"""
Cremona's tables of elliptic curves
Sage includes John Cremona's tables of elliptic curves in an
easy-to-use format. An instance of the class CremonaDatabase()
gives access to the database.
If the optional full CremonaDatabase is not installed, a mini-version
is included by default with Sage. It contains Weierstrass equations,
rank, and torsion for curves up to conductor 10000.
The large database includes all curves of conductor up to 240,000. It
also includes data related to the BSD conjecture and modular degrees
for all of these curves, and generators for the Mordell-Weil
groups. To install it type the following in Sage::
!sage -i database_cremona_ellcurve
This causes the latest version of the database to be downloaded from
the internet. You can also install it from a local copy of the
database spkg file, using a command of the form::
!sage -i database_cremona_ellcurve-*.spkg
Both the mini and full versions of John Cremona's tables are stored in
SAGE_DATA/cremona as SQLite databases. The mini version has the layout::
CREATE TABLE t_class(conductor INTEGER, class TEXT PRIMARY KEY, rank INTEGER);
CREATE TABLE t_curve(class TEXT, curve TEXT PRIMARY KEY, eqn TEXT UNIQUE, tors INTEGER);
CREATE INDEX i_t_class_conductor ON t_class(conductor);
CREATE INDEX i_t_curve_class ON t_curve(class);
while the full version has the layout::
CREATE TABLE t_class(conductor INTEGER, class TEXT PRIMARY KEY, rank INTEGER, L REAL, deg INTEGER);
CREATE TABLE t_curve(class TEXT, curve TEXT PRIMARY KEY, eqn TEXT UNIQUE, gens TEXT, tors INTEGER, cp INTEGER, om REAL, reg REAL, sha);
CREATE INDEX i_t_class_conductor ON t_class(conductor);
CREATE INDEX i_t_curve_class ON t_curve(class);
"""
import os
from sage.misc.prandom import randint
import sage.schemes.elliptic_curves.constructor as elliptic
from sql_db import SQLDatabase, verify_column
from sage.misc.package import optional_packages
from sage.misc.misc import SAGE_DATA, walltime
import re
import string
_cremonaSkeleton = {
't_class': {
'conductor': {'sql':'INTEGER', 'index':True},
'class': {'sql':'TEXT', 'primary_key':True},
'rank': {'sql':'INTEGER'},
'L': {'sql':'REAL'},
'deg': {'sql':'INTEGER'}
},
't_curve': {
'class': {'sql':'TEXT', 'index':True},
'curve': {'sql':'TEXT', 'primary_key':True},
'eqn': {'sql':'TEXT', 'unique':True},
'gens': {'sql':'TEXT'},
'tors': {'sql':'INTEGER'},
'cp': {'sql':'INTEGER'},
'om': {'sql':'REAL'},
'reg': {'sql':'REAL'},
'sha': {'sql':'NOTYPE'}
}
}
_miniCremonaSkeleton = {
't_class': {
'conductor': {'sql':'INTEGER', 'index':True},
'class': {'sql':'TEXT', 'primary_key':True},
'rank': {'sql':'INTEGER'}
},
't_curve': {
'class': {'sql':'TEXT', 'index':True},
'curve': {'sql':'TEXT', 'primary_key':True},
'eqn': {'sql':'TEXT', 'unique':True},
'tors': {'sql':'INTEGER'}
}
}
for t in _cremonaSkeleton:
for c in _cremonaSkeleton[t]:
_cremonaSkeleton[t][c] = verify_column(_cremonaSkeleton[t][c])
for c in _miniCremonaSkeleton[t]:
_miniCremonaSkeleton[t][c] = verify_column(_miniCremonaSkeleton[t][c])
def build(name, data_tgz, largest_conductor=0, mini=False, decompress=True):
"""
Build the CremonaDatabase with given name from scratch
using the data_tgz tarball.
... note::
For data up to level 240000, this function takes about 3
minutes on a AMD Opteron(tm) Processor 6174. The resulting database
occupies 309MB disk space.
To create the large Cremona database from Cremona's data_tgz tarball,
run the following command::
sage: d = sage.databases.cremona.build('cremona','ecdata.tgz') # not tested
"""
t = name.replace(' ','_')
if os.path.exists("%s/cremona/%s.db"%(SAGE_DATA, t)):
raise RuntimeError("Please (re)move %s/cremona/%s.db"%(SAGE_DATA, t)
+ " before rebuilding database.")
if not os.path.exists(data_tgz):
raise IOError, "The data file is not at %s"%data_tgz
t = walltime()
if decompress:
cmd = "tar zxvf %s"%data_tgz
n = os.system(cmd)
if n:
raise RuntimeError, "Error extracting tarball."
if mini:
c = MiniCremonaDatabase(name,False,True)
else:
c = LargeCremonaDatabase(name,False,True)
c._init_from_ftpdata('ecdata', largest_conductor)
print "Total time: ", walltime(t)
def is_optimal_id(id):
"""
Returns true if the Cremona id refers to an optimal curve, and
false otherwise. The curve is optimal if the id, which is of the
form [letter code][number] has number 1.
.. note::
990h3 is the optimal curve in that class, so doesn't obey
this rule.
INPUT:
- ``id`` - str of form letter code followed by an
integer, e.g., a3, bb5, etc.
OUTPUT: bool
EXAMPLES::
sage: from sage.databases.cremona import is_optimal_id
sage: is_optimal_id('b1')
True
sage: is_optimal_id('bb1')
True
sage: is_optimal_id('c1')
True
sage: is_optimal_id('c2')
False
"""
return id[-1] == '1' and not id[-2].isdigit()
def cremona_letter_code(n):
"""
Returns the Cremona letter code corresponding to an integer. For
example, 0 - a 25 - z 26 - ba 51 - bz 52 - ca 53 - cb etc.
.. note::
This is just the base 26 representation of n, where a=0, b=1,
..., z=25. This extends the old Cremona notation (counting from
0) for the first 26 classes, and is different for classes above
26.
INPUT:
- ``n`` (int) -- a non-negative integer
OUTPUT: str
EXAMPLES::
sage: from sage.databases.cremona import cremona_letter_code
sage: cremona_letter_code(0)
'a'
sage: cremona_letter_code(26)
'ba'
sage: cremona_letter_code(27)
'bb'
sage: cremona_letter_code(521)
'ub'
sage: cremona_letter_code(53)
'cb'
sage: cremona_letter_code(2005)
'czd'
TESTS::
sage: cremona_letter_code(QQ)
Traceback (most recent call last):
...
ValueError: Cremona letter codes are only defined for non-negative integers
sage: cremona_letter_code(x)
Traceback (most recent call last):
...
ValueError: Cremona letter codes are only defined for non-negative integers
sage: cremona_letter_code(-1)
Traceback (most recent call last):
...
ValueError: Cremona letter codes are only defined for non-negative integers
sage: cremona_letter_code(3.14159)
Traceback (most recent call last):
...
ValueError: Cremona letter codes are only defined for non-negative integers
"""
try:
m = int(n)
if n == m:
n = m
else:
n = -1
except (ValueError, TypeError):
n = -1
if n<0:
raise ValueError, "Cremona letter codes are only defined for non-negative integers"
if n == 0:
return "a"
s = ""
while n != 0:
s = chr(n%26+97) + s
n //= 26
return s
def old_cremona_letter_code(n):
r"""
Returns the *old* Cremona letter code corresponding to an integer.
integer.
For example::
1 --> A
26 --> Z
27 --> AA
52 --> ZZ
53 --> AAA
etc.
INPUT:
- ``n`` - int
OUTPUT: str
EXAMPLES::
sage: from sage.databases.cremona import old_cremona_letter_code
sage: old_cremona_letter_code(1)
'A'
sage: old_cremona_letter_code(26)
'Z'
sage: old_cremona_letter_code(27)
'AA'
sage: old_cremona_letter_code(521)
'AAAAAAAAAAAAAAAAAAAAA'
sage: old_cremona_letter_code(53)
'AAA'
sage: old_cremona_letter_code(2005)
'CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC'
"""
n -= 1
k = n%26 + 65
label = chr(k)*int(n//26 + 1)
return label
def parse_cremona_label(label):
"""
Given a Cremona label that defines an elliptic
curve, e.g., 11A1 or 37B3, parse the label and return the
conductor, isogeny class label, and number.
The isogeny number may be omitted, in which case it defaults to 1.
If the isogeny number and letter are both omitted, so label is just
a string representing a conductor, then the label defaults to 'A'
and the number to 1.
INPUT:
- ``label`` - str
OUTPUT:
- ``int`` - the conductor
- ``str`` - the isogeny class label
- ``int`` - the number
EXAMPLES::
sage: from sage.databases.cremona import parse_cremona_label
sage: parse_cremona_label('37a2')
(37, 'a', 2)
sage: parse_cremona_label('37b1')
(37, 'b', 1)
sage: parse_cremona_label('10bb2')
(10, 'bb', 2)
"""
if not isinstance(label, str):
label = str(label)
i=0
while i<len(label) and label[i]>='0' and label[i]<='9':
i += 1
j=i+1
if j>len(label):
label += "a"
while j<len(label) and not (label[j]>='0' and label[j]<='9'):
j += 1
if j>=len(label):
label += "1"
conductor, iso, num = int(label[:i]), label[i:j], int(label[j:])
return conductor, iso, num
def split_code(key):
"""
Splits class+curve id string into its two parts.
EXAMPLES::
sage: import sage.databases.cremona as cremona
sage: cremona.split_code('ba2')
('ba', '2')
"""
cu = re.split("[a-z]*",key)[1]
cl = re.split("[0-9]*",key)[0]
return (cl,cu)
def class_to_int(k):
"""
Converts class id string into an integer. Note that this is the
inverse of cremona_letter_code.
EXAMPLES::
sage: import sage.databases.cremona as cremona
sage: cremona.class_to_int('ba')
26
sage: cremona.class_to_int('cremona')
821863562
sage: cremona.cremona_letter_code(821863562)
'cremona'
"""
kk = [string.ascii_lowercase.index(ch) for ch in list(k)]
kk.reverse()
return sum([kk[i]*26**i for i in range(len(kk))])
def cmp_code(key1,key2):
"""
Comparison function for curve id strings.
.. note::
Not the same as standard lexicographic order!
EXAMPLES::
sage: import sage.databases.cremona as cremona
sage: cremona.cmp_code('ba1','z1')
1
By contrast::
sage: cmp('ba1','z1')
-1
"""
cl1,cu1 = split_code(key1)
cl2,cu2 = split_code(key2)
d = class_to_int(cl1)-class_to_int(cl2)
if d!=0: return d
return cmp(cu1,cu2)
class MiniCremonaDatabase(SQLDatabase):
"""
The Cremona database of elliptic curves.
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.allcurves(11)
{'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5],
'a2': [[0, -1, 1, -7820, -263580], 0, 1]}
"""
def __init__(self, name, read_only=True, build=False):
"""
Initialize the database.
TESTS::
sage: c = CremonaDatabase('cremona mini')
sage: c.name
'cremona mini'
"""
if build:
if name is None:
raise RuntimeError('The database must have a name.')
if read_only:
raise RuntimeError('The database must not be read_only.')
self.name = name
name = name.replace(' ','_')
SQLDatabase.__init__(self, '%s/cremona/%s.db'%(SAGE_DATA, name), \
read_only=read_only, skeleton=_miniCremonaSkeleton)
return
self.name = name
name = name.replace(' ','_')
SQLDatabase.__init__(self, '%s/cremona/%s.db'%(SAGE_DATA, name), \
read_only=read_only)
if self.get_skeleton() != _miniCremonaSkeleton:
raise RuntimeError('Database at %s does '%(self.__dblocation__) \
+ 'not appear to be a valid SQL Cremona database.')
def __iter__(self):
"""
Returns an iterator through all EllipticCurve objects in the
Cremona database.
TESTS::
sage: it = CremonaDatabase().__iter__()
sage: it.next().label()
'11a1'
sage: it.next().label()
'11a2'
sage: it.next().label()
'11a3'
sage: it.next().label()
'14a1'
sage: skip = [it.next() for _ in range(100)]
sage: it.next().label()
'45a3'
"""
for c in self.__connection__.cursor().execute('SELECT curve FROM ' \
+ 't_curve'):
yield self.elliptic_curve(c[0])
def __getitem__(self, N):
"""
If N is an integer, return all data about level N in the database.
If N is a string it must be a Cremona label, in which case return
the corresponding elliptic curve, if it is in the database.
INPUT:
- ``N`` - int or str
OUTPUT: dict (if N is an int) or EllipticCurve (if N is a str)
TESTS::
sage: c = CremonaDatabase()
sage: c[11]['allcurves']['a2']
[[0, -1, 1, -7820, -263580], 0, 1]
sage: c['11a2']
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field
"""
if isinstance(N, str):
return self.elliptic_curve(N)
try:
N = int(N)
except ValueError:
raise KeyError("N (=%s) must be a string or positive integer."%N)
if N <= 0:
raise KeyError("N (=%s) must be a string or positive integer."%N)
ret = {'allcurves': self.allcurves(N)}
if hasattr(self, 'allbsd'):
ret['allbsd'] = self.allbsd(N)
if hasattr(self, 'degphi'):
ret['degphi'] = self.degphi(N)
if hasattr(self, 'allgens'):
ret['allgens'] = self.allgens(N)
return ret
def __repr__(self):
"""
String representation of this database.
TESTS::
sage: c = CremonaDatabase('cremona mini')
sage: c.__repr__()
"Cremona's database of elliptic curves with conductor at most 9999"
"""
return "Cremona's database of elliptic curves with conductor at most "\
+ str(self.largest_conductor())
def allcurves(self, N):
"""
Returns the allcurves table of curves of conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:[ainvs, rank, tor], ...
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.allcurves(11)['a3']
[[0, -1, 1, 0, 0], 0, 5]
sage: c.allcurves(12)
{}
sage: c.allcurves(12001)['a1'] # optional - database_cremona_ellcurve
[[1, 0, 0, -101, 382], 1, 1]
"""
ret = {}
for c in self.__connection__.cursor().execute('SELECT curve,eqn,' \
+ 'rank,tors FROM t_curve,t_class USING(class) WHERE ' \
+ 'conductor=?',(int(N),)):
N,iso,num = parse_cremona_label(c[0])
ret[iso+str(num)] = [eval(c[1]),c[2],c[3]]
return ret
def curves(self, N):
"""
Returns the curves table of all *optimal* curves of conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:[ainvs, rank, tor], ...
EXAMPLES:
Optimal curves of conductor 37::
sage: CremonaDatabase().curves(37)
{'a1': [[0, 0, 1, -1, 0], 1, 1], 'b1': [[0, 1, 1, -23, -50], 0, 3]}
Note the 'h3', which is the unique case in the tables where
the optimal curve doesn't have label ending in 1::
sage: list(sorted(CremonaDatabase().curves(990).keys()))
['a1', 'b1', 'c1', 'd1', 'e1', 'f1', 'g1', 'h3', 'i1', 'j1', 'k1', 'l1']
TESTS::
sage: c = CremonaDatabase()
sage: c.curves(12001)['a1'] # optional - database_cremona_ellcurve
[[1, 0, 0, -101, 382], 1, 1]
"""
ret = {}
for c in self.__connection__.cursor().execute('SELECT curve,eqn,' \
+ 'rank,tors FROM t_curve,t_class USING(class) WHERE ' \
+ 'curve=class||1 AND conductor=?',(int(N),)):
N,iso,num = parse_cremona_label(c[0])
ret[iso+str(num)] = [eval(c[1]),c[2],c[3]]
if N == 990:
del ret['h1']
ret['h3'] = [[1,-1,1,-1568,-4669],int(1),int(6)]
return ret
def elliptic_curve_from_ainvs(self, ainvs):
"""
Returns the elliptic curve in the database of with minimal
ainvs, if it exists, or raises a RuntimeError exception
otherwise.
INPUT:
- ``ainvs`` - list (5-tuple of int's); the minimal
Weierstrass model for an elliptic curve
OUTPUT: EllipticCurve
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.elliptic_curve_from_ainvs([0, -1, 1, -10, -20])
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: c.elliptic_curve_from_ainvs([1, 0, 0, -101, 382]) # optional - database_cremona_ellcurve
Elliptic Curve defined by y^2 + x*y = x^3 - 101*x + 382 over Rational Field
"""
q = self.__connection__.cursor().execute("SELECT curve FROM t_curve " \
+ "WHERE eqn=?",(str(ainvs).replace(' ',''),))
try:
return self.elliptic_curve(q.next()[0])
except StopIteration:
raise RuntimeError("No elliptic curve with ainvs (=%s) "%ainvs \
+ "in the database.")
def elliptic_curve(self, label):
"""
Return an elliptic curve with given label with some data about it
from the database pre-filled in.
INPUT:
- ``label`` - str (Cremona label)
OUTPUT: EllipticCurve
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.elliptic_curve('11a1')
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: c.elliptic_curve('12001a1') # optional - database_cremona_ellcurve
Elliptic Curve defined by y^2 + x*y = x^3 - 101*x + 382 over Rational Field
sage: c.elliptic_curve('48c1')
Traceback (most recent call last):
...
ValueError: There is no elliptic curve with label 48c1 in the database (note: use lower case letters!)
"""
N, iso, num = parse_cremona_label(label)
label = str(N)+iso+str(num)
if self.get_skeleton() == _miniCremonaSkeleton:
q = self.__connection__.cursor().execute("SELECT eqn,rank,tors " \
+ 'FROM t_curve,t_class USING(class) WHERE curve=?', (label,))
else:
q = self.__connection__.cursor().execute("SELECT eqn,rank,tors," \
+ "deg,gens,cp,om,L,reg,sha FROM t_curve,t_class " \
+ "USING(class) WHERE curve=?",(label,))
try:
c = q.next()
F = elliptic.EllipticCurve(eval(c[0]))
F._set_cremona_label(label)
F._set_rank(c[1])
F._set_torsion_order(c[2])
F._set_conductor(N)
if len(c) > 3:
if num == 1:
F._set_modular_degree(c[3])
F._set_gens(eval(c[4]))
F.db_extra = list(c[5:])
elif c[1] == 0:
F._set_gens([])
return F
except StopIteration:
if N < self.largest_conductor():
message = "There is no elliptic curve with label " + label \
+ " in the database (note: use lower case letters!)"
elif 'database_cremona_ellcurve' in \
[s.split('-')[0] for s in optional_packages()[0]]:
message = "There is no elliptic curve with label " + label \
+ " in the currently available databases"
else:
message = "There is no elliptic curve with label " \
+ label + " in the default database; try installing " \
+ "the optional package database_cremona_ellcurve which " \
+ "contains the complete Cremona database"
raise ValueError(message)
def iter(self, conductors):
"""
Return an iterator through all curves in the database with given
conductors.
INPUT:
- ``conductors`` - list or generator of ints
OUTPUT: generator that iterates over EllipticCurve objects.
EXAMPLES::
sage: [e.cremona_label() for e in CremonaDatabase().iter([11..15])]
['11a1', '11a2', '11a3', '14a1', '14a2', '14a3', '14a4', '14a5',
'14a6', '15a1', '15a2', '15a3', '15a4', '15a5', '15a6', '15a7', '15a8']
"""
for N in conductors:
for c in self.__connection__.cursor().execute('SELECT curve ' \
+ 'FROM t_curve,t_class USING(class) WHERE conductor=?', \
(int(N),)):
yield self.elliptic_curve(c[0])
def isogeny_classes(self, conductor):
"""
Return the allcurves data (ainvariants, rank and torsion) for the
elliptic curves in the database of given conductor as a list of
lists, one for each isogeny class. The curve with number 1 is
always listed first.
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.isogeny_classes(11)
[[[[0, -1, 1, -10, -20], 0, 5],
[[0, -1, 1, -7820, -263580], 0, 1],
[[0, -1, 1, 0, 0], 0, 5]]]
sage: c.isogeny_classes(12001) # optional - database_cremona_ellcurve
[[[[1, 0, 0, -101, 382], 1, 1]],
[[[0, 0, 1, -247, 1494], 1, 1]],
[[[0, 0, 1, -4, -18], 1, 1]],
[[[0, 1, 1, -10, 18], 1, 1]]]
"""
conductor=int(conductor)
classes = []
A = self.allcurves(conductor)
K = A.keys()
K.sort(cmp_code)
for k in K:
v = A[k]
if not (k[-1] == '1' and k[-2].isalpha()):
classes[len(classes)-1].append(v)
else:
classes.append([v])
return classes
def isogeny_class(self, label):
"""
Returns the isogeny class of elliptic curves that are
isogenous to the curve with given Cremona label.
INPUT:
- ``label`` - string
OUTPUT:
- ``list`` - list of EllipticCurve objects.
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.isogeny_class('11a1')
[Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 7820*x - 263580 over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 - x^2 over Rational Field]
sage: c.isogeny_class('12001a1') # optional - database_cremona_ellcurve
[Elliptic Curve defined by y^2 + x*y = x^3 - 101*x + 382 over Rational Field]
"""
conductor,iso,num=parse_cremona_label(label)
q = self.__connection__.cursor().execute("SELECT curve FROM t_curve " \
+ "WHERE class=?",(str(conductor)+iso,))
return [self.elliptic_curve(c[0]) for c in q]
def iter_optimal(self, conductors):
"""
Return an iterator through all optimal curves in the database with given conductors.
INPUT:
- ``conductors`` - list or generator of ints
OUTPUT:
generator that iterates over EllipticCurve objects.
EXAMPLES:
We list optimal curves with conductor up to 20::
sage: [e.cremona_label() for e in CremonaDatabase().iter_optimal([11..20])]
['11a1', '14a1', '15a1', '17a1', '19a1', '20a1']
Note the unfortunate 990h3 special case::
sage: [e.cremona_label() for e in CremonaDatabase().iter_optimal([990])]
['990a1', '990b1', '990c1', '990d1', '990e1', '990f1', '990g1', '990h3', '990i1', '990j1', '990k1', '990l1']
"""
for N in conductors:
if N == 990:
for c in self.__connection__.cursor().execute('SELECT class ' \
+ 'FROM t_class WHERE conductor=990'):
if c[0][-1] == u'h':
yield self.elliptic_curve(c[0]+u'3')
else:
yield self.elliptic_curve(c[0]+u'1')
continue
for c in self.__connection__.cursor().execute('SELECT curve ' \
+ 'FROM t_curve,t_class USING(class) WHERE curve=class||1 ' \
+ 'AND conductor=?',(int(N),)):
yield self.elliptic_curve(c[0])
def list(self, conductors):
"""
Returns a list of all curves with given conductors.
INPUT:
- ``conductors`` - list or generator of ints
OUTPUT:
- list of EllipticCurve objects.
EXAMPLES::
sage: CremonaDatabase().list([37])
[Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 3*x + 1 over Rational Field]
"""
return list(self.iter(conductors))
def list_optimal(self, conductors):
"""
Returns a list of all optimal curves with given conductors.
INPUT:
- ``conductors`` - list or generator of ints
list of EllipticCurve objects.
OUTPUT:
list of EllipticCurve objects.
EXAMPLES::
sage: CremonaDatabase().list_optimal([37])
[Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field,
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field]
"""
return list(self.iter_optimal(conductors))
def largest_conductor(self):
"""
The largest conductor for which the database is complete.
OUTPUT:
- ``int`` - largest conductor
EXAMPLES::
sage: c = CremonaDatabase('cremona mini')
sage: c.largest_conductor()
9999
"""
if hasattr(self, '__largest_conductor__'):
return self.__largest_conductor__
q = self.__connection__.cursor().execute('SELECT conductor FROM ' \
+ 't_class ORDER BY conductor DESC LIMIT 1')
self.__largest_conductor__ = q.next()[0]
return self.__largest_conductor__
def smallest_conductor(self):
"""
The smallest conductor for which the database is complete: always 1.
OUTPUT:
- ``int`` - smallest conductor
.. note::
This always returns the integer 1, since that is the
smallest conductor for which the database is complete,
although there are no elliptic curves of conductor 1. The
smallest conductor of a curve in the database is 11.
EXAMPLES::
sage: CremonaDatabase().smallest_conductor()
1
"""
return 1
def conductor_range(self):
"""
Return the range of conductors that are covered by the database.
OUTPUT: tuple of ints (N1,N2+1) where N1 is the smallest and
N2 the largest conductor for which the database is complete.
EXAMPLES::
sage: c = CremonaDatabase('cremona mini')
sage: c.conductor_range()
(1, 10000)
"""
return 1, self.largest_conductor()+1
def number_of_curves(self, N=0, i=0):
"""
Returns the number of curves stored in the database with conductor
N. If N = 0, returns the total number of curves in the database.
If i is nonzero, returns the number of curves in the i-th isogeny
class. If i is a Cremona letter code, e.g., 'a' or 'bc', it is
converted to the corresponding number.
INPUT:
- ``N`` - int
- ``i`` - int or str
OUTPUT: int
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.number_of_curves(11)
3
sage: c.number_of_curves(37)
4
sage: c.number_of_curves(990)
42
sage: num = c.number_of_curves()
"""
if N == 0:
if hasattr(self, '__number_of_curves__'):
return self.__number_of_curves__
q = self.__connection__.cursor().execute('SELECT COUNT(curve) ' \
+ 'FROM t_curve')
self.__number_of_curves__ = q.next()[0]
return self.__number_of_curves__
if i == 0:
q = self.__connection__.cursor().execute('SELECT COUNT(curve) ' \
+ 'FROM t_curve,t_class USING(class) WHERE conductor=?', \
(int(N),))
return q.next()[0]
if not isinstance(i, str):
i = cremona_letter_code(i)
q = self.__connection__.cursor().execute('SELECT COUNT(curve) FROM ' \
+ 't_curve WHERE class=?',(str(N)+i,))
return q.next()[0]
def number_of_isogeny_classes(self, N=0):
"""
Returns the number of isogeny classes of curves in the database of
conductor N. If N is 0, return the total number of isogeny classes
of curves in the database.
INPUT:
- ``N`` - int
OUTPUT: int
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.number_of_isogeny_classes(11)
1
sage: c.number_of_isogeny_classes(37)
2
sage: num = c.number_of_isogeny_classes()
"""
if N == 0:
if hasattr(self, '__number_of_isogeny_classes__'):
return self.__number_of_isogeny_classes__
q = self.__connection__.cursor().execute('SELECT COUNT(class) ' \
+ 'FROM t_class')
self.__number_of_isogeny_classes__ = q.next()[0]
return self.__number_of_isogeny_classes__
q = self.__connection__.cursor().execute('SELECT COUNT(class) FROM ' \
+ 't_class WHERE conductor=?',(int(N),))
return q.next()[0]
def random(self):
"""
Returns a random curve from the database.
EXAMPLES::
sage: CremonaDatabase().random() # random -- depends on database installed
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 224*x + 3072 over Rational Field
"""
N = randint(11, self.largest_conductor())
q = self.__connection__.cursor().execute('SELECT conductor FROM ' \
+ 't_class WHERE conductor>=? ORDER BY conductor',(int(N),))
try:
N = q.next()[0]
except StopIteration:
N = 11
iso = randint(0, self.number_of_isogeny_classes(N)-1)
iso = cremona_letter_code(iso)
num = randint(1, self.number_of_curves(N,iso))
return self.elliptic_curve(str(N)+iso+str(num))
def _init_from_ftpdata(self, ftpdata, largest_conductor=0):
"""
Create the SQL Cremona Database from the Cremona data directory,
which is available from Cremona's website. I.e., just wget
Cremona's database to a local directory.
To create the large database from Cremona's text files, see
sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.MiniCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested
sage: d._init_from_ftpdata('.') # not tested
"""
if self.__read_only__:
raise RuntimeError("The database must not be read_only.")
if not os.path.exists(ftpdata):
raise RuntimeError("The cremona ftpdata directory '" + ftpdata \
+ "' does not exist.")
if largest_conductor:
print "largest conductor =", largest_conductor
self.__largest_conductor__ = largest_conductor
num_curves, num_iso_classes = self._init_allcurves(ftpdata, largest_conductor)
self.__number_of_curves__ = num_curves
self.__number_of_isogeny_classes__ = num_iso_classes
if hasattr(self, 'degphi'):
self._init_degphi(ftpdata, largest_conductor)
if hasattr(self, 'allbsd'):
self._init_allbsd(ftpdata, largest_conductor)
if hasattr(self, 'allgens'):
self._init_allgens(ftpdata, largest_conductor)
self.vacuum()
def _init_allcurves(self, ftpdata, largest_conductor=0):
"""
Initialize the allcurves table by reading the corresponding ftpdata
files and importing them into the database.
To create the large database from Cremona's text files, see
sage.databases.cremona.build, do NOT run this method directly.
INPUT:
- ``largest_conductor`` - int (default: 0), if 0,
then only include data up to that conductor.
OUTPUT:
- ``int`` - number_of_curves
- ``int`` - number_of_isogeny_classes
EXAMPLES::
sage: d = sage.databases.cremona.MiniCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested
sage: d._init_allcurves('.', 11) # not tested
(3, 1)
"""
if self.__read_only__:
raise RuntimeError("The database must not be read_only.")
files = os.listdir(ftpdata)
files.sort()
name = 'allcurves'
num_curves = 0
num_iso_classes = 0
con = self.get_connection()
for F in files:
if not F[:len(name)] == name:
continue
print "Inserting", F
class_data = []
curve_data = []
for L in file(ftpdata + "/" + F).readlines():
N, iso, num, ainvs, r, tor = L.split()
if largest_conductor and int(N) > largest_conductor: break
cls = N+iso
cur = cls+num
if num == "1":
class_data.append((N,cls,r))
num_iso_classes += 1
curve_data.append((cur,cls,ainvs,tor))
num_curves += 1
con.executemany('INSERT INTO t_class (conductor,class,rank) ' \
+ 'VALUES (?,?,?)', class_data)
con.executemany('INSERT INTO t_curve (curve,class,eqn,tors) ' \
+ 'VALUES (?,?,?,?)', curve_data)
print "Committing..."
print "num_iso_classes =", num_iso_classes
self.commit()
if largest_conductor and int(N) > largest_conductor: break
return num_curves, num_iso_classes
class LargeCremonaDatabase(MiniCremonaDatabase):
"""
The Cremona database of elliptic curves.
EXAMPLES::
sage: c = CremonaDatabase('cremona') # optional - database_cremona_ellcurve
sage: c.allcurves(11) # optional - database_cremona_ellcurve
{'a1': [[0, -1, 1, -10, -20], 0, 5], 'a3': [[0, -1, 1, 0, 0], 0, 5],
'a2': [[0, -1, 1, -7820, -263580], 0, 1]}
"""
def __init__(self, name, read_only=True, build=False):
"""
Initialize the database.
TESTS::
sage: c = CremonaDatabase('cremona') # optional - database_cremona_ellcurve
sage: c.name # optional - database_cremona_ellcurve
'cremona'
"""
if build:
if name is None:
raise RuntimeError('The database must have a name.')
if read_only:
raise RuntimeError('The database must not be read_only.')
self.name = name
name = name.replace(' ','_')
SQLDatabase.__init__(self, '%s/cremona/%s.db'%(SAGE_DATA, name), \
read_only=read_only, skeleton=_cremonaSkeleton)
return
self.name = name
name = name.replace(' ','_')
SQLDatabase.__init__(self, '%s/cremona/%s.db'%(SAGE_DATA, name), \
read_only=read_only)
if self.get_skeleton() != _cremonaSkeleton:
raise RuntimeError('Database at %s does '%(self.__dblocation__) \
+ 'not appear to be a valid SQL Cremona database.')
def allbsd(self, N):
r"""
Return the allbsd table for conductor N. The entries are::
[id, tamagawa_product, Omega_E, L, Reg_E, Sha_an(E)]
where id is the isogeny class (letter) followed by a number, e.g.,
b3, and L is `L^r(E,1)/r!`, where E has rank r.
INPUT:
- ``N`` - int, the conductor
OUTPUT: dict containing the allbsd table for each isogeny class
in conductor N
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.allbsd(12) # optional - database_cremona_ellcurve
{}
sage: c.allbsd(19)['a3'] # optional - database_cremona_ellcurve
[1, 4.079279200464932, 0.4532532444961036, 1.0, 1]
sage: c.allbsd(12001)['a1'] # optional - database_cremona_ellcurve
[2, 3.2760813524872185, 1.5491014309050595, 0.23642597118795194, 1]
"""
ret = {}
for c in self.__connection__.cursor().execute('SELECT curve,cp,om,L,' \
+ 'reg,sha FROM t_curve,t_class USING(class) WHERE conductor=?', \
(int(N),)):
N,iso,num = parse_cremona_label(c[0])
ret[iso+str(num)] = list(c[1:])
return ret
def allgens(self, N):
"""
Return the allgens table for conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:[points, ...], ...
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.allgens(12) # optional - database_cremona_ellcurve
{}
sage: c.allgens(1001)['a1'] # optional - database_cremona_ellcurve
[[61, 181, 1]]
sage: c.allgens(12001)['a1'] # optional - database_cremona_ellcurve
[[7, 2, 1]]
"""
ret = {}
for c in self.__connection__.cursor().execute('SELECT curve,gens ' \
+ 'FROM t_curve,t_class USING(class) WHERE conductor=?',(int(N),)):
N,iso,num = parse_cremona_label(c[0])
ret[iso+str(num)] = eval(c[1])
return ret
def degphi(self, N):
"""
Return the degphi table for conductor N.
INPUT:
- ``N`` - int, the conductor
OUTPUT:
- ``dict`` - id:degphi, ...
EXAMPLES::
sage: c = CremonaDatabase()
sage: c.degphi(11) # optional - database_cremona_ellcurve
{'a1': 1}
sage: c.degphi(12001)['c1'] # optional - database_cremona_ellcurve
1640
"""
ret = {}
for c in self.__connection__.cursor().execute('SELECT curve,deg FROM' \
+ ' t_curve,t_class USING(class) WHERE curve=class||1 AND ' \
+ 'conductor=?', (int(N),)):
N,iso,num = parse_cremona_label(c[0])
ret[iso+str(num)] = c[1]
return ret
def _init_degphi(self, ftpdata, largest_conductor=0):
"""
Initialize the degphi table by reading the corresponding ftpdata
files and importing them into the database.
To create the large database from Cremona's text files, see
sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.LargeCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested
sage: d._init_degphi('.') # not tested
"""
if self.__read_only__:
raise RuntimeError, "The database must not be read_only."
files = os.listdir(ftpdata)
files.sort()
name = "degphi"
con = self.get_connection()
for F in files:
if not F[:len(name)] == name:
continue
print "Inserting", F
class_data = []
for L in file(ftpdata + "/" + F).readlines():
N, iso, num, degree, primes, curve = L.split()
if largest_conductor and int(N) > largest_conductor: break
class_data.append((degree,N+iso))
con.executemany('UPDATE t_class SET deg=? WHERE class=?', \
class_data)
print "Committing..."
self.commit()
if largest_conductor and int(N) > largest_conductor: break
def _init_allbsd(self, ftpdata, largest_conductor=0):
"""
Initialize the allbsd table by reading the corresponding ftpdata
files and importing them into the database.
To create the large database from Cremona's text files, see
sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.LargeCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested
sage: d._init_allbsd('.') # not tested
"""
if self.__read_only__:
raise RuntimeError, "The database must not be read_only."
files = os.listdir(ftpdata)
files.sort()
name = "allbsd"
con = self.get_connection()
for F in files:
if not F[:len(name)] == name:
continue
print "Inserting", F
curve_data = []
class_data = []
for L in file(ftpdata + "/" + F).readlines():
N, iso, num, eqn, rank, tor, cp, om, L, reg, sha = L.split()
if largest_conductor and int(N) > largest_conductor: break
cls = N+iso
if num == "1":
class_data.append((L,cls))
curve_data.append((cp,om,reg,eval(sha),cls+num))
con.executemany("UPDATE t_class SET L=? WHERE class=?", class_data)
con.executemany("UPDATE t_curve SET cp=?,om=?,reg=?,sha=? WHERE " \
+ "curve=?", curve_data)
print "Committing..."
self.commit()
if largest_conductor and int(N) > largest_conductor: break
def _init_allgens(self, ftpdata, largest_conductor=0):
"""
Initialize the allgens table by reading the corresponding ftpdata
files and importing them into the database.
To create the large database from Cremona's text files, see
sage.databases.cremona.build, do NOT run this method directly.
EXAMPLES::
sage: d = sage.databases.cremona.LargeCremonaDatabase(name='cremona', read_only=False, rebuild=True) # not tested
sage: d._init_allgens('.') # not tested
"""
if self.__read_only__:
raise RuntimeError, "The database must not be read_only."
files = os.listdir(ftpdata)
files.sort()
name = "allgens"
con = self.get_connection()
for F in files:
if not F[:len(name)] == name:
continue
print "Inserting", F
curve_data = []
for L in file(ftpdata + "/" + F).readlines():
v = L.split()
if largest_conductor and int(v[0]) > largest_conductor: break
gens = '['+','.join(v[6:6+int(v[4])]).replace(':',',')+']'
curve_data.append((gens,''.join(v[:3])))
con.executemany("UPDATE t_curve SET gens=? WHERE curve=?", \
curve_data)
print "Committing..."
if largest_conductor and int(v[0]) > largest_conductor: break
_db = None
def CremonaDatabase(name=None,mini=None,set_global=None):
"""
Initializes the Cremona database with name ``name``. If ``name`` is
``None`` it instead initializes large Cremona database (named 'cremona'),
if available or default mini Cremona database (named 'cremona mini').
If the Cremona database in question is in the format of the mini database,
you must set ``mini=True``, otherwise it must be set to ``False``.
If you would like other components of Sage to use this database, mark
``set_global=True``.
TESTS::
sage: c = CremonaDatabase()
sage: isinstance(c, sage.databases.cremona.MiniCremonaDatabase)
True
sage: isinstance(c, sage.databases.cremona.LargeCremonaDatabase) # optional - database_cremona_ellcurve
True
"""
global _db
if set_global is None:
set_global = _db is None and name is None
if name is None and not set_global:
return _db
if set_global and name is None:
if os.path.isfile('%s/cremona/cremona.db'%SAGE_DATA):
name = 'cremona'
elif os.path.isfile('%s/cremona/cremona_mini.db'%SAGE_DATA):
name = 'cremona mini'
else:
raise RuntimeError('Could not find valid cremona database. ' \
+ 'Please make sure SAGE_DATA is set correctly.')
if name == 'cremona':
mini = False
elif name == 'cremona mini':
mini = True
if mini is None:
raise ValueError('mini must be set as either True or False')
if set_global:
if mini:
_db = MiniCremonaDatabase(name)
else:
_db = LargeCremonaDatabase(name)
return _db
if mini:
return MiniCremonaDatabase(name)
return LargeCremonaDatabase(name)
