=0):
"""
x.tan() - tangent of x
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(2).tan()
-2.18503986326152
sage: C.<i> = ComplexField()
sage: pari(i).tan()
0.E-19 + 0.761594155955765*I
"""
pari_catch_sig_on()
return P.new_gen(gtan(x.g, prec_bits_to_words(precision)))
def tanh(gen x, unsigned long precision=0):
"""
x.tanh() - hyperbolic tangent of x
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(1).tanh()
0.761594155955765
sage: C.<i> = ComplexField()
sage: z = pari(i); z
0.E-19 + 1.00000000000000*I
sage: result = z.tanh()
sage: result.real() <= 1e-18
True
sage: result.imag()
1.55740772465490
"""
pari_catch_sig_on()
return P.new_gen(gth(x.g, prec_bits_to_words(precision)))
def teichmuller(gen x):
r"""
teichmuller(x): teichmuller character of p-adic number x.
This is the unique `(p-1)`-st root of unity congruent to
`x/p^{v_p(x)}` modulo `p`.
EXAMPLES::
sage: pari('2+O(7^5)').teichmuller()
2 + 4*7 + 6*7^2 + 3*7^3 + O(7^5)
"""
pari_catch_sig_on()
return P.new_gen(teich(x.g))
def theta(gen q, z, unsigned long precision=0):
"""
q.theta(z): Jacobi sine theta-function.
If `q` or `z` is an exact argument, it is first
converted to a real or complex number using the optional parameter
precision (in bits). If the arguments are inexact (e.g. real), the
smallest of their precisions is used in the computation, and the
parameter precision is ignored.
EXAMPLES::
sage: pari(0.5).theta(2)
1.63202590295260
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(theta(q.g, t0.g, prec_bits_to_words(precision)))
def thetanullk(gen q, long k, unsigned long precision=0):
"""
q.thetanullk(k): return the k-th derivative at z=0 of theta(q,z).
If `q` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `q` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
EXAMPLES::
sage: pari(0.5).thetanullk(1)
0.548978532560341
"""
pari_catch_sig_on()
return P.new_gen(thetanullk(q.g, k, prec_bits_to_words(precision)))
def weber(gen x, long flag=0, unsigned long precision=0):
r"""
x.weber(flag=0): One of Weber's f functions of x. flag is optional,
and can be 0: default, function
f(x)=exp(-i\*Pi/24)\*eta((x+1)/2)/eta(x) such that
`j=(f^{24}-16)^3/f^{24}`, 1: function f1(x)=eta(x/2)/eta(x)
such that `j=(f1^24+16)^3/f2^{24}`, 2: function
f2(x)=sqrt(2)\*eta(2\*x)/eta(x) such that
`j=(f2^{24}+16)^3/f2^{24}`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
TODO: Add further explanation from PARI manual.
EXAMPLES::
sage: C.<i> = ComplexField()
sage: pari(i).weber()
1.18920711500272 + 0.E-19*I # 32-bit
1.18920711500272 + 2.71050543121376 E-20*I # 64-bit
sage: pari(i).weber(1)
1.09050773266526 + 0.E-19*I
sage: pari(i).weber(2)
1.09050773266526
"""
pari_catch_sig_on()
return P.new_gen(weber0(x.g, flag, prec_bits_to_words(precision)))
def zeta(gen s, unsigned long precision=0):
"""
zeta(s): zeta function at s with s a complex or a p-adic number.
If `s` is a complex number, this is the Riemann zeta
function `\zeta(s)=\sum_{n\geq 1} n^{-s}`, computed either
using the Euler-Maclaurin summation formula (if `s` is not
an integer), or using Bernoulli numbers (if `s` is a
negative integer or an even nonnegative integer), or using modular
forms (if `s` is an odd nonnegative integer).
If `s` is a `p`-adic number, this is the
Kubota-Leopoldt zeta function, i.e. the unique continuous
`p`-adic function on the `p`-adic integers that
interpolates the values of `(1-p^{-k})\zeta(k)` at negative
integers `k` such that `k\equiv 1\pmod{p-1}` if
`p` is odd, and at odd `k` if `p=2`.
If `x` is an exact argument, it is first converted to a
real or complex number using the optional parameter precision (in
bits). If `x` is inexact (e.g. real), its own precision is
used in the computation, and the parameter precision is ignored.
INPUT:
- ``s`` - gen (real, complex, or p-adic number)
OUTPUT:
- ``gen`` - value of zeta at s.
EXAMPLES::
sage: pari(2).zeta()
1.64493406684823
sage: x = RR(pi)^2/6
sage: pari(x)
1.64493406684823
sage: pari(3).zeta()
1.20205690315959
sage: pari('1+5*7+2*7^2+O(7^3)').zeta()
4*7^-2 + 5*7^-1 + O(7^0)
"""
pari_catch_sig_on()
return P.new_gen(gzeta(s.g, prec_bits_to_words(precision)))
def bezout(gen x, y):
cdef gen u, v, g
cdef GEN U, V, G
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
G = gbezout(x.g, t0.g, &U, &V)
g = P.new_gen_noclear(G)
u = P.new_gen_noclear(U)
v = P.new_gen(V)
return g, u, v
def binomial(gen x, long k):
"""
binomial(x, k): return the binomial coefficient "x choose k".
INPUT:
- ``x`` - any PARI object (gen)
- ``k`` - integer
EXAMPLES::
sage: pari(6).binomial(2)
15
sage: pari('x+1').binomial(3)
1/6*x^3 - 1/6*x
sage: pari('2+x+O(x^2)').binomial(3)
1/3*x + O(x^2)
"""
pari_catch_sig_on()
return P.new_gen(binomial(x.g, k))
def contfrac(gen x, b=0, long lmax=0):
"""
contfrac(x,b,lmax): continued fraction expansion of x (x rational,
real or rational function). b and lmax are both optional, where b
is the vector of numerators of the continued fraction, and lmax is
a bound for the number of terms in the continued fraction
expansion.
"""
cdef gen t0 = objtogen(b)
pari_catch_sig_on()
return P.new_gen(contfrac0(x.g, t0.g, lmax))
def contfracpnqn(gen x, b=0, long lmax=0):
"""
contfracpnqn(x): [p_n,p_n-1; q_n,q_n-1] corresponding to the
continued fraction x.
"""
pari_catch_sig_on()
return P.new_gen(pnqn(x.g))
def ffgen(gen T, v=-1):
r"""
Return the generator `g=x \bmod T` of the finite field defined
by the polynomial `T`.
INPUT:
- ``T`` -- a gen of type t_POL with coefficients of type t_INTMOD:
a polynomial over a prime finite field
- ``v`` -- string: a variable name or -1 (optional)
If `v` is a string, then `g` will be a polynomial in `v`, else the
variable of the polynomial `T` is used.
EXAMPLES::
sage: x = GF(2)['x'].gen()
sage: pari(x^2+x+2).ffgen()
x
sage: pari(x^2+x+1).ffgen('a')
a
"""
pari_catch_sig_on()
return P.new_gen(ffgen(T.g, P.get_var(v)))
def ffinit(gen p, long n, v=-1):
r"""
Return a monic irreducible polynomial `g` of degree `n` over the
finite field of `p` elements.
INPUT:
- ``p`` -- a gen of type t_INT: a prime number
- ``n`` -- integer: the degree of the polynomial
- ``v`` -- string: a variable name or -1 (optional)
If `v \geq 0', then `g` will be a polynomial in `v`, else the
variable `x` is used.
EXAMPLES::
sage: pari(7).ffinit(11)
Mod(1, 7)*x^11 + Mod(1, 7)*x^10 + Mod(4, 7)*x^9 + Mod(5, 7)*x^8 + Mod(1, 7)*x^7 + Mod(1, 7)*x^2 + Mod(1, 7)*x + Mod(6, 7)
sage: pari(2003).ffinit(3)
Mod(1, 2003)*x^3 + Mod(1, 2003)*x^2 + Mod(1993, 2003)*x + Mod(1995, 2003)
"""
pari_catch_sig_on()
return P.new_gen(ffinit(p.g, n, P.get_var(v)))
def fibonacci(gen x):
r"""
Return the Fibonacci number of index x.
EXAMPLES::
sage: pari(18).fibonacci()
2584
sage: [pari(n).fibonacci() for n in range(10)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
"""
pari_catch_sig_on()
return P.new_gen(fibo(long(x)))
def gcd(gen x, y=None):
"""
Return the greatest common divisor of `x` and `y`.
If `y` is ``None``, then `x` must be a list or tuple, and the
greatest common divisor of its components is returned.
EXAMPLES::
sage: pari(10).gcd(15)
5
sage: pari([5, 'y']).gcd()
1
sage: pari(['x', x^2]).gcd()
x
"""
cdef gen t0
if y is None:
pari_catch_sig_on()
return P.new_gen(ggcd0(x.g, NULL))
else:
t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(ggcd0(x.g, t0.g))
def issquare(gen x, find_root=False):
"""
issquare(x,n): ``True`` if x is a square, ``False`` if not. If
``find_root`` is given, also returns the exact square root.
"""
cdef GEN G
cdef long t
cdef gen g
pari_catch_sig_on()
if find_root:
t = itos(gissquareall(x.g, &G))
if t:
return True, P.new_gen(G)
else:
P.clear_stack()
return False, None
else:
t = itos(gissquare(x.g))
pari_catch_sig_off()
return t != 0
def issquarefree(gen self):
"""
EXAMPLES::
sage: pari(10).issquarefree()
True
sage: pari(20).issquarefree()
False
"""
pari_catch_sig_on()
cdef long t = issquarefree(self.g)
pari_catch_sig_off()
return t != 0
def lcm(gen x, y=None):
"""
Return the least common multiple of `x` and `y`.
If `y` is ``None``, then `x` must be a list or tuple, and the
least common multiple of its components is returned.
EXAMPLES::
sage: pari(10).lcm(15)
30
sage: pari([5, 'y']).lcm()
5*y
sage: pari([10, 'x', x^2]).lcm()
10*x^2
"""
cdef gen t0
if y is None:
pari_catch_sig_on()
return P.new_gen(glcm0(x.g, NULL))
else:
t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(glcm0(x.g, t0.g))
def numdiv(gen n):
"""
Return the number of divisors of the integer n.
EXAMPLES::
sage: pari(10).numdiv()
4
"""
pari_catch_sig_on()
return P.new_gen(gnumbdiv(n.g))
def phi(gen n):
"""
Return the Euler phi function of n. EXAMPLES::
sage: pari(10).phi()
4
"""
pari_catch_sig_on()
return P.new_gen(geulerphi(n.g))
def primepi(gen self):
"""
Return the number of primes less than or equal to self.
EXAMPLES::
sage: pari(7).primepi()
4
sage: pari(100).primepi()
25
sage: pari(1000).primepi()
168
sage: pari(100000).primepi()
9592
sage: pari(0).primepi()
0
sage: pari(-15).primepi()
0
sage: pari(500509).primepi()
41581
"""
pari_catch_sig_on()
if self > P._primelimit():
P.init_primes(self + 10)
if signe(self.g) != 1:
pari_catch_sig_off()
return P.PARI_ZERO
return P.new_gen(primepi(self.g))
def sumdiv(gen n):
"""
Return the sum of the divisors of `n`.
EXAMPLES::
sage: pari(10).sumdiv()
18
"""
pari_catch_sig_on()
return P.new_gen(sumdiv(n.g))
def sumdivk(gen n, long k):
"""
Return the sum of the k-th powers of the divisors of n.
EXAMPLES::
sage: pari(10).sumdivk(2)
130
"""
pari_catch_sig_on()
return P.new_gen(sumdivk(n.g, k))
def xgcd(gen x, y):
"""
Returns u,v,d such that d=gcd(x,y) and u\*x+v\*y=d.
EXAMPLES::
sage: pari(10).xgcd(15)
(5, -1, 1)
"""
return x.bezout(y)
def Zn_issquare(gen self, n):
"""
Return ``True`` if ``self`` is a square modulo `n`, ``False``
if not.
INPUT:
- ``self`` -- integer
- ``n`` -- integer or factorisation matrix
EXAMPLES::
sage: pari(3).Zn_issquare(4)
False
sage: pari(4).Zn_issquare(30.factor())
True
"""
cdef gen t0 = objtogen(n)
pari_catch_sig_on()
cdef long t = Zn_issquare(self.g, t0.g)
pari_catch_sig_off()
return t != 0
def Zn_sqrt(gen self, n):
"""
Return a square root of ``self`` modulo `n`, if such a square
root exists; otherwise, raise a ``ValueError``.
INPUT:
- ``self`` -- integer
- ``n`` -- integer or factorisation matrix
EXAMPLES::
sage: pari(3).Zn_sqrt(4)
Traceback (most recent call last):
...
ValueError: 3 is not a square modulo 4
sage: pari(4).Zn_sqrt(30.factor())
22
"""
cdef gen t0 = objtogen(n)
cdef GEN s
pari_catch_sig_on()
s = Zn_sqrt(self.g, t0.g)
if s == NULL:
pari_catch_sig_off()
raise ValueError("%s is not a square modulo %s" % (self, n))
return P.new_gen(s)
def ellinit(self, long flag=0, unsigned long precision=0):
"""
Return the PARI elliptic curve object with Weierstrass coefficients
given by self, a list with 5 elements.
INPUT:
- ``self`` - a list of 5 coefficients
- ``flag (optional, default: 0)`` - if 0, ask for a PARI ell
structure with 19 components; if 1, ask for a shorted PARI
sell structure with only the first 13 components.
- ``precision (optional, default: 0)`` - the real
precision to be used in the computation of the components of the
PARI (s)ell structure; if 0, use the default 64 bits.
.. note::
The parameter ``precision`` in ``ellinit`` controls not
only the real precision of the resulting (s)ell structure,
but in some cases also the precision of most subsequent
computations with this elliptic curve (if those rely on
the precomputations done by ``ellinit``). You should
therefore set the precision from the start to the value
you require.
OUTPUT:
- ``gen`` - either a PARI ell structure with 19 components
(if flag=0), or a PARI sell structure with 13 components
(if flag=1).
EXAMPLES: An elliptic curve with integer coefficients::
sage: e = pari([0,1,0,1,0]).ellinit(); e
[0, 1, 0, 1, 0, 4, 2, 0, -1, -32, 224, -48, 2048/3, [0.E-28, -0.500000000000000 - 0.866025403784439*I, -0.500000000000000 + 0.866025403784439*I]~, 3.37150070962519, -1.68575035481260 - 2.15651564749964*I, 1.37451455785745 - 1.084202173 E-19*I, -0.687257278928726 + 0.984434956803824*I, 7.27069403586288] # 32-bit
[0, 1, 0, 1, 0, 4, 2, 0, -1, -32, 224, -48, 2048/3, [0.E-38, -0.500000000000000 - 0.866025403784439*I, -0.500000000000000 + 0.866025403784439*I]~, 3.37150070962519, -1.68575035481260 - 2.15651564749964*I, 1.37451455785745 - 5.42101086242752 E-19*I, -0.687257278928726 + 0.984434956803824*I, 7.27069403586288] # 64-bit
Its inexact components have the default precision of 53 bits::
sage: RR(e[14])
3.37150070962519
We can compute this to higher precision::
sage: R = RealField(150)
sage: e = pari([0,1,0,1,0]).ellinit(precision=150)
sage: R(e[14])
3.3715007096251920857424073155981539790016018
Using flag=1 returns a short elliptic curve PARI object::
sage: pari([0,1,0,1,0]).ellinit(flag=1)
[0, 1, 0, 1, 0, 4, 2, 0, -1, -32, 224, -48, 2048/3]
The coefficients can be any ring elements that convert to PARI::
sage: pari([0,1/2,0,-3/4,0]).ellinit(flag=1)
[0, 1/2, 0, -3/4, 0, 2, -3/2, 0, -9/16, 40, -116, 117/4, 256000/117]
sage: pari([0,0.5,0,-0.75,0]).ellinit(flag=1)
[0, 0.500000000000000, 0, -0.750000000000000, 0, 2.00000000000000, -1.50000000000000, 0, -0.562500000000000, 40.0000000000000, -116.000000000000, 29.2500000000000, 2188.03418803419]
sage: pari([0,I,0,1,0]).ellinit(flag=1)
[0, I, 0, 1, 0, 4*I, 2, 0, -1, -64, 352*I, -80, 16384/5]
sage: pari([0,x,0,2*x,1]).ellinit(flag=1)
[0, x, 0, 2*x, 1, 4*x, 4*x, 4, -4*x^2 + 4*x, 16*x^2 - 96*x, -64*x^3 + 576*x^2 - 864, 64*x^4 - 576*x^3 + 576*x^2 - 432, (256*x^6 - 4608*x^5 + 27648*x^4 - 55296*x^3)/(4*x^4 - 36*x^3 + 36*x^2 - 27)]
"""
pari_catch_sig_on()
return P.new_gen(ellinit0(self.g, flag, prec_bits_to_words(precision)))
def ellglobalred(self):
"""
e.ellglobalred(): return information related to the global minimal
model of the elliptic curve e.
INPUT:
- ``e`` - elliptic curve (returned by ellinit)
OUTPUT:
- ``gen`` - the (arithmetic) conductor of e
- ``gen`` - a vector giving the coordinate change over
Q from e to its minimal integral model (see also ellminimalmodel)
- ``gen`` - the product of the local Tamagawa numbers
of e
EXAMPLES::
sage: e = pari([0, 5, 2, -1, 1]).ellinit()
sage: e.ellglobalred()
[20144, [1, -2, 0, -1], 1]
sage: e = pari(EllipticCurve('17a').a_invariants()).ellinit()
sage: e.ellglobalred()
[17, [1, 0, 0, 0], 4]
"""
pari_catch_sig_on()
return P.new_gen(ellglobalred(self.g))
def elladd(self, z0, z1):
"""
e.elladd(z0, z1): return the sum of the points z0 and z1 on this
elliptic curve.
INPUT:
- ``e`` - elliptic curve E
- ``z0`` - point on E
- ``z1`` - point on E
OUTPUT: point on E
EXAMPLES: First we create an elliptic curve::
sage: e = pari([0, 1, 1, -2, 0]).ellinit()
sage: str(e)[:65] # first part of output
'[0, 1, 1, -2, 0, 4, -4, 1, -3, 112, -856, 389, 1404928/389, [0.90'
Next we add two points on the elliptic curve. Notice that the
Python lists are automatically converted to PARI objects so you
don't have to do that explicitly in your code.
::
sage: e.elladd([1,0], [-1,1])
[-3/4, -15/8]
"""
cdef gen t0 = objtogen(z0)
cdef gen t1 = objtogen(z1)
pari_catch_sig_on()
return P.new_gen(addell(self.g, t0.g, t1.g))
def ellak(self, n):
r"""
e.ellak(n): Returns the coefficient `a_n` of the
`L`-function of the elliptic curve e, i.e. the
`n`-th Fourier coefficient of the weight 2 newform
associated to e (according to Shimura-Taniyama).
The curve `e` *must* be a medium or long vector of the type
given by ellinit. For this function to work for every n and not
just those prime to the conductor, e must be a minimal Weierstrass
equation. If this is not the case, use the function ellminimalmodel
first before using ellak (or you will get INCORRECT RESULTS!)
INPUT:
- ``e`` - a PARI elliptic curve.
- ``n`` - integer.
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.ellak(6)
2
sage: e.ellak(2005)
2
sage: e.ellak(-1)
0
sage: e.ellak(0)
0
"""
cdef gen t0 = objtogen(n)
pari_catch_sig_on()
return P.new_gen(akell(self.g, t0.g))
def ellan(self, long n, python_ints=False):
"""
Return the first `n` Fourier coefficients of the modular
form attached to this elliptic curve. See ellak for more details.
INPUT:
- ``n`` - a long integer
- ``python_ints`` - bool (default is False); if True,
return a list of Python ints instead of a PARI gen wrapper.
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.ellan(3)
[1, -2, -1]
sage: e.ellan(20)
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2]
sage: e.ellan(-1)
[]
sage: v = e.ellan(10, python_ints=True); v
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
sage: type(v)
<type 'list'>
sage: type(v[0])
<type 'int'>
"""
pari_catch_sig_on()
cdef GEN g
if python_ints:
g = anell(self.g, n)
v = [gtolong(<GEN> g[i+1]) for i in range(glength(g))]
P.clear_stack()
return v
else:
return P.new_gen(anell(self.g, n))
def ellanalyticrank(self, unsigned long precision=0):
r"""
Returns a 2-component vector with the order of vanishing at
`s = 1` of the L-function of the elliptic curve and the value
of the first non-zero derivative.
EXAMPLE::
sage: E = EllipticCurve('389a1')
sage: pari(E).ellanalyticrank()
[2, 1.51863300057685]
"""
pari_catch_sig_on()
return P.new_gen(ellanalyticrank(self.g, <GEN>0, prec_bits_to_words(precision)))
def ellap(self, p):
r"""
e.ellap(p): Returns the prime-indexed coefficient `a_p` of the
`L`-function of the elliptic curve `e`, i.e. the `p`-th Fourier
coefficient of the newform attached to e.
The computation uses the Shanks--Mestre method, or the SEA
algorithm.
.. WARNING::
For this function to work for every n and not just those prime
to the conductor, e must be a minimal Weierstrass equation.
If this is not the case, use the function ellminimalmodel first
before using ellap (or you will get INCORRECT RESULTS!)
INPUT:
- ``e`` - a PARI elliptic curve.
- ``p`` - prime integer
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.ellap(2)
-2
sage: e.ellap(2003)
4
sage: e.ellak(-1)
0
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(ellap(self.g, t0.g))
def ellaplist(self, long n, python_ints=False):
r"""
e.ellaplist(n): Returns a PARI list of all the prime-indexed
coefficients `a_p` (up to n) of the `L`-function
of the elliptic curve `e`, i.e. the Fourier coefficients of
the newform attached to `e`.
INPUT:
- ``self`` -- an elliptic curve
- ``n`` -- a long integer
- ``python_ints`` -- bool (default is False); if True,
return a list of Python ints instead of a PARI gen wrapper.
.. WARNING::
The curve e must be a medium or long vector of the type given by
ellinit. For this function to work for every n and not just those
prime to the conductor, e must be a minimal Weierstrass equation.
If this is not the case, use the function ellminimalmodel first
before using ellaplist (or you will get INCORRECT RESULTS!)
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: v = e.ellaplist(10); v
[-2, -1, 1, -2]
sage: type(v)
<type 'sage.libs.pari.gen.gen'>
sage: v.type()
't_VEC'
sage: e.ellan(10)
[1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
sage: v = e.ellaplist(10, python_ints=True); v
[-2, -1, 1, -2]
sage: type(v)
<type 'list'>
sage: type(v[0])
<type 'int'>
TESTS::
sage: v = e.ellaplist(1)
sage: print v, type(v)
[] <type 'sage.libs.pari.gen.gen'>
sage: v = e.ellaplist(1, python_ints=True)
sage: print v, type(v)
[] <type 'list'>
"""
if python_ints:
return [int(x) for x in self.ellaplist(n)]
if n < 2:
pari_catch_sig_on()
return P.new_gen(zerovec(0))
P.init_primes(n+1)
cdef gen t0 = objtogen(n)
pari_catch_sig_on()
cdef GEN g = primes(gtolong(primepi(t0.g)))
cdef long i
for i from 0 <= i < glength(g):
set_gel(g, i + 1, ellap(self.g, gel(g, i + 1)))
return P.new_gen(g)
def ellbil(self, z0, z1, unsigned long precision=0):
"""
e.ellbil(z0, z1): return the value of the canonical bilinear form
on z0 and z1.
INPUT:
- ``e`` - elliptic curve (assumed integral given by a
minimal model, as returned by ellminimalmodel)
- ``z0, z1`` - rational points on e
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit().ellminimalmodel()[0]
sage: e.ellbil([1, 0], [-1, 1])
0.418188984498861
"""
cdef gen t0 = objtogen(z0)
cdef gen t1 = objtogen(z1)
pari_catch_sig_on()
return P.new_gen(bilhell(self.g, t0.g, t1.g, prec_bits_to_words(precision)))
def ellchangecurve(self, ch):
"""
e.ellchangecurve(ch): return the new model (equation) for the
elliptic curve e given by the change of coordinates ch.
The change of coordinates is specified by a vector ch=[u,r,s,t]; if
`x'` and `y'` are the new coordinates, then
`x = u^2 x' + r` and `y = u^3 y' + su^2 x' + t`.
INPUT:
- ``e`` - elliptic curve
- ``ch`` - change of coordinates vector with 4
entries
EXAMPLES::
sage: e = pari([1,2,3,4,5]).ellinit()
sage: e.ellglobalred()
[10351, [1, -1, 0, -1], 1]
sage: f = e.ellchangecurve([1,-1,0,-1])
sage: f[:5]
[1, -1, 0, 4, 3]
"""
cdef gen t0 = objtogen(ch)
pari_catch_sig_on()
return P.new_gen(ellchangecurve(self.g, t0.g))
def elleta(self, unsigned long precision=0):
"""
e.elleta(): return the vector [eta1,eta2] of quasi-periods
associated with the period lattice e.omega() of the elliptic curve
e.
EXAMPLES::
sage: e = pari([0,0,0,-82,0]).ellinit()
sage: e.elleta()
[3.60546360143265, 3.60546360143265*I]
sage: w1,w2 = e.omega()
sage: eta1, eta2 = e.elleta()
sage: w1*eta2-w2*eta1
6.28318530717959*I
sage: w1*eta2-w2*eta1 == pari(2*pi*I)
True
sage: pari([0,0,0,-82,0]).ellinit(flag=1).elleta()
Traceback (most recent call last):
...
PariError: incorrect type in elleta
"""
pari_catch_sig_on()
return P.new_gen(elleta(self.g, prec_bits_to_words(precision)))
def ellheight(self, a, long flag=2, unsigned long precision=0):
"""
e.ellheight(a, flag=2): return the global Neron-Tate height of the
point a on the elliptic curve e.
INPUT:
- ``e`` - elliptic curve over `\QQ`,
assumed to be in a standard minimal integral model (as given by
ellminimalmodel)
- ``a`` - rational point on e
- ``flag (optional)`` - specifies which algorithm to
be used for computing the archimedean local height:
- ``0`` - uses sigma- and theta-functions and a trick
due to J. Silverman
- ``1`` - uses Tate's `4^n` algorithm
- ``2`` - uses Mestre's AGM algorithm (this is the
default, being faster than the other two)
- ``precision (optional)`` - the precision of the
result, in bits.
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit().ellminimalmodel()[0]
sage: e.ellheight([1,0])
0.476711659343740
sage: e.ellheight([1,0], flag=0)
0.476711659343740
sage: e.ellheight([1,0], flag=1)
0.476711659343740
sage: e.ellheight([1,0], precision=128).sage()
0.47671165934373953737948605888465305932
"""
cdef gen t0 = objtogen(a)
pari_catch_sig_on()
return P.new_gen(ellheight0(self.g, t0.g, flag, prec_bits_to_words(precision)))
def ellheightmatrix(self, x, unsigned long precision=0):
"""
e.ellheightmatrix(x): return the height matrix for the vector x of
points on the elliptic curve e.
In other words, it returns the Gram matrix of x with respect to the
height bilinear form on e (see ellbil).
INPUT:
- ``e`` - elliptic curve over `\QQ`,
assumed to be in a standard minimal integral model (as given by
ellminimalmodel)
- ``x`` - vector of rational points on e
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit().ellminimalmodel()[0]
sage: e.ellheightmatrix([[1,0], [-1,1]])
[0.476711659343740, 0.418188984498861; 0.418188984498861, 0.686667083305587]
It is allowed to call :meth:`ellinit` with ``flag=1``::
sage: E = pari([0,1,1,-2,0]).ellinit(flag=1)
sage: E.ellheightmatrix([[1,0], [-1,1]], precision=128).sage()
[0.47671165934373953737948605888465305932 0.41818898449886058562988945821587638244]
[0.41818898449886058562988945821587638244 0.68666708330558658572355210295409678904]
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(mathell(self.g, t0.g, prec_bits_to_words(precision)))
def ellisoncurve(self, x):
"""
e.ellisoncurve(x): return True if the point x is on the elliptic
curve e, False otherwise.
If the point or the curve have inexact coefficients, an attempt is
made to take this into account.
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit()
sage: e.ellisoncurve([1,0])
True
sage: e.ellisoncurve([1,1])
False
sage: e.ellisoncurve([1,0.00000000000000001])
False
sage: e.ellisoncurve([1,0.000000000000000001])
True
sage: e.ellisoncurve([0])
True
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
cdef int t = oncurve(self.g, t0.g)
pari_catch_sig_off()
return t != 0
def elllocalred(self, p):
r"""
e.elllocalred(p): computes the data of local reduction at the prime
p on the elliptic curve e
For more details on local reduction and Kodaira types, see IV.8 and
IV.9 in J. Silverman's book "Advanced topics in the arithmetic of
elliptic curves".
INPUT:
- ``e`` - elliptic curve with coefficients in `\ZZ`
- ``p`` - prime number
OUTPUT:
- ``gen`` - the exponent of p in the arithmetic
conductor of e
- ``gen`` - the Kodaira type of e at p, encoded as an
integer:
- ``1`` - type `I_0`: good reduction,
nonsingular curve of genus 1
- ``2`` - type `II`: rational curve with a
cusp
- ``3`` - type `III`: two nonsingular rational
curves intersecting tangentially at one point
- ``4`` - type `IV`: three nonsingular
rational curves intersecting at one point
- ``5`` - type `I_1`: rational curve with a
node
- ``6 or larger`` - think of it as `4+v`, then
it is type `I_v`: `v` nonsingular rational curves
arranged as a `v`-gon
- ``-1`` - type `I_0^*`: nonsingular rational
curve of multiplicity two with four nonsingular rational curves of
multiplicity one attached
- ``-2`` - type `II^*`: nine nonsingular
rational curves in a special configuration
- ``-3`` - type `III^*`: eight nonsingular
rational curves in a special configuration
- ``-4`` - type `IV^*`: seven nonsingular
rational curves in a special configuration
- ``-5 or smaller`` - think of it as `-4-v`,
then it is type `I_v^*`: chain of `v+1`
nonsingular rational curves of multiplicity two, with two
nonsingular rational curves of multiplicity one attached at either
end
- ``gen`` - a vector with 4 components, giving the
coordinate changes done during the local reduction; if the first
component is 1, then the equation for e was already minimal at p
- ``gen`` - the local Tamagawa number `c_p`
EXAMPLES:
Type `I_0`::
sage: e = pari([0,0,0,0,1]).ellinit()
sage: e.elllocalred(7)
[0, 1, [1, 0, 0, 0], 1]
Type `II`::
sage: e = pari(EllipticCurve('27a3').a_invariants()).ellinit()
sage: e.elllocalred(3)
[3, 2, [1, -1, 0, 1], 1]
Type `III`::
sage: e = pari(EllipticCurve('24a4').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, 3, [1, 1, 0, 1], 2]
Type `IV`::
sage: e = pari(EllipticCurve('20a2').a_invariants()).ellinit()
sage: e.elllocalred(2)
[2, 4, [1, 1, 0, 1], 3]
Type `I_1`::
sage: e = pari(EllipticCurve('11a2').a_invariants()).ellinit()
sage: e.elllocalred(11)
[1, 5, [1, 0, 0, 0], 1]
Type `I_2`::
sage: e = pari(EllipticCurve('14a4').a_invariants()).ellinit()
sage: e.elllocalred(2)
[1, 6, [1, 0, 0, 0], 2]
Type `I_6`::
sage: e = pari(EllipticCurve('14a1').a_invariants()).ellinit()
sage: e.elllocalred(2)
[1, 10, [1, 0, 0, 0], 2]
Type `I_0^*`::
sage: e = pari(EllipticCurve('32a3').a_invariants()).ellinit()
sage: e.elllocalred(2)
[5, -1, [1, 1, 1, 0], 1]
Type `II^*`::
sage: e = pari(EllipticCurve('24a5').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, -2, [1, 2, 1, 4], 1]
Type `III^*`::
sage: e = pari(EllipticCurve('24a2').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, -3, [1, 2, 1, 4], 2]
Type `IV^*`::
sage: e = pari(EllipticCurve('20a1').a_invariants()).ellinit()
sage: e.elllocalred(2)
[2, -4, [1, 0, 1, 2], 3]
Type `I_1^*`::
sage: e = pari(EllipticCurve('24a1').a_invariants()).ellinit()
sage: e.elllocalred(2)
[3, -5, [1, 0, 1, 2], 4]
Type `I_6^*`::
sage: e = pari(EllipticCurve('90c2').a_invariants()).ellinit()
sage: e.elllocalred(3)
[2, -10, [1, 96, 1, 316], 4]
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(elllocalred(self.g, t0.g))
def elllseries(self, s, A=1, unsigned long precision=0):
"""
e.elllseries(s, A=1): return the value of the `L`-series of
the elliptic curve e at the complex number s.
This uses an `O(N^{1/2})` algorithm in the conductor N of
e, so it is impractical for large conductors (say greater than
`10^{12}`).
INPUT:
- ``e`` - elliptic curve defined over `\QQ`
- ``s`` - complex number
- ``A (optional)`` - cutoff point for the integral,
which must be chosen close to 1 for best speed.
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit()
sage: e.elllseries(2.1)
0.402838047956645
sage: e.elllseries(1, precision=128)
2.87490929644255 E-38
sage: e.elllseries(1, precision=256)
3.00282377034977 E-77
sage: e.elllseries(-2)
0
sage: e.elllseries(2.1, A=1.1)
0.402838047956645
"""
cdef gen t0 = objtogen(s)
cdef gen t1 = objtogen(A)
pari_catch_sig_on()
return P.new_gen(elllseries(self.g, t0.g, t1.g, prec_bits_to_words(precision)))
def ellminimalmodel(self):
"""
ellminimalmodel(e): return the standard minimal integral model of
the rational elliptic curve e and the corresponding change of
variables. INPUT:
- ``e`` - gen (that defines an elliptic curve)
OUTPUT:
- ``gen`` - minimal model
- ``gen`` - change of coordinates
EXAMPLES::
sage: e = pari([1,2,3,4,5]).ellinit()
sage: F, ch = e.ellminimalmodel()
sage: F[:5]
[1, -1, 0, 4, 3]
sage: ch
[1, -1, 0, -1]
sage: e.ellchangecurve(ch)[:5]
[1, -1, 0, 4, 3]
"""
cdef GEN x, y
cdef gen model, change
cdef pari_sp t
pari_catch_sig_on()
x = ellminimalmodel(self.g, &y)
change = P.new_gen_noclear(y)
model = P.new_gen(x)
return model, change
def ellorder(self, x):
"""
e.ellorder(x): return the order of the point x on the elliptic
curve e (return 0 if x is not a torsion point)
INPUT:
- ``e`` - elliptic curve defined over `\QQ`
- ``x`` - point on e
EXAMPLES::
sage: e = pari(EllipticCurve('65a1').a_invariants()).ellinit()
A point of order two::
sage: e.ellorder([0,0])
2
And a point of infinite order::
sage: e.ellorder([1,0])
0
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(orderell(self.g, t0.g))
def ellordinate(self, x, unsigned long precision=0):
"""
e.ellordinate(x): return the `y`-coordinates of the points
on the elliptic curve e having x as `x`-coordinate.
INPUT:
- ``e`` - elliptic curve
- ``x`` - x-coordinate (can be a complex or p-adic
number, or a more complicated object like a power series)
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit()
sage: e.ellordinate(0)
[0, -1]
sage: e.ellordinate(I)
[0.582203589721741 - 1.38606082464177*I, -1.58220358972174 + 1.38606082464177*I]
sage: e.ellordinate(I, precision=128)[0].sage()
0.58220358972174117723338947874993600727 - 1.3860608246417697185311834209833653345*I
sage: e.ellordinate(1+3*5^1+O(5^3))
[4*5 + 5^2 + O(5^3), 4 + 3*5^2 + O(5^3)]
sage: e.ellordinate('z+2*z^2+O(z^4)')
[-2*z - 7*z^2 - 23*z^3 + O(z^4), -1 + 2*z + 7*z^2 + 23*z^3 + O(z^4)]
The field in which PARI looks for the point depends on the
input field::
sage: e.ellordinate(5)
[]
sage: e.ellordinate(5.0)
[11.3427192823270, -12.3427192823270]
sage: e.ellordinate(RR(-3))
[-1/2 + 3.42782730020052*I, -1/2 - 3.42782730020052*I]
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(ellordinate(self.g, t0.g, prec_bits_to_words(precision)))
def ellpointtoz(self, pt, unsigned long precision=0):
"""
e.ellpointtoz(pt): return the complex number (in the fundamental
parallelogram) corresponding to the point ``pt`` on the elliptic curve
e, under the complex uniformization of e given by the Weierstrass
p-function.
The complex number z returned by this function lies in the
parallelogram formed by the real and complex periods of e, as given
by e.omega().
EXAMPLES::
sage: e = pari([0,0,0,1,0]).ellinit()
sage: e.ellpointtoz([0,0])
1.85407467730137
The point at infinity is sent to the complex number 0::
sage: e.ellpointtoz([0])
0
"""
cdef gen t0 = objtogen(pt)
pari_catch_sig_on()
return P.new_gen(zell(self.g, t0.g, prec_bits_to_words(precision)))
def ellpow(self, z, n):
"""
e.ellpow(z, n): return `n` times the point `z` on the elliptic
curve `e`.
INPUT:
- ``e`` - elliptic curve
- ``z`` - point on `e`
- ``n`` - integer, or a complex quadratic integer of complex
multiplication for `e`. Complex multiplication currently
only works if `e` is defined over `Q`.
EXAMPLES: We consider a curve with CM by `Z[i]`::
sage: e = pari([0,0,0,3,0]).ellinit()
sage: p = [1,2] # Point of infinite order
Multiplication by two::
sage: e.ellpow([0,0], 2)
[0]
sage: e.ellpow(p, 2)
[1/4, -7/8]
Complex multiplication::
sage: q = e.ellpow(p, 1+I); q
[-2*I, 1 + I]
sage: e.ellpow(q, 1-I)
[1/4, -7/8]
TESTS::
sage: for D in [-7, -8, -11, -12, -16, -19, -27, -28]: # long time (1s)
....: hcpol = hilbert_class_polynomial(D)
....: j = hcpol.roots(multiplicities=False)[0]
....: t = (1728-j)/(27*j)
....: E = EllipticCurve([4*t,16*t^2])
....: P = E.point([0, 4*t])
....: assert(E.j_invariant() == j)
....: #
....: # Compute some CM number and its minimal polynomial
....: #
....: cm = pari('cm = (3*quadgen(%s)+2)'%D)
....: cm_minpoly = pari('minpoly(cm)')
....: #
....: # Evaluate cm_minpoly(cm)(P), which should be zero
....: #
....: e = pari(E) # Convert E to PARI
....: P2 = e.ellpow(P, cm_minpoly[2]*cm + cm_minpoly[1])
....: P0 = e.elladd(e.ellpow(P, cm_minpoly[0]), e.ellpow(P2, cm))
....: assert(P0 == E(0))
"""
cdef gen t0 = objtogen(z)
cdef gen t1 = objtogen(n)
pari_catch_sig_on()
return P.new_gen(powell(self.g, t0.g, t1.g))
def ellrootno(self, p=1):
"""
e.ellrootno(p): return the (local or global) root number of the
`L`-series of the elliptic curve e
If p is a prime number, the local root number at p is returned. If
p is 1, the global root number is returned. Note that the global
root number is the sign of the functional equation of the
`L`-series, and therefore conjecturally equal to the parity
of the rank of e.
INPUT:
- ``e`` - elliptic curve over `\QQ`
- ``p (default = 1)`` - 1 or a prime number
OUTPUT: 1 or -1
EXAMPLES: Here is a curve of rank 3::
sage: e = pari([0,0,0,-82,0]).ellinit()
sage: e.ellrootno()
-1
sage: e.ellrootno(2)
1
sage: e.ellrootno(1009)
1
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
rootno = ellrootno(self.g, t0.g)
pari_catch_sig_off()
return rootno
def ellsigma(self, z, long flag=0, unsigned long precision=0):
"""
e.ellsigma(z, flag=0): return the value at the complex point z of
the Weierstrass `\sigma` function associated to the
elliptic curve e.
EXAMPLES::
sage: e = pari([0,0,0,1,0]).ellinit()
sage: C.<i> = ComplexField()
sage: e.ellsigma(2+i)
1.43490215804166 + 1.80307856719256*I
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(ellsigma(self.g, t0.g, flag, prec_bits_to_words(precision)))
def ellsub(self, z0, z1):
"""
e.ellsub(z0, z1): return z0-z1 on this elliptic curve.
INPUT:
- ``e`` - elliptic curve E
- ``z0`` - point on E
- ``z1`` - point on E
OUTPUT: point on E
EXAMPLES::
sage: e = pari([0, 1, 1, -2, 0]).ellinit()
sage: e.ellsub([1,0], [-1,1])
[0, 0]
"""
cdef gen t0 = objtogen(z0)
cdef gen t1 = objtogen(z1)
pari_catch_sig_on()
return P.new_gen(subell(self.g, t0.g, t1.g))
def elltaniyama(self):
pari_catch_sig_on()
return P.new_gen(taniyama(self.g))
def elltors(self, long flag=0):
"""
e.elltors(flag = 0): return information about the torsion subgroup
of the elliptic curve e
INPUT:
- ``e`` - elliptic curve over `\QQ`
- ``flag (optional)`` - specify which algorithm to
use:
- ``0 (default)`` - use Doud's algorithm: bound
torsion by computing the cardinality of e(GF(p)) for small primes
of good reduction, then look for torsion points using Weierstrass
parametrization and Mazur's classification
- ``1`` - use algorithm given by the Nagell-Lutz
theorem (this is much slower)
OUTPUT:
- ``gen`` - the order of the torsion subgroup, a.k.a.
the number of points of finite order
- ``gen`` - vector giving the structure of the torsion
subgroup as a product of cyclic groups, sorted in non-increasing
order
- ``gen`` - vector giving points on e generating these
cyclic groups
EXAMPLES::
sage: e = pari([1,0,1,-19,26]).ellinit()
sage: e.elltors()
[12, [6, 2], [[-2, 8], [3, -2]]]
"""
pari_catch_sig_on()
return P.new_gen(elltors0(self.g, flag))
def ellzeta(self, z, unsigned long precision=0):
"""
e.ellzeta(z): return the value at the complex point z of the
Weierstrass `\zeta` function associated with the elliptic
curve e.
.. note::
This function has infinitely many poles (one of which is at
z=0); attempting to evaluate it too close to one of the
poles will result in a PariError.
INPUT:
- ``e`` - elliptic curve
- ``z`` - complex number
EXAMPLES::
sage: e = pari([0,0,0,1,0]).ellinit()
sage: e.ellzeta(1)
1.06479841295883 + 0.E-19*I # 32-bit
1.06479841295883 + 5.42101086242752 E-20*I # 64-bit
sage: C.<i> = ComplexField()
sage: e.ellzeta(i-1)
-0.350122658523049 - 0.350122658523049*I
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(ellzeta(self.g, t0.g, prec_bits_to_words(precision)))
def ellztopoint(self, z, unsigned long precision=0):
"""
e.ellztopoint(z): return the point on the elliptic curve e
corresponding to the complex number z, under the usual complex
uniformization of e by the Weierstrass p-function.
INPUT:
- ``e`` - elliptic curve
- ``z`` - complex number
OUTPUT point on e
EXAMPLES::
sage: e = pari([0,0,0,1,0]).ellinit()
sage: C.<i> = ComplexField()
sage: e.ellztopoint(1+i)
[0.E-19 - 1.02152286795670*I, -0.149072813701096 - 0.149072813701096*I] # 32-bit
[7.96075508054992 E-21 - 1.02152286795670*I, -0.149072813701096 - 0.149072813701096*I] # 64-bit
Complex numbers belonging to the period lattice of e are of course
sent to the point at infinity on e::
sage: e.ellztopoint(0)
[0]
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(pointell(self.g, t0.g, prec_bits_to_words(precision)))
def omega(self):
"""
e.omega(): return basis for the period lattice of the elliptic
curve e.
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.omega()
[1.26920930427955, -0.634604652139777 - 1.45881661693850*I]
"""
return self[14:16]
def disc(self):
"""
e.disc(): return the discriminant of the elliptic curve e.
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.disc()
-161051
sage: _.factor()
[-1, 1; 11, 5]
"""
return self[11]
def j(self):
"""
e.j(): return the j-invariant of the elliptic curve e.
EXAMPLES::
sage: e = pari([0, -1, 1, -10, -20]).ellinit()
sage: e.j()
-122023936/161051
sage: _.factor()
[-1, 1; 2, 12; 11, -5; 31, 3]
"""
return self[12]
def ellj(self, unsigned long precision=0):
"""
Elliptic `j`-invariant of ``self``.
EXAMPLES::
sage: pari(I).ellj()
1728.00000000000
sage: pari(3*I).ellj()
153553679.396729
sage: pari('quadgen(-3)').ellj()
0.E-54
sage: pari('quadgen(-7)').ellj(precision=256).sage()
-3375.000000000000000000000000000000000000000000000000000000000000000000000000
sage: pari(-I).ellj()
Traceback (most recent call last):
...
PariError: argument '-I' does not belong to upper half-plane
"""
pari_catch_sig_on()
return P.new_gen(jell(self.g, prec_bits_to_words(precision)))
def bnfcertify(self):
r"""
``bnf`` being as output by ``bnfinit``, checks whether the result is
correct, i.e. whether the calculation of the contents of ``self``
are correct without assuming the Generalized Riemann Hypothesis.
If it is correct, the answer is 1. If not, the program may output
some error message or loop indefinitely.
For more information about PARI and the Generalized Riemann
Hypothesis, see [PariUsers], page 120.
REFERENCES:
.. [PariUsers] User's Guide to PARI/GP,
http://pari.math.u-bordeaux.fr/pub/pari/manuals/2.5.1/users.pdf
"""
pari_catch_sig_on()
n = bnfcertify(self.g)
pari_catch_sig_off()
return n
def bnfinit(self, long flag=0, tech=None, unsigned long precision=0):
cdef gen t0
if tech is None:
pari_catch_sig_on()
return P.new_gen(bnfinit0(self.g, flag, NULL, prec_bits_to_words(precision)))
else:
t0 = objtogen(tech)
pari_catch_sig_on()
return P.new_gen(bnfinit0(self.g, flag, t0.g, prec_bits_to_words(precision)))
def bnfisintnorm(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(bnfisintnorm(self.g, t0.g))
def bnfisnorm(self, x, long flag=0):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(bnfisnorm(self.g, t0.g, flag))
def bnfisprincipal(self, x, long flag=1):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(bnfisprincipal0(self.g, t0.g, flag))
def bnfnarrow(self):
pari_catch_sig_on()
return P.new_gen(buchnarrow(self.g))
def bnfsunit(self, S, unsigned long precision=0):
cdef gen t0 = objtogen(S)
pari_catch_sig_on()
return P.new_gen(bnfsunit(self.g, t0.g, prec_bits_to_words(precision)))
def bnfunit(self):
pari_catch_sig_on()
return P.new_gen(bnf_get_fu(self.g))
def bnfisunit(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(bnfisunit(self.g, t0.g))
def bnrclassno(self, I):
r"""
Return the order of the ray class group of self modulo ``I``.
INPUT:
- ``self``: a pari "BNF" object representing a number field
- ``I``: a pari "BID" object representing an ideal of self
OUTPUT: integer
TESTS::
sage: K.<z> = QuadraticField(-23)
sage: p = K.primes_above(3)[0]
sage: K.pari_bnf().bnrclassno(p._pari_bid_())
3
"""
cdef gen t0 = objtogen(I)
pari_catch_sig_on()
return P.new_gen(bnrclassno(self.g, t0.g))
def bnfissunit(self, sunit_data, x):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(sunit_data)
pari_catch_sig_on()
return P.new_gen(bnfissunit(self.g, t1.g, t0.g))
def dirzetak(self, n):
cdef gen t0 = objtogen(n)
pari_catch_sig_on()
return P.new_gen(dirzetak(self.g, t0.g))
def galoisapply(self, aut, x):
cdef gen t0 = objtogen(aut)
cdef gen t1 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(galoisapply(self.g, t0.g, t1.g))
def galoisinit(self, den=None):
"""
galoisinit(K{,den}): calculate Galois group of number field K; see PARI manual
for meaning of den
"""
cdef gen t0
if den is None:
pari_catch_sig_on()
return P.new_gen(galoisinit(self.g, NULL))
else:
t0 = objtogen(den)
pari_catch_sig_on()
return P.new_gen(galoisinit(self.g, t0.g))
def galoispermtopol(self, perm):
cdef gen t0 = objtogen(perm)
pari_catch_sig_on()
return P.new_gen(galoispermtopol(self.g, t0.g))
def galoisfixedfield(self, perm, long flag=0, v=-1):
cdef gen t0 = objtogen(perm)
pari_catch_sig_on()
return P.new_gen(galoisfixedfield(self.g, t0.g, flag, P.get_var(v)))
def idealred(self, I, vdir=0):
cdef gen t0 = objtogen(I)
cdef gen t1 = objtogen(vdir)
pari_catch_sig_on()
return P.new_gen(idealred0(self.g, t0.g, t1.g if vdir else NULL))
def idealadd(self, x, y):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(idealadd(self.g, t0.g, t1.g))
def idealaddtoone(self, x, y):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(idealaddtoone0(self.g, t0.g, t1.g))
def idealappr(self, x, long flag=0):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(idealappr0(self.g, t0.g, flag))
def idealcoprime(self, x, y):
"""
Given two integral ideals x and y of a pari number field self,
return an element a of the field (expressed in the integral
basis of self) such that a*x is an integral ideal coprime to
y.
EXAMPLES::
sage: F = NumberField(x^3-2, 'alpha')
sage: nf = F._pari_()
sage: x = pari('[1, -1, 2]~')
sage: y = pari('[1, -1, 3]~')
sage: nf.idealcoprime(x, y)
[1, 0, 0]~
sage: y = pari('[2, -2, 4]~')
sage: nf.idealcoprime(x, y)
[5/43, 9/43, -1/43]~
"""
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(idealcoprime(self.g, t0.g, t1.g))
def idealdiv(self, x, y, long flag=0):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(idealdiv0(self.g, t0.g, t1.g, flag))
def idealfactor(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(idealfactor(self.g, t0.g))
def idealhnf(self, a, b=None):
cdef gen t0 = objtogen(a)
cdef gen t1
if b is None:
pari_catch_sig_on()
return P.new_gen(idealhnf(self.g, t0.g))
else:
t1 = objtogen(b)
pari_catch_sig_on()
return P.new_gen(idealhnf0(self.g, t0.g, t1.g))
def idealintersection(self, x, y):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(idealintersect(self.g, t0.g, t1.g))
def ideallist(self, long bound, long flag = 4):
"""
Vector of vectors `L` of all idealstar of all ideals of `norm <= bound`.
The binary digits of flag mean:
- 1: give generators;
- 2: add units;
- 4: (default) give only the ideals and not the bid.
EXAMPLES::
sage: R.<x> = PolynomialRing(QQ)
sage: K.<a> = NumberField(x^2 + 1)
sage: L = K.pari_nf().ideallist(100)
Now we have our list `L`. Entry `L[n-1]` contains all ideals of
norm `n`::
sage: L[0] # One ideal of norm 1.
[[1, 0; 0, 1]]
sage: L[64] # 4 ideals of norm 65.
[[65, 8; 0, 1], [65, 47; 0, 1], [65, 18; 0, 1], [65, 57; 0, 1]]
"""
pari_catch_sig_on()
return P.new_gen(ideallist0(self.g, bound, flag))
def ideallog(self, x, bid):
"""
Return the discrete logarithm of the unit x in (ring of integers)/bid.
INPUT:
- ``self`` - a pari number field
- ``bid`` - a big ideal structure (corresponding to an ideal I
of self) output by idealstar
- ``x`` - an element of self with valuation zero at all
primes dividing I
OUTPUT:
- the discrete logarithm of x on the generators given in bid[2]
EXAMPLE::
sage: F = NumberField(x^3-2, 'alpha')
sage: nf = F._pari_()
sage: I = pari('[1, -1, 2]~')
sage: bid = nf.idealstar(I)
sage: x = pari('5')
sage: nf.ideallog(x, bid)
[25]~
"""
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(bid)
pari_catch_sig_on()
return P.new_gen(ideallog(self.g, t0.g, t1.g))
def idealmul(self, x, y, long flag=0):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
if flag == 0:
return P.new_gen(idealmul(self.g, t0.g, t1.g))
else:
return P.new_gen(idealmulred(self.g, t0.g, t1.g))
def idealnorm(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(idealnorm(self.g, t0.g))
def idealprimedec(nf, p):
"""
Prime ideal decomposition of the prime number `p` in the number
field `nf` as a vector of 5 component vectors `[p,a,e,f,b]`
representing the prime ideals `p O_K + a O_K`, `e` ,`f` as usual,
`a` as vector of components on the integral basis, `b` Lenstra's
constant.
EXAMPLES::
sage: K.<i> = QuadraticField(-1)
sage: F = pari(K).idealprimedec(5); F
[[5, [-2, 1]~, 1, 1, [2, 1]~], [5, [2, 1]~, 1, 1, [-2, 1]~]]
sage: F[0].pr_get_p()
5
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(idealprimedec(nf.g, t0.g))
def idealstar(self, I, long flag=1):
"""
Return the big ideal (bid) structure of modulus I.
INPUT:
- ``self`` - a pari number field
- ``I`` -- an ideal of self, or a row vector whose first
component is an ideal and whose second component
is a row vector of r_1 0 or 1.
- ``flag`` - determines the amount of computation and the shape
of the output:
- ``1`` (default): return a bid structure without
generators
- ``2``: return a bid structure with generators (slower)
- ``0`` (deprecated): only outputs units of (ring of integers/I)
as an abelian group, i.e as a 3-component
vector [h,d,g]: h is the order, d is the vector
of SNF cyclic components and g the corresponding
generators. This flag is deprecated: it is in
fact slightly faster to compute a true bid
structure, which contains much more information.
EXAMPLE::
sage: F = NumberField(x^3-2, 'alpha')
sage: nf = F._pari_()
sage: I = pari('[1, -1, 2]~')
sage: nf.idealstar(I)
[[[43, 9, 5; 0, 1, 0; 0, 0, 1], [0]], [42, [42]], Mat([[43, [9, 1, 0]~, 1, 1, [-5, -9, 1]~], 1]), [[[[42], [[3, 0, 0]~], [[3, 0, 0]~], [Vecsmall([])], 1]], [[], [], []]], Mat(1)]
"""
cdef gen t0 = objtogen(I)
pari_catch_sig_on()
return P.new_gen(idealstar0(self.g, t0.g, flag))
def idealtwoelt(self, x, a=None):
cdef gen t0 = objtogen(x)
cdef gen t1
if a is None:
pari_catch_sig_on()
return P.new_gen(idealtwoelt0(self.g, t0.g, NULL))
else:
t1 = objtogen(a)
pari_catch_sig_on()
return P.new_gen(idealtwoelt0(self.g, t0.g, t1.g))
def idealval(self, x, p):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(p)
pari_catch_sig_on()
v = idealval(self.g, t0.g, t1.g)
pari_catch_sig_off()
return v
def elementval(self, x, p):
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(p)
pari_catch_sig_on()
v = nfval(self.g, t0.g, t1.g)
pari_catch_sig_off()
return v
def modreverse(self):
"""
modreverse(x): reverse polymod of the polymod x, if it exists.
EXAMPLES:
"""
pari_catch_sig_on()
return P.new_gen(modreverse(self.g))
def nfbasis(self, long flag=0, fa=None):
"""
nfbasis(x, flag, fa): integral basis of the field QQ[a], where ``a`` is
a root of the polynomial x.
Binary digits of ``flag`` mean:
- 1: assume that no square of a prime>primelimit divides the
discriminant of ``x``.
- 2: use round 2 algorithm instead of round 4.
If present, ``fa`` provides the matrix of a partial factorization of
the discriminant of ``x``, useful if one wants only an order maximal at
certain primes only.
EXAMPLES::
sage: pari('x^3 - 17').nfbasis()
[1, x, 1/3*x^2 - 1/3*x + 1/3]
We test ``flag`` = 1, noting it gives a wrong result when the
discriminant (-4 * `p`^2 * `q` in the example below) has a big square
factor::
sage: p = next_prime(10^10); q = next_prime(p)
sage: x = polygen(QQ); f = x^2 + p^2*q
sage: pari(f).nfbasis(1) # Wrong result
[1, x]
sage: pari(f).nfbasis() # Correct result
[1, 1/10000000019*x]
sage: pari(f).nfbasis(fa = "[2,2; %s,2]"%p) # Correct result and faster
[1, 1/10000000019*x]
TESTS:
``flag`` = 2 should give the same result::
sage: pari('x^3 - 17').nfbasis(flag = 2)
[1, x, 1/3*x^2 - 1/3*x + 1/3]
"""
cdef gen t0
if not fa:
pari_catch_sig_on()
return P.new_gen(nfbasis0(self.g, flag, NULL))
else:
t0 = objtogen(fa)
pari_catch_sig_on()
return P.new_gen(nfbasis0(self.g, flag, t0.g))
def nfbasis_d(self, long flag=0, fa=None):
"""
nfbasis_d(x): Return a basis of the number field defined over QQ
by x and its discriminant.
EXAMPLES::
sage: F = NumberField(x^3-2,'alpha')
sage: F._pari_()[0].nfbasis_d()
([1, y, y^2], -108)
::
sage: G = NumberField(x^5-11,'beta')
sage: G._pari_()[0].nfbasis_d()
([1, y, y^2, y^3, y^4], 45753125)
::
sage: pari([-2,0,0,1]).Polrev().nfbasis_d()
([1, x, x^2], -108)
"""
cdef gen t0
cdef GEN disc
if not fa:
pari_catch_sig_on()
B = P.new_gen_noclear(nfbasis(self.g, &disc, flag, NULL))
else:
t0 = objtogen(fa)
pari_catch_sig_on()
B = P.new_gen_noclear(nfbasis(self.g, &disc, flag, t0.g))
D = P.new_gen(disc)
return B, D
def nfbasistoalg(nf, x):
r"""
Transforms the column vector ``x`` on the integral basis into an
algebraic number.
INPUT:
- ``nf`` -- a number field
- ``x`` -- a column of rational numbers of length equal to the
degree of ``nf`` or a single rational number
OUTPUT:
- A POLMOD representing the element of ``nf`` whose coordinates
are ``x`` in the Z-basis of ``nf``.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^3 - 17)
sage: Kpari = K.pari_nf()
sage: Kpari.getattr('zk')
[1, 1/3*y^2 - 1/3*y + 1/3, y]
sage: Kpari.nfbasistoalg(42)
Mod(42, y^3 - 17)
sage: Kpari.nfbasistoalg("[3/2, -5, 0]~")
Mod(-5/3*y^2 + 5/3*y - 1/6, y^3 - 17)
sage: Kpari.getattr('zk') * pari("[3/2, -5, 0]~")
-5/3*y^2 + 5/3*y - 1/6
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(basistoalg(nf.g, t0.g))
def nfbasistoalg_lift(nf, x):
r"""
Transforms the column vector ``x`` on the integral basis into a
polynomial representing the algebraic number.
INPUT:
- ``nf`` -- a number field
- ``x`` -- a column of rational numbers of length equal to the
degree of ``nf`` or a single rational number
OUTPUT:
- ``nf.nfbasistoalg(x).lift()``
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^3 - 17)
sage: Kpari = K.pari_nf()
sage: Kpari.getattr('zk')
[1, 1/3*y^2 - 1/3*y + 1/3, y]
sage: Kpari.nfbasistoalg_lift(42)
42
sage: Kpari.nfbasistoalg_lift("[3/2, -5, 0]~")
-5/3*y^2 + 5/3*y - 1/6
sage: Kpari.getattr('zk') * pari("[3/2, -5, 0]~")
-5/3*y^2 + 5/3*y - 1/6
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(gel(basistoalg(nf.g, t0.g), 2))
def nfdisc(self, long flag=0, p=0):
"""
nfdisc(x): Return the discriminant of the number field defined over
QQ by x.
EXAMPLES::
sage: F = NumberField(x^3-2,'alpha')
sage: F._pari_()[0].nfdisc()
-108
::
sage: G = NumberField(x^5-11,'beta')
sage: G._pari_()[0].nfdisc()
45753125
::
sage: f = x^3-2
sage: f._pari_()
x^3 - 2
sage: f._pari_().nfdisc()
-108
"""
cdef gen _p
cdef GEN g
if p != 0:
_p = self.pari(p)
g = _p.g
else:
g = <GEN>NULL
pari_catch_sig_on()
return P.new_gen(nfdisc0(self.g, flag, g))
def nfeltdiveuc(self, x, y):
"""
Given `x` and `y` in the number field ``self``, return `q` such
that `x - q y` is "small".
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 5)
sage: x = 10
sage: y = a + 1
sage: pari(k).nfeltdiveuc(pari(x), pari(y))
[2, -2]~
"""
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(nfdiveuc(self.g, t0.g, t1.g))
def nfeltreduce(self, x, I):
"""
Given an ideal I in Hermite normal form and an element x of the pari
number field self, finds an element r in self such that x-r belongs
to the ideal and r is small.
EXAMPLES::
sage: k.<a> = NumberField(x^2 + 5)
sage: I = k.ideal(a)
sage: kp = pari(k)
sage: kp.nfeltreduce(12, I.pari_hnf())
[2, 0]~
sage: 12 - k(kp.nfeltreduce(12, I.pari_hnf())) in I
True
"""
cdef gen t0 = objtogen(x)
cdef gen t1 = objtogen(I)
pari_catch_sig_on()
return P.new_gen(nfreduce(self.g, t0.g, t1.g))
def nffactor(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(nffactor(self.g, t0.g))
def nfgenerator(self):
f = self[0]
x = f.variable()
return x.Mod(f)
def nfhilbert(self, a, b, p=None):
"""
nfhilbert(nf,a,b,{p}): if p is omitted, global Hilbert symbol (a,b)
in nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z)
in nf, -1 otherwise. Otherwise compute the local symbol modulo the
prime ideal p.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<t> = NumberField(x^3 - x + 1)
sage: pari(K).nfhilbert(t, t + 2)
-1
sage: P = K.ideal(t^2 + t - 2) # Prime ideal above 5
sage: pari(K).nfhilbert(t, t + 2, P.pari_prime())
-1
sage: P = K.ideal(t^2 + 3*t - 1) # Prime ideal above 23, ramified
sage: pari(K).nfhilbert(t, t + 2, P.pari_prime())
1
"""
cdef gen t0 = objtogen(a)
cdef gen t1 = objtogen(b)
cdef gen t2
if p:
t2 = objtogen(p)
pari_catch_sig_on()
r = nfhilbert0(self.g, t0.g, t1.g, t2.g)
else:
pari_catch_sig_on()
r = nfhilbert(self.g, t0.g, t1.g)
pari_catch_sig_off()
return r
def nfhnf(self,x):
"""
nfhnf(nf,x) : given a pseudo-matrix (A, I) or an integral pseudo-matrix (A,I,J), finds a
pseudo-basis in Hermite normal form of the module it generates.
A pseudo-matrix is a 2-component row vector (A, I) where A is a relative m x n matrix and
I an ideal list of length n. An integral pseudo-matrix is a 3-component row vector (A, I, J).
.. NOTE::
The definition of a pseudo-basis ([Cohen]_):
Let M be a finitely generated, torsion-free R-module, and set V = KM. If `\mathfrak{a}_i` are
fractional ideals of R and `w_i` are elements of V, we say that
`(w_i, \mathfrak{a}_k)_{1 \leq i \leq k}`
is a pseudo-basis of M if
`M = \mathfrak{a}_1 w_1 \oplus \cdots \oplus \mathfrak{a}_k w_k.`
REFERENCES:
.. [Cohen] Cohen, "Advanced Topics in Computational Number Theory"
EXAMPLES::
sage: F.<a> = NumberField(x^2-x-1)
sage: Fp = pari(F)
sage: A = matrix(F,[[1,2,a,3],[3,0,a+2,0],[0,0,a,2],[3+a,a,0,1]])
sage: I = [F.ideal(-2*a+1),F.ideal(7), F.ideal(3),F.ideal(1)]
sage: Fp.nfhnf([pari(A),[pari(P) for P in I]])
[[1, [-969/5, -1/15]~, [15, -2]~, [-1938, -3]~; 0, 1, 0, 0; 0, 0, 1, 0;
0, 0, 0, 1], [[3997, 1911; 0, 7], [15, 6; 0, 3], [1, 0; 0, 1], [1, 0; 0,
1]]]
sage: K.<b> = NumberField(x^3-2)
sage: Kp = pari(K)
sage: A = matrix(K,[[1,0,0,5*b],[1,2*b^2,b,57],[0,2,1,b^2-3],[2,0,0,b]])
sage: I = [K.ideal(2),K.ideal(3+b^2),K.ideal(1),K.ideal(1)]
sage: Kp.nfhnf([pari(A),[pari(P) for P in I]])
[[1, -225, 72, -31; 0, 1, [0, -1, 0]~, [0, 0, -1/2]~; 0, 0, 1, [0, 0,
-1/2]~; 0, 0, 0, 1], [[1116, 756, 612; 0, 18, 0; 0, 0, 18], [2, 0, 0; 0,
2, 0; 0, 0, 2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [2, 0, 0; 0, 1, 0; 0, 0,
1]]]
An example where the ring of integers of the number field is not a PID::
sage: K.<b> = NumberField(x^2+5)
sage: Kp = pari(K)
sage: A = matrix(K,[[1,0,0,5*b],[1,2*b^2,b,57],[0,2,1,b^2-3],[2,0,0,b]])
sage: I = [K.ideal(2),K.ideal(3+b^2),K.ideal(1),K.ideal(1)]
sage: Kp.nfhnf([pari(A),[pari(P) for P in I]])
[[1, [15, 6]~, [0, -54]~, [113, 72]~; 0, 1, [-4, -1]~, [0, -1]~; 0, 0,
1, 0; 0, 0, 0, 1], [[360, 180; 0, 180], [6, 4; 0, 2], [1, 0; 0, 1], [1,
0; 0, 1]]]
sage: A = matrix(K,[[1,0,0,5*b],[1,2*b,b,57],[0,2,1,b-3],[2,0,b,b]])
sage: I = [K.ideal(2).factor()[0][0],K.ideal(3+b),K.ideal(1),K.ideal(1)]
sage: Kp.nfhnf([pari(A),[pari(P) for P in I]])
[[1, [7605, 4]~, [5610, 5]~, [7913, -6]~; 0, 1, 0, -1; 0, 0, 1, 0; 0, 0,
0, 1], [[19320, 13720; 0, 56], [2, 1; 0, 1], [1, 0; 0, 1], [1, 0; 0,
1]]]
AUTHORS:
- Aly Deines (2012-09-19)
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(nfhnf(self.g, t0.g))
def nfinit(self, long flag=0, unsigned long precision=0):
"""
nfinit(pol, {flag=0}): ``pol`` being a nonconstant irreducible
polynomial, gives a vector containing all the data necessary for PARI
to compute in this number field.
``flag`` is optional and can be set to:
- 0: default
- 1: do not compute different
- 2: first use polred to find a simpler polynomial
- 3: outputs a two-element vector [nf,Mod(a,P)], where nf is as in 2
and Mod(a,P) is a polmod equal to Mod(x,pol) and P=nf.pol
EXAMPLES::
sage: pari('x^3 - 17').nfinit()
[x^3 - 17, [1, 1], -867, 3, [[1, 1.68006..., 2.57128...; 1, -0.340034... + 2.65083...*I, -1.28564... - 2.22679...*I], [1, 1.68006..., 2.57128...; 1, 2.31080..., -3.51243...; 1, -2.99087..., 0.941154...], [1, 2, 3; 1, 2, -4; 1, -3, 1], [3, 1, 0; 1, -11, 17; 0, 17, 0], [51, 0, 16; 0, 17, 3; 0, 0, 1], [17, 0, -1; 0, 0, 3; -1, 3, 2], [51, [-17, 6, -1; 0, -18, 3; 1, 0, -16]]], [2.57128..., -1.28564... - 2.22679...*I], [1, 1/3*x^2 - 1/3*x + 1/3, x], [1, 0, -1; 0, 0, 3; 0, 1, 1], [1, 0, 0, 0, -4, 6, 0, 6, -1; 0, 1, 0, 1, 1, -1, 0, -1, 3; 0, 0, 1, 0, 2, 0, 1, 0, 1]]
TESTS:
This example only works after increasing precision::
sage: pari('x^2 + 10^100 + 1').nfinit(precision=64)
Traceback (most recent call last):
...
PariError: precision too low in floorr (precision loss in truncation)
sage: pari('x^2 + 10^100 + 1').nfinit()
[...]
Throw a PARI error which is not of type ``precer``::
sage: pari('1.0').nfinit()
Traceback (most recent call last):
...
PariError: incorrect type in checknf
"""
if precision:
pari_catch_sig_on()
return P.new_gen(nfinit0(self.g, flag, prec_bits_to_words(precision)))
precision = 64
while True:
try:
return self.nfinit(flag, precision)
except PariError as err:
if err.errnum() == precer:
precision *= 2
else:
raise
def nfisisom(self, gen other):
"""
nfisisom(x, y): Determine if the number fields defined by x and y
are isomorphic. According to the PARI documentation, this is much
faster if at least one of x or y is a number field. If they are
isomorphic, it returns an embedding for the generators. If not,
returns 0.
EXAMPLES::
sage: F = NumberField(x^3-2,'alpha')
sage: G = NumberField(x^3-2,'beta')
sage: F._pari_().nfisisom(G._pari_())
[y]
::
sage: GG = NumberField(x^3-4,'gamma')
sage: F._pari_().nfisisom(GG._pari_())
[1/2*y^2]
::
sage: F._pari_().nfisisom(GG.pari_nf())
[1/2*y^2]
::
sage: F.pari_nf().nfisisom(GG._pari_()[0])
[y^2]
::
sage: H = NumberField(x^2-2,'alpha')
sage: F._pari_().nfisisom(H._pari_())
0
"""
pari_catch_sig_on()
return P.new_gen(nfisisom(self.g, other.g))
def nfrootsof1(self):
"""
nf.nfrootsof1()
number of roots of unity and primitive root of unity in the number
field nf.
EXAMPLES::
sage: nf = pari('x^2 + 1').nfinit()
sage: nf.nfrootsof1()
[4, -x]
"""
pari_catch_sig_on()
return P.new_gen(rootsof1(self.g))
def nfsubfields(self, long d=0):
"""
Find all subfields of degree d of number field nf (all subfields if
d is null or omitted). Result is a vector of subfields, each being
given by [g,h], where g is an absolute equation and h expresses one
of the roots of g in terms of the root x of the polynomial defining
nf.
INPUT:
- ``self`` - nf number field
- ``d`` - C long integer
"""
pari_catch_sig_on()
return P.new_gen(nfsubfields(self.g, d))
def rnfcharpoly(self, T, a, v='x'):
cdef gen t0 = objtogen(T)
cdef gen t1 = objtogen(a)
cdef gen t2 = objtogen(v)
pari_catch_sig_on()
return P.new_gen(rnfcharpoly(self.g, t0.g, t1.g, gvar(t2.g)))
def rnfdisc(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfdiscf(self.g, t0.g))
def rnfeltabstorel(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfelementabstorel(self.g, t0.g))
def rnfeltreltoabs(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfelementreltoabs(self.g, t0.g))
def rnfequation(self, poly, long flag=0):
cdef gen t0 = objtogen(poly)
pari_catch_sig_on()
return P.new_gen(rnfequation0(self.g, t0.g, flag))
def rnfidealabstorel(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfidealabstorel(self.g, t0.g))
def rnfidealdown(self, x):
r"""
rnfidealdown(rnf,x): finds the intersection of the ideal x with the base field.
EXAMPLES:
sage: x = ZZ['xx1'].0; pari(x)
xx1
sage: y = ZZ['yy1'].0; pari(y)
yy1
sage: nf = pari(y^2 - 6*y + 24).nfinit()
sage: rnf = nf.rnfinit(x^2 - pari(y))
This is the relative HNF of the inert ideal (2) in rnf::
sage: P = pari('[[[1, 0]~, [0, 0]~; [0, 0]~, [1, 0]~], [[2, 0; 0, 2], [2, 0; 0, 1/2]]]')
And this is the HNF of the inert ideal (2) in nf:
sage: rnf.rnfidealdown(P)
[2, 0; 0, 2]
"""
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfidealdown(self.g, t0.g))
def rnfidealhnf(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfidealhermite(self.g, t0.g))
def rnfidealnormrel(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfidealnormrel(self.g, t0.g))
def rnfidealreltoabs(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfidealreltoabs(self.g, t0.g))
def rnfidealtwoelt(self, x):
cdef gen t0 = objtogen(x)
pari_catch_sig_on()
return P.new_gen(rnfidealtwoelement(self.g, t0.g))
def rnfinit(self, poly):
"""
EXAMPLES: We construct a relative number field.
::
sage: f = pari('y^3+y+1')
sage: K = f.nfinit()
sage: x = pari('x'); y = pari('y')
sage: g = x^5 - x^2 + y
sage: L = K.rnfinit(g)
"""
cdef gen t0 = objtogen(poly)
pari_catch_sig_on()
return P.new_gen(rnfinit(self.g, t0.g))
def rnfisfree(self, poly):
cdef gen t0 = objtogen(poly)
pari_catch_sig_on()
r = rnfisfree(self.g, t0.g)
pari_catch_sig_off()
return r
def quadhilbert(self):
r"""
Returns a polynomial over `\QQ` whose roots generate the
Hilbert class field of the quadratic field of discriminant
``self`` (which must be fundamental).
EXAMPLES::
sage: pari(-23).quadhilbert()
x^3 - x^2 + 1
sage: pari(145).quadhilbert()
x^4 - x^3 - 3*x^2 + x + 1
sage: pari(-12).quadhilbert() # Not fundamental
Traceback (most recent call last):
...
PariError: quadray needs a fundamental discriminant
"""
pari_catch_sig_on()
return P.new_gen(quadhilbert(self.g, DEFAULTPREC))
def reverse(self):
"""
Return the polynomial obtained by reversing the coefficients of
this polynomial.
"""
return self.Vec().Polrev()
def content(self):
"""
Greatest common divisor of all the components of ``self``.
EXAMPLES::
sage: R.<x> = PolynomialRing(ZZ)
sage: pari(2*x^2 + 2).content()
2
sage: pari("4*x^3 - 2*x/3 + 2/5").content()
2/15
"""
pari_catch_sig_on()
return P.new_gen(content(self.g))
def deriv(self, v=-1):
pari_catch_sig_on()
return P.new_gen(deriv(self.g, P.get_var(v)))
def eval(self, *args, **kwds):
"""
Evaluate ``self`` with the given arguments.
This is currently implemented in 3 cases:
- univariate polynomials (using a single unnamed argument or
keyword arguments),
- any PARI object supporting the PARI function ``substvec``
(in particular, multivariate polynomials) using keyword
arguments,
- objects of type ``t_CLOSURE`` (functions in GP bytecode form)
using unnamed arguments.
In no case is mixing unnamed and keyword arguments allowed.
EXAMPLES::
sage: f = pari('x^2 + 1')
sage: f.type()
't_POL'
sage: f.eval(I)
0
sage: f.eval(x=2)
5
The notation ``f(x)`` is an alternative for ``f.eval(x)``::
sage: f(3) == f.eval(3)
True
Evaluating multivariate polynomials::
sage: f = pari('y^2 + x^3')
sage: f(1) # Dangerous, depends on PARI variable ordering
y^2 + 1
sage: f(x=1) # Safe
y^2 + 1
sage: f(y=1)
x^3 + 1
sage: f(1, 2)
Traceback (most recent call last):
...
TypeError: evaluating a polynomial takes exactly 1 argument (2 given)
sage: f(y='x', x='2*y')
x^2 + 8*y^3
sage: f()
x^3 + y^2
It's not an error to substitute variables which do not appear::
sage: f.eval(z=37)
x^3 + y^2
sage: pari(42).eval(t=0)
42
We can define and evaluate closures as follows::
sage: T = pari('n -> n + 2')
sage: T.type()
't_CLOSURE'
sage: T.eval(3)
5
sage: T = pari('() -> 42')
sage: T()
42
sage: pr = pari('s -> print(s)')
sage: pr.eval('"hello world"')
hello world
sage: f = pari('myfunc(x,y) = x*y')
sage: f.eval(5, 6)
30
Default arguments work, missing arguments are treated as zero
(like in GP)::
sage: f = pari("(x, y, z=1.0) -> [x, y, z]")
sage: f(1, 2, 3)
[1, 2, 3]
sage: f(1, 2)
[1, 2, 1.00000000000000]
sage: f(1)
[1, 0, 1.00000000000000]
sage: f()
[0, 0, 1.00000000000000]
Using keyword arguments, we can substitute in more complicated
objects, for example a number field::
sage: K.<a> = NumberField(x^2 + 1)
sage: nf = K._pari_()
sage: nf
[y^2 + 1, [0, 1], -4, 1, [Mat([1, 0.E-19 - 1.00000000000000*I]), [1, -1.00000000000000; 1, 1.00000000000000], [1, -1; 1, 1], [2, 0; 0, -2], [2, 0; 0, 2], [1, 0; 0, -1], [1, [0, -1; 1, 0]]], [0.E-19 - 1.00000000000000*I], [1, y], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, 0]]
sage: nf(y='x')
[x^2 + 1, [0, 1], -4, 1, [Mat([1, 0.E-19 - 1.00000000000000*I]), [1, -1.00000000000000; 1, 1.00000000000000], [1, -1; 1, 1], [2, 0; 0, -2], [2, 0; 0, 2], [1, 0; 0, -1], [1, [0, -1; 1, 0]]], [0.E-19 - 1.00000000000000*I], [1, x], [1, 0; 0, 1], [1, 0, 0, -1; 0, 1, 1, 0]]
"""
cdef long t = typ(self.g)
cdef gen t0
cdef GEN result
cdef long arity
cdef long nargs = len(args)
cdef long nkwds = len(kwds)
if t == t_CLOSURE:
if nkwds > 0:
raise TypeError("cannot evaluate a PARI closure using keyword arguments")
arity = self.g[1]
if nargs > arity:
raise TypeError("PARI closure takes at most %d argument%s (%d given)"%(
arity, "s" if (arity!=1) else "", nargs))
t0 = objtogen(args)
pari_catch_sig_on()
result = closure_callgenvec(self.g, t0.g)
if result == gnil:
P.clear_stack()
return None
return P.new_gen(result)
if nargs > 0:
if nkwds > 0:
raise TypeError("mixing unnamed and keyword arguments not allowed when evaluating a PARI object")
if t == t_POL:
if nargs != 1:
raise TypeError("evaluating a polynomial takes exactly 1 argument (%d given)"%nargs)
t0 = objtogen(args[0])
pari_catch_sig_on()
return P.new_gen(poleval(self.g, t0.g))
raise TypeError("cannot evaluate %s using unnamed arguments"%self.type())
vstr = kwds.keys()
t0 = objtogen(kwds.values())
pari_catch_sig_on()
cdef GEN v = cgetg(nkwds+1, t_VEC)
cdef long i
for i in range(nkwds):
set_gel(v, i+1, pol_x(P.get_var(vstr[i])))
return P.new_gen(gsubstvec(self.g, v, t0.g))
def __call__(self, *args, **kwds):
"""
Evaluate ``self`` with the given arguments.
This has the same effect as :meth:`eval`.
EXAMPLES::
sage: R.<x> = GF(3)[]
sage: f = (x^2 + x + 1)._pari_()
sage: f.type()
't_POL'
sage: f(2)
Mod(1, 3)
TESTS::
sage: T = pari('n -> 1/n')
sage: T.type()
't_CLOSURE'
sage: T(0)
Traceback (most recent call last):
...
PariError: _/_: division by zero
sage: pari('() -> 42')(1,2,3)
Traceback (most recent call last):
...
TypeError: PARI closure takes at most 0 arguments (3 given)
sage: pari('n -> n')(n=2)
Traceback (most recent call last):
...
TypeError: cannot evaluate a PARI closure using keyword arguments
sage: pari('x + y')(4, y=1)
Traceback (most recent call last):
...
TypeError: mixing unnamed and keyword arguments not allowed when evaluating a PARI object
sage: pari("12345")(4)
Traceback (most recent call last):
...
TypeError: cannot evaluate t_INT using unnamed arguments
"""
return self.eval(*args, **kwds)
def factornf(self, t):
"""
Factorization of the polynomial ``self`` over the number field
defined by the polynomial ``t``. This does not require that `t`
is integral, nor that the discriminant of the number field can be
factored.
EXAMPLES::
sage: x = polygen(QQ)
sage: K.<a> = NumberField(x^2 - 1/8)
sage: pari(x^2 - 2).factornf(K.pari_polynomial("a"))
[x + Mod(-4*a, 8*a^2 - 1), 1; x + Mod(4*a, 8*a^2 - 1), 1]
"""
cdef gen t0 = objtogen(t)
pari_catch_sig_on()
return P.new_gen(polfnf(self.g, t0.g))
def factorpadic(self, p, long r=20, long flag=0):
"""
self.factorpadic(p,r=20,flag=0): p-adic factorization of the
polynomial x to precision r. flag is optional and may be set to 0
(use round 4) or 1 (use Buchmann-Lenstra)
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(factorpadic0(self.g, t0.g, r, flag))
def factormod(self, p, long flag=0):
"""
x.factormod(p,flag=0): factorization mod p of the polynomial x
using Berlekamp. flag is optional, and can be 0: default or 1:
simple factormod, same except that only the degrees of the
irreducible factors are given.
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(factormod0(self.g, t0.g, flag))
def intformal(self, y=-1):
"""
x.intformal(y): formal integration of x with respect to the main
variable of y, or to the main variable of x if y is omitted
"""
pari_catch_sig_on()
return P.new_gen(integ(self.g, P.get_var(y)))
def padicappr(self, a):
"""
x.padicappr(a): p-adic roots of the polynomial x congruent to a mod
p
"""
cdef gen t0 = objtogen(a)
pari_catch_sig_on()
return P.new_gen(padicappr(self.g, t0.g))
def newtonpoly(self, p):
"""
x.newtonpoly(p): Newton polygon of polynomial x with respect to the
prime p.
EXAMPLES::
sage: x = pari('y^8+6*y^6-27*y^5+1/9*y^2-y+1')
sage: x.newtonpoly(3)
[1, 1, -1/3, -1/3, -1/3, -1/3, -1/3, -1/3]
"""
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(newtonpoly(self.g, t0.g))
def polcoeff(self, long n, var=-1):
"""
EXAMPLES::
sage: f = pari("x^2 + y^3 + x*y")
sage: f
x^2 + y*x + y^3
sage: f.polcoeff(1)
y
sage: f.polcoeff(3)
0
sage: f.polcoeff(3, "y")
1
sage: f.polcoeff(1, "y")
x
"""
pari_catch_sig_on()
return P.new_gen(polcoeff0(self.g, n, P.get_var(var)))
def polcompositum(self, pol2, long flag=0):
cdef gen t0 = objtogen(pol2)
pari_catch_sig_on()
return P.new_gen(polcompositum0(self.g, t0.g, flag))
def poldegree(self, var=-1):
"""
f.poldegree(var=x): Return the degree of this polynomial.
"""
pari_catch_sig_on()
n = poldegree(self.g, P.get_var(var))
pari_catch_sig_off()
return n
def poldisc(self, var=-1):
"""
Return the discriminant of this polynomial.
EXAMPLES::
sage: pari("x^2 + 1").poldisc()
-4
Before :trac:`15654`, this used to take a very long time.
Now it takes much less than a second::
sage: pari.allocatemem(200000)
PARI stack size set to 200000 bytes
sage: x = polygen(ZpFM(3,10))
sage: pol = ((x-1)^50 + x)
sage: pari(pol).poldisc()
2*3 + 3^4 + 2*3^6 + 3^7 + 2*3^8 + 2*3^9 + O(3^10)
"""
pari_catch_sig_on()
return P.new_gen(poldisc0(self.g, P.get_var(var)))
def poldiscreduced(self):
pari_catch_sig_on()
return P.new_gen(reduceddiscsmith(self.g))
def polgalois(self, unsigned long precision=0):
"""
f.polgalois(): Galois group of the polynomial f
"""
pari_catch_sig_on()
return P.new_gen(polgalois(self.g, prec_bits_to_words(precision)))
def nfgaloisconj(self, long flag=0, denom=None, unsigned long precision=0):
r"""
Edited from the pari documentation:
nfgaloisconj(nf): list of conjugates of a root of the
polynomial x=nf.pol in the same number field.
Uses a combination of Allombert's algorithm and nfroots.
EXAMPLES::
sage: x = QQ['x'].0; nf = pari(x^2 + 2).nfinit()
sage: nf.nfgaloisconj()
[-x, x]~
sage: nf = pari(x^3 + 2).nfinit()
sage: nf.nfgaloisconj()
[x]~
sage: nf = pari(x^4 + 2).nfinit()
sage: nf.nfgaloisconj()
[-x, x]~
"""
cdef gen t0
if denom is None:
pari_catch_sig_on()
return P.new_gen(galoisconj0(self.g, flag, NULL, prec_bits_to_words(precision)))
else:
t0 = objtogen(denom)
pari_catch_sig_on()
return P.new_gen(galoisconj0(self.g, flag, t0.g, prec_bits_to_words(precision)))
def nfroots(self, poly):
r"""
Return the roots of `poly` in the number field self without
multiplicity.
EXAMPLES::
sage: y = QQ['yy'].0; _ = pari(y) # pari has variable ordering rules
sage: x = QQ['zz'].0; nf = pari(x^2 + 2).nfinit()
sage: nf.nfroots(y^2 + 2)
[Mod(-zz, zz^2 + 2), Mod(zz, zz^2 + 2)]
sage: nf = pari(x^3 + 2).nfinit()
sage: nf.nfroots(y^3 + 2)
[Mod(zz, zz^3 + 2)]
sage: nf = pari(x^4 + 2).nfinit()
sage: nf.nfroots(y^4 + 2)
[Mod(-zz, zz^4 + 2), Mod(zz, zz^4 + 2)]
"""
cdef gen t0 = objtogen(poly)
pari_catch_sig_on()
return P.new_gen(nfroots(self.g, t0.g))
def polhensellift(self, y, p, long e):
"""
self.polhensellift(y, p, e): lift the factorization y of self
modulo p to a factorization modulo `p^e` using Hensel lift.
The factors in y must be pairwise relatively prime modulo p.
"""
cdef gen t0 = objtogen(y)
cdef gen t1 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(polhensellift(self.g, t0.g, t1.g, e))
def polisirreducible(self):
"""
f.polisirreducible(): Returns True if f is an irreducible
non-constant polynomial, or False if f is reducible or constant.
"""
pari_catch_sig_on()
cdef long t = itos(gisirreducible(self.g))
P.clear_stack()
return t != 0
def pollead(self, v=-1):
"""
self.pollead(v): leading coefficient of polynomial or series self,
or self itself if self is a scalar. Error otherwise. With respect
to the main variable of self if v is omitted, with respect to the
variable v otherwise
"""
pari_catch_sig_on()
return P.new_gen(pollead(self.g, P.get_var(v)))
def polrecip(self):
pari_catch_sig_on()
return P.new_gen(polrecip(self.g))
def polred(self, long flag=0, fa=None):
cdef gen t0
if fa is None:
pari_catch_sig_on()
return P.new_gen(polred0(self.g, flag, NULL))
else:
t0 = objtogen(fa)
pari_catch_sig_on()
return P.new_gen(polred0(self.g, flag, t0.g))
def polredabs(self, long flag=0):
pari_catch_sig_on()
return P.new_gen(polredabs0(self.g, flag))
def polredbest(self, long flag=0):
pari_catch_sig_on()
return P.new_gen(polredbest(self.g, flag))
def polresultant(self, y, var=-1, long flag=0):
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(polresultant0(self.g, t0.g, P.get_var(var), flag))
def polroots(self, long flag=0, unsigned long precision=0):
"""
polroots(x,flag=0): complex roots of the polynomial x. flag is
optional, and can be 0: default, uses Schonhage's method modified
by Gourdon, or 1: uses a modified Newton method.
"""
pari_catch_sig_on()
return P.new_gen(roots0(self.g, flag, prec_bits_to_words(precision)))
def polrootsmod(self, p, long flag=0):
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(rootmod0(self.g, t0.g, flag))
def polrootspadic(self, p, r=20):
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(rootpadic(self.g, t0.g, r))
def polrootspadicfast(self, p, r=20):
cdef gen t0 = objtogen(p)
pari_catch_sig_on()
return P.new_gen(rootpadicfast(self.g, t0.g, r))
def polsturm(self, a, b):
cdef gen t0 = objtogen(a)
cdef gen t1 = objtogen(b)
pari_catch_sig_on()
n = sturmpart(self.g, t0.g, t1.g)
pari_catch_sig_off()
return n
def polsturm_full(self):
pari_catch_sig_on()
n = sturmpart(self.g, NULL, NULL)
pari_catch_sig_off()
return n
def polsylvestermatrix(self, g):
cdef gen t0 = objtogen(g)
pari_catch_sig_on()
return P.new_gen(sylvestermatrix(self.g, t0.g))
def polsym(self, long n):
pari_catch_sig_on()
return P.new_gen(polsym(self.g, n))
def serconvol(self, g):
cdef gen t0 = objtogen(g)
pari_catch_sig_on()
return P.new_gen(convol(self.g, t0.g))
def serlaplace(self):
pari_catch_sig_on()
return P.new_gen(laplace(self.g))
def serreverse(self):
"""
serreverse(f): reversion of the power series f.
If f(t) is a series in t with valuation 1, find the series g(t)
such that g(f(t)) = t.
EXAMPLES::
sage: f = pari('x+x^2+x^3+O(x^4)'); f
x + x^2 + x^3 + O(x^4)
sage: g = f.serreverse(); g
x - x^2 + x^3 + O(x^4)
sage: f.subst('x',g)
x + O(x^4)
sage: g.subst('x',f)
x + O(x^4)
"""
pari_catch_sig_on()
return P.new_gen(recip(self.g))
def thueinit(self, long flag=0, unsigned long precision=0):
pari_catch_sig_on()
return P.new_gen(thueinit(self.g, flag, prec_bits_to_words(precision)))
def rnfisnorminit(self, polrel, long flag=2):
cdef gen t0 = objtogen(polrel)
pari_catch_sig_on()
return P.new_gen(rnfisnorminit(self.g, t0.g, flag))
def rnfisnorm(self, T, long flag=0):
cdef gen t0 = objtogen(T)
pari_catch_sig_on()
return P.new_gen(rnfisnorm(t0.g, self.g, flag))
def vecextract(self, y, z=None):
r"""
self.vecextract(y,z): extraction of the components of the matrix or
vector x according to y and z. If z is omitted, y designates
columns, otherwise y corresponds to rows and z to columns. y and z
can be vectors (of indices), strings (indicating ranges as
in"1..10") or masks (integers whose binary representation indicates
the indices to extract, from left to right 1, 2, 4, 8, etc.)
.. note::
This function uses the PARI row and column indexing, so the
first row or column is indexed by 1 instead of 0.
"""
cdef gen t0 = objtogen(y)
cdef gen t1
if z is None:
pari_catch_sig_on()
return P.new_gen(shallowextract(self.g, t0.g))
else:
t1 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(extract0(self.g, t0.g, t1.g))
def ncols(self):
"""
Return the number of columns of self.
EXAMPLES::
sage: pari('matrix(19,8)').ncols()
8
"""
cdef long n
pari_catch_sig_on()
n = glength(self.g)
pari_catch_sig_off()
return n
def nrows(self):
"""
Return the number of rows of self.
EXAMPLES::
sage: pari('matrix(19,8)').nrows()
19
"""
cdef long n
pari_catch_sig_on()
if self.ncols() == 0:
pari_catch_sig_off()
return 0
n = glength(<GEN>(self.g[1]))
pari_catch_sig_off()
return n
def mattranspose(self):
"""
Transpose of the matrix self.
EXAMPLES::
sage: pari('[1,2,3; 4,5,6; 7,8,9]').mattranspose()
[1, 4, 7; 2, 5, 8; 3, 6, 9]
"""
pari_catch_sig_on()
return P.new_gen(gtrans(self.g)).Mat()
def matadjoint(self):
"""
matadjoint(x): adjoint matrix of x.
EXAMPLES::
sage: pari('[1,2,3; 4,5,6; 7,8,9]').matadjoint()
[-3, 6, -3; 6, -12, 6; -3, 6, -3]
sage: pari('[a,b,c; d,e,f; g,h,i]').matadjoint()
[(i*e - h*f), (-i*b + h*c), (f*b - e*c); (-i*d + g*f), i*a - g*c, -f*a + d*c; (h*d - g*e), -h*a + g*b, e*a - d*b]
"""
pari_catch_sig_on()
return P.new_gen(adj(self.g)).Mat()
def qflll(self, long flag=0):
"""
qflll(x,flag=0): LLL reduction of the vectors forming the matrix x
(gives the unimodular transformation matrix). The columns of x must
be linearly independent, unless specified otherwise below. flag is
optional, and can be 0: default, 1: assumes x is integral, columns
may be dependent, 2: assumes x is integral, returns a partially
reduced basis, 4: assumes x is integral, returns [K,I] where K is
the integer kernel of x and I the LLL reduced image, 5: same as 4
but x may have polynomial coefficients, 8: same as 0 but x may have
polynomial coefficients.
"""
pari_catch_sig_on()
return P.new_gen(qflll0(self.g,flag)).Mat()
def qflllgram(self, long flag=0):
"""
qflllgram(x,flag=0): LLL reduction of the lattice whose gram matrix
is x (gives the unimodular transformation matrix). flag is optional
and can be 0: default,1: lllgramint algorithm for integer matrices,
4: lllgramkerim giving the kernel and the LLL reduced image, 5:
lllgramkerimgen same when the matrix has polynomial coefficients,
8: lllgramgen, same as qflllgram when the coefficients are
polynomials.
"""
pari_catch_sig_on()
return P.new_gen(qflllgram0(self.g,flag)).Mat()
def lllgram(self):
return self.qflllgram(0)
def lllgramint(self):
return self.qflllgram(1)
def qfminim(self, b=None, m=None, long flag=0, unsigned long precision=0):
"""
Return vectors with bounded norm for this quadratic form.
INPUT:
- ``self`` -- a quadratic form
- ``b`` -- a bound on vector norm (finds minimal non-zero
vectors if b=0)
- ``m`` -- maximum number of vectors to return. If ``None``
(default), return all vectors of norm at most B
- flag (optional) --
- 0: default;
- 1: return only the first minimal vector found (ignore ``max``);
- 2: as 0 but uses a more robust, slower implementation,
valid for non integral quadratic forms.
OUTPUT:
A triple consisting of
- the number of vectors of norm <= b,
- the actual maximum norm of vectors listed
- a matrix whose columns are vectors with norm less than or
equal to b for the definite quadratic form. Only one of `v`
and `-v` is returned and the zero vector is never returned.
.. note::
If max is specified then only max vectors will be output,
but all vectors withing the given norm bound will be computed.
EXAMPLES::
sage: A = Matrix(3,3,[1,2,3,2,5,5,3,5,11])
sage: A.is_positive_definite()
True
The first 5 vectors of norm at most 10::
sage: pari(A).qfminim(10, 5).python()
[
[-17 -14 -15 -16 -13]
[ 4 3 3 3 2]
146, 10, [ 3 3 3 3 3]
]
All vectors of minimal norm::
sage: pari(A).qfminim(0).python()
[
[-5 -2 1]
[ 1 1 0]
6, 1, [ 1 0 0]
]
Use flag=2 for non-integral input::
sage: pari(A.change_ring(RR)).qfminim(5, m=5, flag=2).python()
[
[ -5 -10 -2 -7 3]
[ 1 2 1 2 0]
10, 5.00000000023283..., [ 1 2 0 1 -1]
]
"""
cdef gen t0, t1
cdef GEN g0, g1
if b is None:
g0 = NULL
else:
t0 = objtogen(b)
g0 = t0.g
if m is None:
g1 = NULL
else:
t1 = objtogen(m)
g1 = t1.g
pari_catch_sig_on()
return P.new_gen(qfminim0(self.g, g0, g1, flag, prec_bits_to_words(precision)))
def qfrep(self, B, long flag=0):
"""
qfrep(x,B,flag=0): vector of (half) the number of vectors of norms
from 1 to B for the integral and definite quadratic form x. Binary
digits of flag mean 1: count vectors of even norm from 1 to 2B, 2:
return a t_VECSMALL instead of a t_VEC.
"""
cdef gen t0 = objtogen(B)
pari_catch_sig_on()
return P.new_gen(qfrep0(self.g, t0.g, flag))
def matsolve(self, B):
"""
matsolve(B): Solve the linear system Mx=B for an invertible matrix
M
matsolve(B) uses Gaussian elimination to solve Mx=B, where M is
invertible and B is a column vector.
The corresponding pari library routine is gauss. The gp-interface
name matsolve has been given preference here.
INPUT:
- ``B`` - a column vector of the same dimension as the
square matrix self
EXAMPLES::
sage: pari('[1,1;1,-1]').matsolve(pari('[1;0]'))
[1/2; 1/2]
"""
cdef gen t0 = objtogen(B)
pari_catch_sig_on()
return P.new_gen(gauss(self.g, t0.g))
def matsolvemod(self, D, B, long flag = 0):
r"""
For column vectors `D=(d_i)` and `B=(b_i)`, find a small integer
solution to the system of linear congruences
.. math::
R_ix=b_i\text{ (mod }d_i),
where `R_i` is the ith row of ``self``. If `d_i=0`, the equation is
considered over the integers. The entries of ``self``, ``D``, and
``B`` should all be integers (those of ``D`` should also be
non-negative).
If ``flag`` is 1, the output is a two-component row vector whose first
component is a solution and whose second component is a matrix whose
columns form a basis of the solution set of the homogeneous system.
For either value of ``flag``, the output is 0 if there is no solution.
Note that if ``D`` or ``B`` is an integer, then it will be considered
as a vector all of whose entries are that integer.
EXAMPLES::
sage: D = pari('[3,4]~')
sage: B = pari('[1,2]~')
sage: M = pari('[1,2;3,4]')
sage: M.matsolvemod(D, B)
[-2, 0]~
sage: M.matsolvemod(3, 1)
[-1, 1]~
sage: M.matsolvemod(pari('[3,0]~'), pari('[1,2]~'))
[6, -4]~
sage: M2 = pari('[1,10;9,18]')
sage: M2.matsolvemod(3, pari('[2,3]~'), 1)
[[0, -1]~, [-1, -2; 1, -1]]
sage: M2.matsolvemod(9, pari('[2,3]~'))
0
sage: M2.matsolvemod(9, pari('[2,45]~'), 1)
[[1, 1]~, [-1, -4; 1, -5]]
"""
cdef gen t0 = objtogen(D)
cdef gen t1 = objtogen(B)
pari_catch_sig_on()
return P.new_gen(matsolvemod0(self.g, t0.g, t1.g, flag))
def matker(self, long flag=0):
"""
Return a basis of the kernel of this matrix.
INPUT:
- ``flag`` - optional; may be set to 0: default;
non-zero: x is known to have integral entries.
EXAMPLES::
sage: pari('[1,2,3;4,5,6;7,8,9]').matker()
[1; -2; 1]
With algorithm 1, even if the matrix has integer entries the kernel
need not be saturated (which is weird)::
sage: pari('[1,2,3;4,5,6;7,8,9]').matker(1)
[3; -6; 3]
sage: pari('matrix(3,3,i,j,i)').matker()
[-1, -1; 1, 0; 0, 1]
sage: pari('[1,2,3;4,5,6;7,8,9]*Mod(1,2)').matker()
[Mod(1, 2); Mod(0, 2); Mod(1, 2)]
"""
pari_catch_sig_on()
return P.new_gen(matker0(self.g, flag))
def matkerint(self, long flag=0):
"""
Return the integer kernel of a matrix.
This is the LLL-reduced Z-basis of the kernel of the matrix x with
integral entries.
INPUT:
- ``flag`` - optional, and may be set to 0: default,
uses a modified LLL, 1: uses matrixqz.
EXAMPLES::
sage: pari('[2,1;2,1]').matker()
[-1/2; 1]
sage: pari('[2,1;2,1]').matkerint()
[1; -2]
sage: pari('[2,1;2,1]').matkerint(1)
[1; -2]
"""
pari_catch_sig_on()
return P.new_gen(matkerint0(self.g, flag))
def matdet(self, long flag=0):
"""
Return the determinant of this matrix.
INPUT:
- ``flag`` - (optional) flag 0: using Gauss-Bareiss.
1: use classical Gaussian elimination (slightly better for integer
entries)
EXAMPLES::
sage: pari('[1,2; 3,4]').matdet(0)
-2
sage: pari('[1,2; 3,4]').matdet(1)
-2
"""
pari_catch_sig_on()
return P.new_gen(det0(self.g, flag))
def trace(self):
"""
Return the trace of this PARI object.
EXAMPLES::
sage: pari('[1,2; 3,4]').trace()
5
"""
pari_catch_sig_on()
return P.new_gen(gtrace(self.g))
def mathnf(self, long flag=0):
"""
A.mathnf(flag=0): (upper triangular) Hermite normal form of A,
basis for the lattice formed by the columns of A.
INPUT:
- ``flag`` - optional, value range from 0 to 4 (0 if
omitted), meaning : 0: naive algorithm
- ``1: Use Batut's algorithm`` - output 2-component
vector [H,U] such that H is the HNF of A, and U is a unimodular
matrix such that xU=H. 3: Use Batut's algorithm. Output [H,U,P]
where P is a permutation matrix such that P A U = H. 4: As 1, using
a heuristic variant of LLL reduction along the way.
EXAMPLES::
sage: pari('[1,2,3; 4,5,6; 7,8,9]').mathnf()
[6, 1; 3, 1; 0, 1]
"""
pari_catch_sig_on()
return P.new_gen(mathnf0(self.g, flag))
def mathnfmod(self, d):
"""
Returns the Hermite normal form if d is a multiple of the
determinant
Beware that PARI's concept of a Hermite normal form is an upper
triangular matrix with the same column space as the input matrix.
INPUT:
- ``d`` - multiple of the determinant of self
EXAMPLES::
sage: M=matrix([[1,2,3],[4,5,6],[7,8,11]])
sage: d=M.det()
sage: pari(M).mathnfmod(d)
[6, 4, 3; 0, 1, 0; 0, 0, 1]
Note that d really needs to be a multiple of the discriminant, not
just of the exponent of the cokernel::
sage: M=matrix([[1,0,0],[0,2,0],[0,0,6]])
sage: pari(M).mathnfmod(6)
[1, 0, 0; 0, 1, 0; 0, 0, 6]
sage: pari(M).mathnfmod(12)
[1, 0, 0; 0, 2, 0; 0, 0, 6]
"""
cdef gen t0 = objtogen(d)
pari_catch_sig_on()
return P.new_gen(hnfmod(self.g, t0.g))
def mathnfmodid(self, d):
"""
Returns the Hermite Normal Form of M concatenated with d\*Identity
Beware that PARI's concept of a Hermite normal form is a maximal
rank upper triangular matrix with the same column space as the
input matrix.
INPUT:
- ``d`` - Determines
EXAMPLES::
sage: M=matrix([[1,0,0],[0,2,0],[0,0,6]])
sage: pari(M).mathnfmodid(6)
[1, 0, 0; 0, 2, 0; 0, 0, 6]
This routine is not completely equivalent to mathnfmod::
sage: pari(M).mathnfmod(6)
[1, 0, 0; 0, 1, 0; 0, 0, 6]
"""
cdef gen t0 = objtogen(d)
pari_catch_sig_on()
return P.new_gen(hnfmodid(self.g, t0.g))
def matsnf(self, long flag=0):
"""
x.matsnf(flag=0): Smith normal form (i.e. elementary divisors) of
the matrix x, expressed as a vector d. Binary digits of flag mean
1: returns [u,v,d] where d=u\*x\*v, otherwise only the diagonal d
is returned, 2: allow polynomial entries, otherwise assume x is
integral, 4: removes all information corresponding to entries equal
to 1 in d.
EXAMPLES::
sage: pari('[1,2,3; 4,5,6; 7,8,9]').matsnf()
[0, 3, 1]
"""
pari_catch_sig_on()
return P.new_gen(matsnf0(self.g, flag))
def matfrobenius(self, long flag=0):
r"""
M.matfrobenius(flag=0): Return the Frobenius form of the square
matrix M. If flag is 1, return only the elementary divisors (a list
of polynomials). If flag is 2, return a two-components vector [F,B]
where F is the Frobenius form and B is the basis change so that
`M=B^{-1} F B`.
EXAMPLES::
sage: a = pari('[1,2;3,4]')
sage: a.matfrobenius()
[0, 2; 1, 5]
sage: a.matfrobenius(flag=1)
[x^2 - 5*x - 2]
sage: a.matfrobenius(2)
[[0, 2; 1, 5], [1, -1/3; 0, 1/3]]
sage: v = a.matfrobenius(2)
sage: v[0]
[0, 2; 1, 5]
sage: v[1]^(-1)*v[0]*v[1]
[1, 2; 3, 4]
We let t be the matrix of `T_2` acting on modular symbols
of level 43, which was computed using
``ModularSymbols(43,sign=1).T(2).matrix()``::
sage: t = pari('[3, -2, 0, 0; 0, -2, 0, 1; 0, -1, -2, 2; 0, -2, 0, 2]')
sage: t.matfrobenius()
[0, 0, 0, -12; 1, 0, 0, -2; 0, 1, 0, 8; 0, 0, 1, 1]
sage: t.charpoly('x')
x^4 - x^3 - 8*x^2 + 2*x + 12
sage: t.matfrobenius(1)
[x^4 - x^3 - 8*x^2 + 2*x + 12]
AUTHORS:
- Martin Albrect (2006-04-02)
"""
pari_catch_sig_on()
return P.new_gen(matfrobenius(self.g, flag, 0))
def factor(gen self, limit=-1, bint proof=1):
"""
Return the factorization of x.
INPUT:
- ``limit`` -- (default: -1) is optional and can be set
whenever x is of (possibly recursive) rational type. If limit is
set return partial factorization, using primes up to limit (up to
primelimit if limit=0).
- ``proof`` -- (default: True) optional. If False (not the default),
returned factors larger than `2^{64}` may only be pseudoprimes.
.. note::
In the standard PARI/GP interpreter and C-library the
factor command *always* has proof=False, so beware!
EXAMPLES::
sage: pari('x^10-1').factor()
[x - 1, 1; x + 1, 1; x^4 - x^3 + x^2 - x + 1, 1; x^4 + x^3 + x^2 + x + 1, 1]
sage: pari(2^100-1).factor()
[3, 1; 5, 3; 11, 1; 31, 1; 41, 1; 101, 1; 251, 1; 601, 1; 1801, 1; 4051, 1; 8101, 1; 268501, 1]
sage: pari(2^100-1).factor(proof=False)
[3, 1; 5, 3; 11, 1; 31, 1; 41, 1; 101, 1; 251, 1; 601, 1; 1801, 1; 4051, 1; 8101, 1; 268501, 1]
We illustrate setting a limit::
sage: pari(next_prime(10^50)*next_prime(10^60)*next_prime(10^4)).factor(10^5)
[10007, 1; 100000000000000000000000000000000000000000000000151000000000700000000000000000000000000000000000000000000001057, 1]
PARI doesn't have an algorithm for factoring multivariate
polynomials::
sage: pari('x^3 - y^3').factor()
Traceback (most recent call last):
...
PariError: sorry, factor for general polynomials is not yet implemented
"""
cdef int r
cdef GEN t0
cdef GEN cutoff
if limit == -1 and typ(self.g) == t_INT and proof:
pari_catch_sig_on()
cutoff = mkintn(3, 1, 0, 0)
r = factorint_withproof_sage(&t0, self.g, cutoff)
z = P.new_gen(t0)
if not r:
return z
else:
return _factor_int_when_pari_factor_failed(self, z)
pari_catch_sig_on()
return P.new_gen(factor0(self.g, limit))
def hilbert(x, y, p):
cdef gen t0 = objtogen(y)
cdef gen t1 = objtogen(p)
pari_catch_sig_on()
ret = hilbert0(x.g, t0.g, t1.g)
pari_catch_sig_off()
return ret
def chinese(self, y):
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(chinese(self.g, t0.g))
def order(self):
pari_catch_sig_on()
return P.new_gen(order(self.g))
def znprimroot(self):
"""
Return a primitive root modulo self, whenever it exists.
This is a generator of the group `(\ZZ/n\ZZ)^*`, whenever
this group is cyclic, i.e. if `n=4` or `n=p^k` or
`n=2p^k`, where `p` is an odd prime and `k`
is a natural number.
INPUT:
- ``self`` - positive integer equal to 4, or a power
of an odd prime, or twice a power of an odd prime
OUTPUT: gen
EXAMPLES::
sage: pari(4).znprimroot()
Mod(3, 4)
sage: pari(10007^3).znprimroot()
Mod(5, 1002101470343)
sage: pari(2*109^10).znprimroot()
Mod(236736367459211723407, 473472734918423446802)
"""
pari_catch_sig_on()
return P.new_gen(znprimroot0(self.g))
def __abs__(self):
return self.abs()
def norm(gen self):
pari_catch_sig_on()
return P.new_gen(gnorm(self.g))
def nextprime(gen self, bint add_one=0):
"""
nextprime(x): smallest pseudoprime greater than or equal to `x`.
If ``add_one`` is non-zero, return the smallest pseudoprime
strictly greater than `x`.
EXAMPLES::
sage: pari(1).nextprime()
2
sage: pari(2).nextprime()
2
sage: pari(2).nextprime(add_one = 1)
3
sage: pari(2^100).nextprime()
1267650600228229401496703205653
"""
pari_catch_sig_on()
if add_one:
return P.new_gen(gnextprime(gaddsg(1,self.g)))
return P.new_gen(gnextprime(self.g))
def change_variable_name(self, var):
"""
In ``self``, which must be a ``t_POL`` or ``t_SER``, set the
variable to ``var``. If the variable of ``self`` is already
``var``, then return ``self``.
.. WARNING::
You should be careful with variable priorities when
applying this on a polynomial or series of which the
coefficients have polynomial components. To be safe, only
use this function on polynomials with integer or rational
coefficients. For a safer alternative, use :meth:`subst`.
EXAMPLES::
sage: f = pari('x^3 + 17*x + 3')
sage: f.change_variable_name("y")
y^3 + 17*y + 3
sage: f = pari('1 + 2*y + O(y^10)')
sage: f.change_variable_name("q")
1 + 2*q + O(q^10)
sage: f.change_variable_name("y") is f
True
In PARI, ``I`` refers to the square root of -1, so it cannot be
used as variable name. Note the difference with :meth:`subst`::
sage: f = pari('x^2 + 1')
sage: f.change_variable_name("I")
Traceback (most recent call last):
...
PariError: I already exists with incompatible valence
sage: f.subst("x", "I")
0
"""
pari_catch_sig_on()
cdef long n = P.get_var(var)
pari_catch_sig_off()
if varn(self.g) == n:
return self
if typ(self.g) != t_POL and typ(self.g) != t_SER:
raise TypeError, "set_variable() only works for polynomials or power series"
cdef gen newg = P.new_gen_noclear(self.g)
setvarn(newg.g, n)
return newg
def subst(self, var, z):
"""
In ``self``, replace the variable ``var`` by the expression `z`.
EXAMPLES::
sage: x = pari("x"); y = pari("y")
sage: f = pari('x^3 + 17*x + 3')
sage: f.subst(x, y)
y^3 + 17*y + 3
sage: f.subst(x, "z")
z^3 + 17*z + 3
sage: f.subst(x, "z")^2
z^6 + 34*z^4 + 6*z^3 + 289*z^2 + 102*z + 9
sage: f.subst(x, "x+1")
x^3 + 3*x^2 + 20*x + 21
sage: f.subst(x, "xyz")
xyz^3 + 17*xyz + 3
sage: f.subst(x, "xyz")^2
xyz^6 + 34*xyz^4 + 6*xyz^3 + 289*xyz^2 + 102*xyz + 9
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(gsubst(self.g, P.get_var(var), t0.g))
def substpol(self, y, z):
cdef gen t0 = objtogen(y)
cdef gen t1 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(gsubstpol(self.g, t0.g, t1.g))
def nf_subst(self, z):
"""
Given a PARI number field ``self``, return the same PARI
number field but in the variable ``z``.
INPUT:
- ``self`` -- A PARI number field being the output of ``nfinit()``,
``bnfinit()`` or ``bnrinit()``.
EXAMPLES::
sage: x = polygen(QQ)
sage: K = NumberField(x^2 + 5, 'a')
We can substitute in a PARI ``nf`` structure::
sage: Kpari = K.pari_nf()
sage: Kpari.nf_get_pol()
y^2 + 5
sage: Lpari = Kpari.nf_subst('a')
sage: Lpari.nf_get_pol()
a^2 + 5
We can also substitute in a PARI ``bnf`` structure::
sage: Kpari = K.pari_bnf()
sage: Kpari.nf_get_pol()
y^2 + 5
sage: Kpari.bnf_get_cyc() # Structure of class group
[2]
sage: Lpari = Kpari.nf_subst('a')
sage: Lpari.nf_get_pol()
a^2 + 5
sage: Lpari.bnf_get_cyc() # We still have a bnf after substituting
[2]
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(gsubst(self.g, gvar(self.g), t0.g))
def taylor(self, v=-1):
pari_catch_sig_on()
return P.new_gen(tayl(self.g, P.get_var(v), precdl))
def thue(self, rhs, ne):
cdef gen t0 = objtogen(rhs)
cdef gen t1 = objtogen(ne)
pari_catch_sig_on()
return P.new_gen(thue(self.g, t0.g, t1.g))
def charpoly(self, var=-1, long flag=0):
"""
charpoly(A,v=x,flag=0): det(v\*Id-A) = characteristic polynomial of
A using the comatrix. flag is optional and may be set to 1 (use
Lagrange interpolation) or 2 (use Hessenberg form), 0 being the
default.
"""
pari_catch_sig_on()
return P.new_gen(charpoly0(self.g, P.get_var(var), flag))
def kronecker(gen self, y):
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(gkronecker(self.g, t0.g))
def type(gen self):
"""
Return the PARI type of self as a string.
.. note::
In Cython, it is much faster to simply use typ(self.g) for
checking PARI types.
EXAMPLES::
sage: pari(7).type()
't_INT'
sage: pari('x').type()
't_POL'
"""
cdef long t = typ(self.g)
if t == t_INT: return 't_INT'
elif t == t_REAL: return 't_REAL'
elif t == t_INTMOD: return 't_INTMOD'
elif t == t_FRAC: return 't_FRAC'
elif t == t_FFELT: return 't_FFELT'
elif t == t_COMPLEX: return 't_COMPLEX'
elif t == t_PADIC: return 't_PADIC'
elif t == t_QUAD: return 't_QUAD'
elif t == t_POLMOD: return 't_POLMOD'
elif t == t_POL: return 't_POL'
elif t == t_SER: return 't_SER'
elif t == t_RFRAC: return 't_RFRAC'
elif t == t_QFR: return 't_QFR'
elif t == t_QFI: return 't_QFI'
elif t == t_VEC: return 't_VEC'
elif t == t_COL: return 't_COL'
elif t == t_MAT: return 't_MAT'
elif t == t_LIST: return 't_LIST'
elif t == t_STR: return 't_STR'
elif t == t_VECSMALL: return 't_VECSMALL'
elif t == t_CLOSURE: return 't_CLOSURE'
else:
raise TypeError, "Unknown PARI type: %s"%t
def polinterpolate(self, ya, x):
"""
self.polinterpolate(ya,x,e): polynomial interpolation at x
according to data vectors self, ya (i.e. return P such that
P(self[i]) = ya[i] for all i). Also return an error estimate on the
returned value.
"""
cdef gen t0 = objtogen(ya)
cdef gen t1 = objtogen(x)
cdef GEN dy, g
pari_catch_sig_on()
g = polint(self.g, t0.g, t1.g, &dy)
dif = P.new_gen_noclear(dy)
return P.new_gen(g), dif
def algdep(self, long n):
"""
EXAMPLES::
sage: n = pari.set_real_precision(210)
sage: w1 = pari('z1=2-sqrt(26); (z1+I)/(z1-I)')
sage: f = w1.algdep(12); f
545*x^11 - 297*x^10 - 281*x^9 + 48*x^8 - 168*x^7 + 690*x^6 - 168*x^5 + 48*x^4 - 281*x^3 - 297*x^2 + 545*x
sage: f(w1).abs() < 1.0e-200
True
sage: f.factor()
[x, 1; x + 1, 2; x^2 + 1, 1; x^2 + x + 1, 1; 545*x^4 - 1932*x^3 + 2790*x^2 - 1932*x + 545, 1]
sage: pari.set_real_precision(n)
210
"""
pari_catch_sig_on()
return P.new_gen(algdep(self.g, n))
def concat(self, y):
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(concat(self.g, t0.g))
def lindep(self, long flag=0):
pari_catch_sig_on()
return P.new_gen(lindep0(self.g, flag))
def listinsert(self, obj, long n):
cdef gen t0 = objtogen(obj)
pari_catch_sig_on()
return P.new_gen(listinsert(self.g, t0.g, n))
def listput(self, obj, long n):
cdef gen t0 = objtogen(obj)
pari_catch_sig_on()
return P.new_gen(listput(self.g, t0.g, n))
def elleisnum(self, long k, long flag=0, unsigned long precision=0):
"""
om.elleisnum(k, flag=0): om=[om1,om2] being a 2-component vector
giving a basis of a lattice L and k an even positive integer,
computes the numerical value of the Eisenstein series of weight k.
When flag is non-zero and k=4 or 6, this gives g2 or g3 with the
correct normalization.
INPUT:
- ``om`` - gen, 2-component vector giving a basis of a
lattice L
- ``k`` - int (even positive)
- ``flag`` - int (default 0)
OUTPUT:
- ``gen`` - numerical value of E_k
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit()
sage: om = e.omega()
sage: om
[2.49021256085506, -1.97173770155165*I]
sage: om.elleisnum(2) # was: -5.28864933965426
10.0672605281120
sage: om.elleisnum(4)
112.000000000000
sage: om.elleisnum(100)
2.15314248576078 E50
"""
pari_catch_sig_on()
return P.new_gen(elleisnum(self.g, k, flag, prec_bits_to_words(precision)))
def ellwp(gen self, z='z', long n=20, long flag=0, unsigned long precision=0):
"""
Return the value or the series expansion of the Weierstrass
`P`-function at `z` on the lattice `self` (or the lattice
defined by the elliptic curve `self`).
INPUT:
- ``self`` -- an elliptic curve created using ``ellinit`` or a
list ``[om1, om2]`` representing generators for a lattice.
- ``z`` -- (default: 'z') a complex number or a variable name
(as string or PARI variable).
- ``n`` -- (default: 20) if 'z' is a variable, compute the
series expansion up to at least `O(z^n)`.
- ``flag`` -- (default = 0): If ``flag`` is 0, compute only
`P(z)`. If ``flag`` is 1, compute `[P(z), P'(z)]`.
OUTPUT:
- `P(z)` (if ``flag`` is 0) or `[P(z), P'(z)]` (if ``flag`` is 1).
numbers
EXAMPLES:
We first define the elliptic curve X_0(11)::
sage: E = pari([0,-1,1,-10,-20]).ellinit()
Compute P(1)::
sage: E.ellwp(1)
13.9658695257485 + 0.E-18*I
Compute P(1+i), where i = sqrt(-1)::
sage: C.<i> = ComplexField()
sage: E.ellwp(pari(1+i))
-1.11510682565555 + 2.33419052307470*I
sage: E.ellwp(1+i)
-1.11510682565555 + 2.33419052307470*I
The series expansion, to the default `O(z^20)` precision::
sage: E.ellwp()
z^-2 + 31/15*z^2 + 2501/756*z^4 + 961/675*z^6 + 77531/41580*z^8 + 1202285717/928746000*z^10 + 2403461/2806650*z^12 + 30211462703/43418875500*z^14 + 3539374016033/7723451736000*z^16 + 413306031683977/1289540602350000*z^18 + O(z^20)
Compute the series for wp to lower precision::
sage: E.ellwp(n=4)
z^-2 + 31/15*z^2 + O(z^4)
Next we use the version where the input is generators for a
lattice::
sage: pari([1.2692, 0.63 + 1.45*i]).ellwp(1)
13.9656146936689 + 0.000644829272810...*I
With flag=1, compute the pair P(z) and P'(z)::
sage: E.ellwp(1, flag=1)
[13.9658695257485 + 0.E-18*I, 50.5619300880073 ... E-18*I]
"""
cdef gen t0 = objtogen(z)
pari_catch_sig_on()
return P.new_gen(ellwp0(self.g, t0.g, flag, n+2, prec_bits_to_words(precision)))
def ellchangepoint(self, y):
"""
self.ellchangepoint(y): change data on point or vector of points
self on an elliptic curve according to y=[u,r,s,t]
EXAMPLES::
sage: e = pari([0,1,1,-2,0]).ellinit()
sage: x = pari([1,0])
sage: e.ellisoncurve([1,4])
False
sage: e.ellisoncurve(x)
True
sage: f = e.ellchangecurve([1,2,3,-1])
sage: f[:5] # show only first five entries
[6, -2, -1, 17, 8]
sage: x.ellchangepoint([1,2,3,-1])
[-1, 4]
sage: f.ellisoncurve([-1,4])
True
"""
cdef gen t0 = objtogen(y)
pari_catch_sig_on()
return P.new_gen(ellchangepoint(self.g, t0.g))
def debug(gen self, long depth = -1):
r"""
Show the internal structure of self (like the ``\x`` command in gp).
EXAMPLE::
sage: pari('[1/2, 1.0*I]').debug() # random addresses
[&=0000000004c5f010] VEC(lg=3):2200000000000003 0000000004c5eff8 0000000004c5efb0
1st component = [&=0000000004c5eff8] FRAC(lg=3):0800000000000003 0000000004c5efe0 0000000004c5efc8
num = [&=0000000004c5efe0] INT(lg=3):0200000000000003 (+,lgefint=3):4000000000000003 0000000000000001
den = [&=0000000004c5efc8] INT(lg=3):0200000000000003 (+,lgefint=3):4000000000000003 0000000000000002
2nd component = [&=0000000004c5efb0] COMPLEX(lg=3):0c00000000000003 00007fae8a2eb840 0000000004c5ef90
real = gen_0
imag = [&=0000000004c5ef90] REAL(lg=4):0400000000000004 (+,expo=0):6000000000000000 8000000000000000 0000000000000000
"""
pari_catch_sig_on()
dbgGEN(self.g, depth)
pari_catch_sig_off()
return
def init_pari_stack(s=8000000):
"""
Deprecated, use ``pari.allocatemem()`` instead.
EXAMPLES::
sage: from sage.libs.pari.gen import init_pari_stack
sage: init_pari_stack()
doctest:...: DeprecationWarning: init_pari_stack() is deprecated; use pari.allocatemem() instead.
See http://trac.sagemath.org/10018 for details.
sage: pari.stacksize()
8000000
"""
from sage.misc.superseded import deprecation
deprecation(10018, 'init_pari_stack() is deprecated; use pari.allocatemem() instead.')
P.allocatemem(s, silent=True)
cdef gen objtogen(s):
"""Convert any Sage/Python object to a PARI gen"""
cdef GEN g
cdef Py_ssize_t length, i
cdef mpz_t mpz_int
cdef gen v
if isinstance(s, gen):
return s
try:
return s._pari_()
except AttributeError:
pass
if PyString_Check(s):
pari_catch_sig_on()
g = gp_read_str(PyString_AsString(s))
if g == gnil:
P.clear_stack()
return None
return P.new_gen(g)
if PyInt_Check(s):
pari_catch_sig_on()
return P.new_gen(stoi(PyInt_AS_LONG(s)))
if PyBool_Check(s):
return P.PARI_ONE if s else P.PARI_ZERO
if PyLong_Check(s):
pari_catch_sig_on()
mpz_init(mpz_int)
mpz_set_pylong(mpz_int, s)
g = P._new_GEN_from_mpz_t(mpz_int)
mpz_clear(mpz_int)
return P.new_gen(g)
if PyFloat_Check(s):
pari_catch_sig_on()
return P.new_gen(dbltor(PyFloat_AS_DOUBLE(s)))
if PyComplex_Check(s):
pari_catch_sig_on()
g = cgetg(3, t_COMPLEX)
set_gel(g, 1, dbltor(PyComplex_RealAsDouble(s)))
set_gel(g, 2, dbltor(PyComplex_ImagAsDouble(s)))
return P.new_gen(g)
if isinstance(s, (types.ListType, types.XRangeType,
types.TupleType, types.GeneratorType)):
length = len(s)
v = P._empty_vector(length)
for i from 0 <= i < length:
v[i] = objtogen(s[i])
return v
return objtogen(str(s))
cdef GEN _Vec_append(GEN v, GEN a, long n):
"""
This implements appending zeros (or another constant GEN ``a``) to
the result of :meth:`Vec` and similar functions.
This is a shallow function, copying ``a`` and entries of ``v`` to
the result. The result is simply stored on the PARI stack.
INPUT:
- ``v`` -- GEN of type ``t_VEC`` or ``t_COL``
- ``a`` -- GEN which will be used for the added entries.
Normally, this would be ``gen_0``.
- ``n`` -- Make the vector of minimal length `|n|`. If `n > 0`,
append zeros; if `n < 0`, prepend zeros.
OUTPUT:
A GEN of the same type as ``v``.
"""
cdef long lenv = lg(v)-1
cdef GEN w
cdef long i
if n > lenv:
w = cgetg(n+1, typ(v))
for i from 1 <= i <= lenv:
set_gel(w, i, gel(v, i))
for i from 1 <= i <= n-lenv:
set_gel(w, i+lenv, a)
return w
elif n < -lenv:
n = -n
w = cgetg(n+1, typ(v))
for i from 1 <= i <= lenv:
set_gel(w, i+(n-lenv), gel(v, i))
for i from 1 <= i <= n-lenv:
set_gel(w, i, a)
return w
else:
return v
class PariError(RuntimeError):
"""
Error raised by PARI
"""
def errnum(self):
r"""
Return the PARI error number corresponding to this exception.
EXAMPLES::
sage: try:
....: pari('1/0')
....: except PariError as err:
....: print err.errnum()
27
"""
return self.args[0]
def errtext(self):
"""
Return the message output by PARI when this error occurred.
EXAMPLE::
sage: try:
....: pari('pi()')
....: except PariError as e:
....: print e.errtext()
....:
*** at top-level: pi()
*** ^----
*** not a function in function call
"""
return self.args[1]
def __repr__(self):
r"""
TESTS::
sage: PariError(11)
PariError(11)
"""
return "PariError(%d)"%self.errnum()
def __str__(self):
r"""
Return a suitable message for displaying this exception.
This is the last line of ``self.errtext()``, with the leading
``" *** "`` and trailing periods and colons (if any) removed.
An exception is syntax errors, where the "syntax error" line is
shown.
EXAMPLES::
sage: try:
....: pari('1/0')
....: except PariError as err:
....: print err
_/_: division by zero
A syntax error::
sage: pari('!@#$%^&*()')
Traceback (most recent call last):
...
PariError: syntax error, unexpected $undefined: !@#$%^&*()
"""
lines = self.errtext().split('\n')
if self.errnum() == syntaxer:
for line in lines:
if "syntax error" in line:
return line.lstrip(" *").rstrip(" .:")
return lines[-1].lstrip(" *").rstrip(" .:")
cdef _factor_int_when_pari_factor_failed(x, failed_factorization):
"""
This is called by factor when PARI's factor tried to factor, got
the failed_factorization, and it turns out that one of the factors
in there is not proved prime. At this point, we don't care too much
about speed (so don't write everything below using the PARI C
library), since the probability this function ever gets called is
infinitesimal. (That said, we of course did test this function by
forcing a fake failure in the code in misc.h.)
"""
P = failed_factorization[0]
E = failed_factorization[1]
if len(P) == 1 and E[0] == 1:
print "BIG WARNING: The number %s wasn't split at all by PARI, but it's definitely composite."%(P[0])
print "This is probably an infinite loop..."
w = []
for i in range(len(P)):
p = P[i]
e = E[i]
if not p.isprime():
F = p.factor(proof=True)
for j in range(len(F[0])):
w.append((F[0][j], F[1][j]))
else:
w.append((p, e))
m = P.matrix(len(w), 2)
for i in range(len(w)):
m[i,0] = w[i][0]
m[i,1] = w[i][1]
return m
