"""
Cython interface to Cremona's ``eclib`` library (also known as ``mwrank``)
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata, _mw
sage: c = _Curvedata(1,2,3,4,5)
sage: print c
[1,2,3,4,5]
b2 = 9 b4 = 11 b6 = 29 b8 = 35
c4 = -183 c6 = -3429
disc = -10351 (# real components = 1)
#torsion not yet computed
sage: t= _mw(c)
sage: t.search(10)
sage: t
[[1:2:1]]
"""
import os
import sys
include 'sage/ext/interrupt.pxi'
include 'sage/ext/stdsage.pxi'
"""
SAGE_DOCTEST_ALLOW_TABS
"""
cdef extern from "wrap.h":
long mwrank_get_precision()
void mwrank_set_precision(long n)
void mwrank_initprimes(char* pfilename, int verb)
struct bigint
bigint* new_bigint()
void del_bigint(bigint* x)
bigint* str_to_bigint(char* s)
char* bigint_to_str(bigint* x)
struct Curvedata
Curvedata* Curvedata_new(bigint* a1, bigint* a2,
bigint* a3, bigint* a4,
bigint* a6, int min_on_init)
void Curvedata_del(Curvedata* curve)
char* Curvedata_repr(Curvedata* curve)
double Curvedata_silverman_bound(Curvedata* curve)
double Curvedata_cps_bound(Curvedata* curve)
double Curvedata_height_constant(Curvedata* curve)
char* Curvedata_getdiscr(Curvedata* curve)
char* Curvedata_conductor(Curvedata* m)
char* Curvedata_isogeny_class(Curvedata* E, int verbose)
struct mw
mw* mw_new(Curvedata* curve, int verb, int pp, int maxr)
void mw_del(mw* m)
int mw_process(Curvedata* curve, mw* m,
bigint* x, bigint* y,
bigint* z, int sat)
char* mw_getbasis(mw* m)
char* mw_regulator(mw* m)
int mw_rank(mw* m)
int mw_saturate(mw* m, bigint* index, char** unsat,
long sat_bd, int odd_primes_only)
void mw_search(mw* m, char* h_lim, int moduli_option, int verb)
struct two_descent
two_descent* two_descent_new(Curvedata* curve,
int verb, int sel,
long firstlim, long secondlim,
long n_aux, int second_descent)
void two_descent_del(two_descent* t)
int two_descent_ok(two_descent* t)
long two_descent_get_certain(two_descent* t)
char* two_descent_get_basis(two_descent* t)
char* two_descent_regulator(two_descent* t)
long two_descent_get_rank(two_descent* t)
long two_descent_get_rank_bound(two_descent* t)
long two_descent_get_selmer_rank(two_descent* t)
void two_descent_saturate(two_descent* t, long sat_bd)
cdef object string_sigoff(char* s):
sig_off()
t = str(s)
sage_free(s)
return t
class __init:
pass
_INIT = __init()
mwrank_set_precision(50)
def get_precision():
"""
Returns the working floating point precision of mwrank.
OUTPUT:
(int) The current precision in decimal digits.
EXAMPLE::
sage: from sage.libs.mwrank.mwrank import get_precision
sage: get_precision()
50
"""
return mwrank_get_precision()
def set_precision(n):
"""
Sets the working floating point precision of mwrank.
INPUT:
- ``n`` (int) -- a positive integer: the number of decimal digits.
OUTPUT:
None.
EXAMPLE::
sage: from sage.libs.mwrank.mwrank import set_precision
sage: set_precision(50)
"""
mwrank_set_precision(n)
def initprimes(filename, verb=False):
"""
Initialises mwrank/eclib's internal prime list.
INPUT:
- ``filename`` (string) -- the name of a file of primes.
- ``verb`` (bool: default ``False``) -- verbose or not?
EXAMPLES::
sage: file = os.path.join(SAGE_TMP, 'PRIMES')
sage: open(file,'w').write(' '.join([str(p) for p in prime_range(10^7,10^7+20)]))
sage: mwrank_initprimes(file, verb=True)
Computed 78519 primes, largest is 1000253
reading primes from file ...
read extra prime 10000019
finished reading primes from file ...
Extra primes in list: 10000019
sage: mwrank_initprimes("x" + file, True)
Traceback (most recent call last):
...
IOError: No such file or directory: ...
"""
if not os.path.exists(filename):
raise IOError, 'No such file or directory: %s'%filename
mwrank_initprimes(filename, verb)
if verb:
sys.stdout.flush()
sys.stderr.flush()
cdef class _bigint:
"""
Cython class wrapping eclib's bigint class.
"""
cdef bigint* x
def __init__(self, x="0"):
"""
Constructor for bigint class.
INPUT:
- ``x`` -- string or int: a string representing a decimal
integer, or a Sage integer
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _bigint
sage: _bigint(123)
123
sage: _bigint('123')
123
sage: type(_bigint(123))
<type 'sage.libs.mwrank.mwrank._bigint'>
"""
if not (x is _INIT):
s = str(x)
if s.isdigit() or s[0] == "-" and s[1:].isdigit():
self.x = str_to_bigint(s)
else:
raise ValueError, "invalid _bigint: %s"%x
def __dealloc__(self):
"""
Destructor for bigint class (releases memory).
"""
del_bigint(self.x)
def __repr__(self):
"""
String representation of bigint.
OUTPUT:
(string) the bigint as a string (decimal)
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _bigint
sage: a = _bigint('123')
sage: a.__repr__()
'123'
sage: a = _bigint('-456')
sage: a.__repr__()
'-456'
"""
sig_on()
return string_sigoff(bigint_to_str(self.x))
cdef make_bigint(bigint* x):
cdef _bigint y
sig_off()
y = _bigint(_INIT)
y.x = x
return y
cdef class _Curvedata:
cdef Curvedata* x
def __init__(self, a1, a2, a3,
a4, a6, min_on_init=0):
"""
Constructor for Curvedata class.
INPUT:
- ``a1``,``a2``,``a3``,``a4``,``a6`` (int) -- integer
coefficients of a Weierstrass equation (must be
nonsingular).
- ``min_on_init`` (int, default 0) -- flag controlling whether
the constructed curve is replaced by a global minimal model.
If nonzero then this minimisation does take place.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: _Curvedata(1,2,3,4,5)
[1,2,3,4,5]
b2 = 9 b4 = 11 b6 = 29 b8 = 35
c4 = -183 c6 = -3429
disc = -10351 (# real components = 1)
#torsion not yet computed
A non-minimal example::
sage: _Curvedata(0,0,0,0,64)
[0,0,0,0,64]
b2 = 0 b4 = 0 b6 = 256 b8 = 0
c4 = 0 c6 = -55296
disc = -1769472 (# real components = 1)
#torsion not yet computed
sage: _Curvedata(0,0,0,0,64,min_on_init=1)
[0,0,0,0,1] (reduced minimal model)
b2 = 0 b4 = 0 b6 = 4 b8 = 0
c4 = 0 c6 = -864
disc = -432 (# real components = 1)
#torsion not yet computed
"""
cdef _bigint _a1, _a2, _a3, _a4, _a6
_a1 = _bigint(a1)
_a2 = _bigint(a2)
_a3 = _bigint(a3)
_a4 = _bigint(a4)
_a6 = _bigint(a6)
self.x = Curvedata_new(_a1.x, _a2.x, _a3.x, _a4.x, _a6.x, min_on_init)
if self.discriminant() == 0:
raise ArithmeticError, "Invariants (= %s) do not describe an elliptic curve."%([a1,a2,a3,a4,a6])
def __dealloc__(self):
"""
Destructor for Curvedata class.
"""
Curvedata_del(self.x)
def __repr__(self):
"""
String representation of Curvedata
OUTPUT:
(string) the Curvedata as a string
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,2,3,4,5)
sage: E.__repr__()
'[1,2,3,4,5]\nb2 = 9\t b4 = 11\t b6 = 29\t b8 = 35\nc4 = -183\t\tc6 = -3429\ndisc = -10351\t(# real components = 1)\n#torsion not yet computed'
sage: E
[1,2,3,4,5]
b2 = 9 b4 = 11 b6 = 29 b8 = 35
c4 = -183 c6 = -3429
disc = -10351 (# real components = 1)
#torsion not yet computed
"""
sig_on()
return string_sigoff(Curvedata_repr(self.x))[:-1]
def silverman_bound(self):
"""
The Silverman height bound for this elliptic curve.
OUTPUT:
(float) A non-negative real number `B` such that for every
rational point on this elliptic curve `E`, `h(P)\le\hat{h}(P)
+ B`, where `h(P)` is the naive height and `\hat{h}(P)` the
canonical height.
TODO:
Since eclib can compute this to arbitrary precision it would
make sense to return a Sage real.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,2,3,4,5)
sage: E.silverman_bound()
6.52226179519101...
sage: type(E.silverman_bound())
<type 'float'>
"""
return Curvedata_silverman_bound(self.x)
def cps_bound(self):
"""
The Cremona-Prickett-Siksek height bound for this elliptic curve.
OUTPUT:
(float) A non-negative real number `B` such that for every
rational point on this elliptic curve `E`, `h(P)\le\hat{h}(P)
+ B`, where `h(P)` is the naive height and `\hat{h}(P)` the
canonical height.
TODO:
Since eclib can compute this to arbitrary precision it would
make sense to return a Sage real.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,2,3,4,5)
sage: E.cps_bound()
0.11912451909250982
Note that this is a better bound than Silverman's in this case::
sage: E.silverman_bound()
6.52226179519101...
"""
cdef double x
sig_on()
x = Curvedata_cps_bound(self.x)
sig_off()
return x
def height_constant(self):
"""
A height bound for this elliptic curve.
OUTPUT:
(float) A non-negative real number `B` such that for every
rational point on this elliptic curve `E`, `h(P)\le\hat{h}(P)
+ B`, where `h(P)` is the naive height and `\hat{h}(P)` the
canonical height. This is the minimum of the Silverman and
Cremona_Prickett-Siksek height bounds.
TODO:
Since eclib can compute this to arbitrary precision it would
make sense to return a Sage real.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,2,3,4,5)
sage: E.height_constant()
0.11912451909250982
"""
return Curvedata_height_constant(self.x)
def discriminant(self):
"""
The discriminant of this elliptic curve.
OUTPUT:
(Integer) The discriminant.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,2,3,4,5)
sage: E.discriminant()
-10351
sage: E = _Curvedata(100,200,300,400,500)
sage: E.discriminant()
-1269581104000000
sage: ZZ(E.discriminant())
-1269581104000000
"""
sig_on()
from sage.rings.all import Integer
return Integer(string_sigoff(Curvedata_getdiscr(self.x)))
def conductor(self):
"""
The conductor of this elliptic curve.
OUTPUT:
(Integer) The conductor.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,2,3,4,5)
sage: E.discriminant()
-10351
sage: E = _Curvedata(100,200,300,400,500)
sage: E.conductor()
126958110400
"""
sig_on()
from sage.rings.all import Integer
return Integer(string_sigoff(Curvedata_conductor(self.x)))
def isogeny_class(self, verbose=False):
"""
The isogeny class of this elliptic curve.
OUTPUT:
(tuple) A tuple consisting of (1) a list of the curves in the
isogeny class, each as a list of its Weierstrass coefficients;
(2) a matrix of the degrees of the isogenies between the
curves in the class (prime degrees only).
.. warning::
The list may not be complete, if the precision is too low.
Use ``mwrank_set_precision()`` to increase the precision.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,0,1,4,-6)
sage: E.conductor()
14
sage: E.isogeny_class()
([[1, 0, 1, 4, -6], [1, 0, 1, -36, -70], [1, 0, 1, -1, 0], [1, 0, 1, -171, -874], [1, 0, 1, -11, 12], [1, 0, 1, -2731, -55146]], [[0, 2, 3, 3, 0, 0], [2, 0, 0, 0, 3, 3], [3, 0, 0, 0, 2, 0], [3, 0, 0, 0, 0, 2], [0, 3, 2, 0, 0, 0], [0, 3, 0, 2, 0, 0]])
"""
sig_on()
s = string_sigoff(Curvedata_isogeny_class(self.x, verbose))
if verbose:
sys.stdout.flush()
sys.stderr.flush()
from sage.misc.all import preparse
from sage.rings.all import Integer
return eval(s)
cdef class _mw:
"""
Cython class wrapping eclib's mw class.
"""
cdef mw* x
cdef Curvedata* curve
cdef int verb
def __init__(self, _Curvedata curve, verb=False, pp=1, maxr=999):
"""
Constructor for mw class.
INPUT:
- ``curve`` (_Curvedata) -- an elliptic curve
- ``verb`` (bool, default False) -- verbosity flag (controls
amount of output produced in point searches)
- ``pp`` (int, default 1) -- process points flag (if nonzero,
the points found are processed, so that at all times only a
`\ZZ`-basis for the subgroup generated by the points found
so far is stored; if zero, no processing is done and all
points found are stored).
- ``maxr`` (int, default 999) -- maximum rank (quit point
searching once the points found generate a subgroup of this
rank; useful if an upper bound for the rank is already
known).
EXAMPLE::
sage: from sage.libs.mwrank.mwrank import _mw
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(1,0,1,4,-6)
sage: EQ = _mw(E)
sage: EQ
[]
sage: type(EQ)
<type 'sage.libs.mwrank.mwrank._mw'>
sage: E = _Curvedata(0,0,1,-7,6)
sage: EQ = _mw(E)
sage: EQ.search(2)
sage: EQ
[[1:-1:1], [-2:3:1], [-14:25:8]]
Example to illustrate the verbose parameter::
sage: from sage.libs.mwrank.mwrank import _mw
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(0,0,1,-7,6)
sage: EQ = _mw(E, verb=False)
sage: EQ.search(1)
sage: EQ = _mw(E, verb=True)
sage: EQ.search(1)
P1 = [0:1:0] is torsion point, order 1
P1 = [-3:0:1] is generator number 1
...
P4 = [12:35:27] = 1*P1 + -1*P2 + -1*P3 (mod torsion)
The previous command produces the following output::
P1 = [0:1:0] is torsion point, order 1
P1 = [-3:0:1] is generator number 1
saturating up to 20...Checking 2-saturation
Points have successfully been 2-saturated (max q used = 7)
Checking 3-saturation
Points have successfully been 3-saturated (max q used = 7)
Checking 5-saturation
Points have successfully been 5-saturated (max q used = 23)
Checking 7-saturation
Points have successfully been 7-saturated (max q used = 41)
Checking 11-saturation
Points have successfully been 11-saturated (max q used = 17)
Checking 13-saturation
Points have successfully been 13-saturated (max q used = 43)
Checking 17-saturation
Points have successfully been 17-saturated (max q used = 31)
Checking 19-saturation
Points have successfully been 19-saturated (max q used = 37)
done
P2 = [-2:3:1] is generator number 2
saturating up to 20...Checking 2-saturation
possible kernel vector = [1,1]
This point may be in 2E(Q): [14:-52:1]
...and it is!
Replacing old generator #1 with new generator [1:-1:1]
Points have successfully been 2-saturated (max q used = 7)
Index gain = 2^1
Checking 3-saturation
Points have successfully been 3-saturated (max q used = 13)
Checking 5-saturation
Points have successfully been 5-saturated (max q used = 67)
Checking 7-saturation
Points have successfully been 7-saturated (max q used = 53)
Checking 11-saturation
Points have successfully been 11-saturated (max q used = 73)
Checking 13-saturation
Points have successfully been 13-saturated (max q used = 103)
Checking 17-saturation
Points have successfully been 17-saturated (max q used = 113)
Checking 19-saturation
Points have successfully been 19-saturated (max q used = 47)
done (index = 2).
Gained index 2, new generators = [ [1:-1:1] [-2:3:1] ]
P3 = [-14:25:8] is generator number 3
saturating up to 20...Checking 2-saturation
Points have successfully been 2-saturated (max q used = 11)
Checking 3-saturation
Points have successfully been 3-saturated (max q used = 13)
Checking 5-saturation
Points have successfully been 5-saturated (max q used = 71)
Checking 7-saturation
Points have successfully been 7-saturated (max q used = 101)
Checking 11-saturation
Points have successfully been 11-saturated (max q used = 127)
Checking 13-saturation
Points have successfully been 13-saturated (max q used = 151)
Checking 17-saturation
Points have successfully been 17-saturated (max q used = 139)
Checking 19-saturation
Points have successfully been 19-saturated (max q used = 179)
done (index = 1).
P4 = [-1:3:1] = -1*P1 + -1*P2 + -1*P3 (mod torsion)
P4 = [0:2:1] = 2*P1 + 0*P2 + 1*P3 (mod torsion)
P4 = [2:13:8] = -3*P1 + 1*P2 + -1*P3 (mod torsion)
P4 = [1:0:1] = -1*P1 + 0*P2 + 0*P3 (mod torsion)
P4 = [2:0:1] = -1*P1 + 1*P2 + 0*P3 (mod torsion)
P4 = [18:7:8] = -2*P1 + -1*P2 + -1*P3 (mod torsion)
P4 = [3:3:1] = 1*P1 + 0*P2 + 1*P3 (mod torsion)
P4 = [4:6:1] = 0*P1 + -1*P2 + -1*P3 (mod torsion)
P4 = [36:69:64] = 1*P1 + -2*P2 + 0*P3 (mod torsion)
P4 = [68:-25:64] = -2*P1 + -1*P2 + -2*P3 (mod torsion)
P4 = [12:35:27] = 1*P1 + -1*P2 + -1*P3 (mod torsion)
sage: EQ
[[1:-1:1], [-2:3:1], [-14:25:8]]
Example to illustrate the process points ``pp`` parameter::
sage: from sage.libs.mwrank.mwrank import _mw
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: E = _Curvedata(0,0,1,-7,6)
sage: EQ = _mw(E, pp=1)
sage: EQ.search(1); EQ
[[1:-1:1], [-2:3:1], [-14:25:8]]
sage: EQ = _mw(E, pp=0)
sage: EQ.search(1); EQ
[[-3:0:1], [-2:3:1], [-14:25:8], [-1:3:1], [0:2:1], [2:13:8], [1:0:1], [2:0:1], [18:7:8], [3:3:1], [4:6:1], [36:69:64], [68:-25:64], [12:35:27]]
"""
self.curve = curve.x
self.x = mw_new(curve.x, verb, pp, maxr)
self.verb = verb
def __dealloc__(self):
"""
Destructor for mw class.
"""
mw_del(self.x)
def __repr__(self):
"""
String representation of the current basis of this mw group.
OUTPUT:
(string) the current basis of this mw as a string
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,0,1,-7,6)
sage: EQ = _mw(E)
sage: EQ # indirect doctest
[]
sage: EQ.search(2)
sage: EQ
[[1:-1:1], [-2:3:1], [-14:25:8]]
"""
sig_on()
return string_sigoff(mw_getbasis(self.x))
def process(self, point, sat=0):
"""
Processes the given point, adding it to the mw group.
INPUT:
- ``point`` (tuple or list) -- tuple or list of 3 integers.
An ``ArithmeticError`` is raised if the point is not on the
curve.
- ``sat`` (int, default 0) --saturate at primes up to ``sat``.
No saturation is done if ``sat=0``. (Note that it is more
efficient to add several points at once and then saturate
just once at the end).
.. note::
The eclib function which implements this only carries out
any saturation if the rank of the points increases upon
adding the new point. This is because it is assumed that
one saturates as ones goes along.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,1,1,-2,0)
sage: EQ = _mw(E)
Initially the list of gens is empty::
sage: EQ
[]
We process a point of infinite order::
sage: EQ.process([-1,1,1])
sage: EQ
[[-1:1:1]]
And another independent one::
sage: EQ.process([0,-1,1])
sage: EQ
[[-1:1:1], [0:-1:1]]
Processing a point dependent on the current basis will not
change the basis::
sage: EQ.process([4,8,1])
sage: EQ
[[-1:1:1], [0:-1:1]]
"""
if not isinstance(point, (tuple, list)) and len(point) == 3:
raise TypeError, "point must be a list or tuple of length 3."
cdef _bigint x,y,z
sig_on()
x,y,z = _bigint(point[0]), _bigint(point[1]), _bigint(point[2])
r = mw_process(self.curve, self.x, x.x, y.x, z.x, sat)
sig_off()
if r != 0:
raise ArithmeticError, "point (=%s) not on curve."%point
def getbasis(self):
"""
Returns the current basis of the mw structure.
OUTPUT:
(string) String representation of the points in the basis of
the mw group.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,1,1,-2,0)
sage: EQ = _mw(E)
sage: EQ.search(3)
sage: EQ.getbasis()
'[[0:-1:1], [-1:1:1]]'
sage: EQ.rank()
2
"""
sig_on()
s = string_sigoff(mw_getbasis(self.x))
return s
def regulator(self):
"""
Returns the regulator of the current basis of the mw group.
OUTPUT:
(float) The current regulator.
TODO:
``eclib`` computes the regulator to arbitrary precision, and
the full precision value should be returned.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,1,1,-2,0)
sage: EQ = _mw(E)
sage: EQ.search(3)
sage: EQ.getbasis()
'[[0:-1:1], [-1:1:1]]'
sage: EQ.rank()
2
sage: EQ.regulator()
0.15246017277240753
"""
cdef float f
sig_on()
f = float(string_sigoff(mw_regulator(self.x)))
return f
def rank(self):
"""
Returns the rank of the current basis of the mw group.
OUTPUT:
(Integer) The current rank.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,1,1,-2,0)
sage: EQ = _mw(E)
sage: EQ.search(3)
sage: EQ.getbasis()
'[[0:-1:1], [-1:1:1]]'
sage: EQ.rank()
2
"""
sig_on()
r = mw_rank(self.x)
sig_off()
from sage.rings.all import Integer
return Integer(r)
def saturate(self, int sat_bd=-1, int odd_primes_only=0):
"""
Saturates the current subgroup of the mw group.
INPUT:
- ``sat_bnd`` (int, default -1) -- bound on primes at which to
saturate. If -1 (default), compute a bound for the primes
which may not be saturated, and use that.
- ``odd_primes_only`` (bool, default False) -- only do
saturation at odd primes. (If the points have been found
via 2-descent they should already be 2-saturated.)
OUTPUT:
(tuple) (success flag, index, list) The success flag will be 1
unless something failed (usually an indication that the points
were not saturated but the precision is not high enough to
divide out successfully). The index is the index of the mw
group before saturation in the mw group after. The list is a
string representation of the primes at which saturation was
not proved or achieved.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,1,1,-2,0)
sage: EQ = _mw(E)
sage: EQ.process([494, -5720, 6859]) # 3 times another point
sage: EQ
[[494:-5720:6859]]
sage: EQ.saturate()
(1, 3, '[ ]')
sage: EQ
[[-1:1:1]]
If we set the saturation bound at 2, then saturation will fail::
sage: EQ = _mw(E)
sage: EQ.process([494, -5720, 6859]) # 3 times another point
sage: EQ.saturate(sat_bd=2)
Saturation index bound = 10
WARNING: saturation at primes p > 2 will not be done;
points may be unsaturated at primes between 2 and index bound
Failed to saturate MW basis at primes [ ]
(0, 1, '[ ]')
sage: EQ
[[494:-5720:6859]]
The following output is also seen in the preceding example::
Saturation index bound = 10
WARNING: saturation at primes p > 2 will not be done;
points may be unsaturated at primes between 2 and index bound
Failed to saturate MW basis at primes [ ]
"""
cdef _bigint index
cdef char* s
cdef int ok
sig_on()
index = _bigint()
ok = mw_saturate(self.x, index.x, &s, sat_bd, odd_primes_only)
unsat = string_sigoff(s)
return ok, index, unsat
def search(self, h_lim, int moduli_option=0, int verb=0):
"""
Search for points in the mw group.
INPUT:
- ``h_lim`` (int) -- bound on logarithmic naive height of points
- ``moduli_option`` (int, default 0) -- option for sieving
strategy. The default (0) uses an adapted version of
Stoll's ratpoints code and is recommended.
- ``verb`` (int, default 0) -- level of verbosity. If 0, no
output. If positive, the points are output as found and
some details of the processing, finding linear relations,
and partial saturation are output.
.. note::
The effect of the search is also governed by the class
options, notably whether the points found are processed:
meaning that linear relations are found and saturation is
carried out, with the result that the list of generators
will always contain a `\ZZ`-span of the saturation of the
points found, modulo torsion.
OUTPUT:
None. The effect of the search is to update the list of
generators.
EXAMPLE::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: from sage.libs.mwrank.mwrank import _mw
sage: E = _Curvedata(0,0,1,-19569,-4064513) # 873c1
sage: EQ = _mw(E)
sage: EQ = _mw(E)
sage: for i in [1..11]: print i, EQ.search(i), EQ
1 None []
2 None []
3 None []
4 None []
5 None []
6 None []
7 None []
8 None []
9 None []
10 None []
11 None [[3639568:106817593:4096]]
"""
cdef char* _h_lim
h_lim = str(h_lim)
_h_lim = h_lim
sig_on()
mw_search(self.x, _h_lim, moduli_option, verb)
if verb:
sys.stdout.flush()
sys.stderr.flush()
sig_off()
cdef class _two_descent:
"""
Cython class wrapping eclib's two_descent class.
"""
cdef two_descent* x
def __init__(self):
"""
Constructor for two_descent class.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
"""
self.x = <two_descent*> 0
def __dealloc__(self):
"""
Destructor for two_descent class.
"""
if self.x:
two_descent_del(self.x)
def do_descent(self, _Curvedata curve,
int verb = 1,
int sel = 0,
int firstlim = 20,
int secondlim = 8,
int n_aux = -1,
int second_descent = 1):
"""
Carry out a 2-descent.
INPUT:
- ``curvedata`` (_Curvedata) -- the curve on which to do descent.
- ``verb`` (int, default 1) -- verbosity level.
- ``sel`` (int, default 0) -- Selmer-only flag. If 1, only
the 2-Selmer group will be computed, with no rational
points. Useful as a faster way of getting an upper bound on
the rank.
- ``firstlim`` (int, default 20) -- naive height bound on
first point search on quartic homogeneous spaces (before
testing local solubility; very simple search with no
overheads).
- ``secondlim`` (int, default 8) -- naive height bound on
second point search on quartic homogeneous spaces (after
testing local solubility; sieve-assisted search)
- ``n_aux`` (int, default -1) -- If positive, the number of
auxiliary primes used in sieve-assisted search for quartics.
If -1 (the default) use a default value (set in the eclib
code in ``src/qrank/mrank1.cc`` in DEFAULT_NAUX: currently 8).
Only relevant for curves with no 2-torsion, where full
2-descent is carried out. Worth increasing for curves
expected to be of of rank>6 to one or two more than the
expected rank.
- ``second_descent`` (int, default 1) -- flag specifying
whether or not a second descent will be carried out (yes if
1, the default; no if 0). Only relevant for curves with
2-torsion. Recommended left as the default except for
experts interested in details of Selmer groups.
OUTPUT:
None
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.getrank()
3
sage: D2.getcertain()
1
sage: D2.ok()
1
"""
sig_on()
self.x = two_descent_new(curve.x, verb, sel, firstlim, secondlim, n_aux, second_descent)
if verb:
sys.stdout.flush()
sys.stderr.flush()
sig_off()
def getrank(self):
"""
Returns the rank (after doing a 2-descent).
OUTPUT:
(Integer) the rank (or an upper bound).
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.getrank()
3
"""
cdef int r
sig_on()
r = two_descent_get_rank(self.x)
sig_off()
from sage.rings.all import Integer
return Integer(r)
def getrankbound(self):
"""
Returns the rank upper bound (after doing a 2-descent).
OUTPUT:
(Integer) an upper bound on the rank.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.getrankbound()
3
"""
cdef int r
sig_on()
r = two_descent_get_rank_bound(self.x)
sig_off()
from sage.rings.all import Integer
return Integer(r)
def getselmer(self):
"""
Returns the 2-Selmer rank (after doing a 2-descent).
OUTPUT:
(Integer) The 2-Selmer rank.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.getselmer()
3
"""
sig_on()
r = two_descent_get_selmer_rank(self.x)
sig_off()
from sage.rings.all import Integer
return Integer(r)
def ok(self):
"""
Returns the success flag (after doing a 2-descent).
OUTPUT:
(bool) Flag indicating whether or not 2-descent was successful.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.ok()
1
"""
return two_descent_ok(self.x)
def getcertain(self):
"""
Returns the certainty flag (after doing a 2-descent).
OUTPUT:
(bool) True if the rank upper and lower bounds are equal.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.getcertain()
1
"""
return two_descent_get_certain(self.x)
def saturate(self, saturation_bound=0):
"""
Carries out saturation of the points found by a 2-descent.
OUTPUT:
None.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.saturate()
Searching for points (bound = 8)...done:
found points which generate a subgroup of rank 3
and regulator 0.417143558758383969817119544618093396749810106098479
Processing points found during 2-descent...done:
now regulator = 0.417143558758383969817119544618093396749810106098479
No saturation being done
sage: D2.getbasis()
'[[1:-1:1], [-2:3:1], [-14:25:8]]'
"""
sig_on()
two_descent_saturate(self.x, saturation_bound)
sig_off()
def getbasis(self):
"""
Returns the basis of points found by doing a 2-descent.
If the success and certain flags are 1, this will be a
`\ZZ/2\ZZ`-basis for `E(\QQ)/2E(\QQ)` (modulo torsion),
otherwise possibly only for a proper subgroup.
.. note::
You must call ``saturate()`` first, or a RunTimeError will be raised.
OUTPUT:
(string) String representation of the list of points after
saturation.
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
sage: D2.saturate()
Searching for points (bound = 8)...done:
found points which generate a subgroup of rank 3
and regulator 0.417143558758383969817119544618093396749810106098479
Processing points found during 2-descent...done:
now regulator = 0.417143558758383969817119544618093396749810106098479
No saturation being done
sage: D2.getbasis()
'[[1:-1:1], [-2:3:1], [-14:25:8]]'
"""
sig_on()
return string_sigoff(two_descent_get_basis(self.x))
def regulator(self):
"""
Returns the regulator of the points found by doing a 2-descent.
OUTPUT:
(float) The regulator (of the subgroup found by 2-descent).
EXAMPLES::
sage: from sage.libs.mwrank.mwrank import _Curvedata
sage: CD = _Curvedata(0,0,1,-7,6)
sage: from sage.libs.mwrank.mwrank import _two_descent
sage: D2 = _two_descent()
sage: D2.do_descent(CD)
Basic pair: I=336, J=-10800
disc=35092224
...
Mordell rank contribution from B=im(eps) = 3
Selmer rank contribution from B=im(eps) = 3
Sha rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer rank contribution from A=ker(eps) = 0
Sha rank contribution from A=ker(eps) = 0
If called before calling ``saturate()``, a bogus value of 1.0
is returned::
sage: D2.regulator()
1.0
After saturation, both ``getbasis()`` and ``regulator()``
return the basis and regulator of the subgroup found by
2-descent::
sage: D2.saturate()
Searching for points (bound = 8)...done:
found points which generate a subgroup of rank 3
and regulator 0.417143558758383969817119544618093396749810106098479
Processing points found during 2-descent...done:
now regulator = 0.417143558758383969817119544618093396749810106098479
No saturation being done
sage: D2.getbasis()
'[[1:-1:1], [-2:3:1], [-14:25:8]]'
sage: D2.regulator()
0.417143558758384
"""
sig_on()
return float(string_sigoff(two_descent_regulator(self.x)))
