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""" 

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 in John Cremona's tables. 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, run the following in the shell:: 

 

sage -i database_cremona_ellcurve 

 

This causes the latest version of the database to be downloaded from 

the internet. 

 

Both the mini and full versions of John Cremona's tables are stored in 

SAGE_SHARE/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); 

""" 

#***************************************************************************** 

# Copyright (C) 2014 John Cremona <john.cremona@gmail.com> 

# Copyright (C) 2011 R. Andrew Ohana <andrew.ohana@gmail.com> 

# Copyright (C) 2005 William Stein <wstein@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

 

from __future__ import print_function 

from __future__ import absolute_import 

 

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 is_package_installed 

from sage.env import SAGE_SHARE 

from sage.misc.all import 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 350000, this function takes about 

3m40s. The resulting database occupies 426MB disk space. 

 

To create the large Cremona database from Cremona's data_tgz 

tarball, obtainable from 

http://homepages.warwick.ac.uk/staff/J.E.Cremona/ftp/data/, run 

the following command:: 

 

sage: d = sage.databases.cremona.build('cremona','ecdata.tgz') # not tested 

""" 

db_path = os.path.join(SAGE_SHARE,'cremona',name.replace(' ','_')+'.db') 

if os.path.exists(db_path): 

raise RuntimeError('Please (re)move %s before building '%db_path \ 

+ '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) 

# The following line assumes that the tarball extracts to a 

# directory called 'ecdata' 

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 

 

old_cremona_label_regex = re.compile(r'(\d+)([A-Z]*)(\d*)$') 

cremona_label_regex = re.compile(r'(\d+)([a-z]*)(\d*)$') 

lmfdb_label_regex = re.compile(r'(\d+)\.([a-z]+)(\d*)$') 

 

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. 

 

For this function, the curve number may be omitted, in which case 

it defaults to 1. If the curve number and isogeny class are both 

omitted (label is just a string representing a conductor), then 

the isogeny class defaults to 'a' and the number to 1. Valid 

labels consist of one or more digits, followed by zero or more 

letters (either all in upper case for an old Cremona label, or all 

in lower case), followed by zero or more digits. 

 

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) 

sage: parse_cremona_label('11a') 

(11, 'a', 1) 

sage: parse_cremona_label('11') 

(11, 'a', 1) 

 

Valid old Cremona labels are allowed:: 

 

sage: parse_cremona_label('17CCCC') 

(17, 'dc', 1) 

sage: parse_cremona_label('5AB2') 

Traceback (most recent call last): 

... 

ValueError: 5AB2 is not a valid Cremona label 

 

TESTS:: 

 

sage: from sage.databases.cremona import parse_cremona_label 

sage: parse_cremona_label('x11') 

Traceback (most recent call last): 

... 

ValueError: x11 is not a valid Cremona label 

""" 

m = cremona_label_regex.match(str(label)) 

if m is None: 

m = old_cremona_label_regex.match(str(label)) 

if m is None: 

raise ValueError(label + " is not a valid Cremona label") 

 

conductor, iso, num = m.groups() 

if len(iso) == 0: 

iso = "a" 

if len(num) == 0: 

num = "1" 

 

# convert old cremona labels to new ones 

if iso.upper() == iso and iso[0]*len(iso) == iso: 

iso = cremona_letter_code((len(iso)-1)*26+ord(iso[0])-ord('A')) 

 

# verify cremona label is valid 

if iso.lower() != iso: 

raise ValueError('%s is not a valid Cremona label'%label) 

 

return int(conductor), iso, int(num) 

 

def parse_lmfdb_label(label): 

""" 

Given an LMFDB label that defines an elliptic curve, e.g., 11.a1 

or 37.b3, parse the label and return the conductor, isogeny class 

label, and number. 

 

The LMFDB label (named after the L-functions and modular forms 

database), is determined by the following two orders: 

 

- Isogeny classes with the same conductor are ordered 

lexicographically by the coefficients in the q-expansion of the 

associated modular form. 

 

- Curves within the same isogeny class are ordered 

lexicographically by the a-invariants of the minimal model. 

 

The format is <conductor>.<iso><curve>, where the isogeny class is 

encoded using the same base-26 encoding into letters used in 

Cremona's labels. For example, 990.h3 is the same as Cremona's 990j1 

 

For this function, the curve number may be omitted, in which case 

it defaults to 1. If the curve number and isogeny class are both 

omitted (label is just a string representing a conductor), then 

the isogeny class 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_lmfdb_label 

sage: parse_lmfdb_label('37.a2') 

(37, 'a', 2) 

sage: parse_lmfdb_label('37.b') 

(37, 'b', 1) 

sage: parse_lmfdb_label('10.bb2') 

(10, 'bb', 2) 

""" 

m = lmfdb_label_regex.match(str(label).lower()) 

if m is None: 

raise ValueError(label + " is not a valid LMFDB label") 

conductor, iso, num = m.groups() 

if len(iso) == 0: 

iso = "a" 

if len(num) == 0: 

num = "1" 

return int(conductor), iso, int(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 sort_key(key1): 

""" 

Comparison key for curve id strings. 

 

.. note:: 

 

Not the same as standard lexicographic order! 

 

EXAMPLES:: 

 

sage: from sage.databases.cremona import sort_key 

sage: l = ['ba1', 'z1'] 

sage: sorted(l, key=sort_key) 

['z1', 'ba1'] 

""" 

cl1, cu1 = split_code(key1) 

return (class_to_int(cl1), cu1) 

 

 

def cremona_to_lmfdb(cremona_label, CDB=None): 

""" 

Converts a Cremona label into an LMFDB label. 

 

See :func:`parse_lmfdb_label` for an explanation of LMFDB labels. 

 

INPUT: 

 

- ``cremona_label`` -- a string, the Cremona label of a curve. 

This can be the label of a curve (e.g. '990j1') or of an isogeny 

class (e.g. '990j') 

- ``CDB`` -- the Cremona database in which to look up the isogeny 

classes of the same conductor. 

 

OUTPUT: 

 

- ``lmfdb_label`` -- a string, the corresponding LMFDB label. 

 

EXAMPLES:: 

 

sage: from sage.databases.cremona import cremona_to_lmfdb, lmfdb_to_cremona 

sage: cremona_to_lmfdb('990j1') 

'990.h3' 

sage: lmfdb_to_cremona('990.h3') 

'990j1' 

 

TESTS:: 

 

sage: for label in ['5077a1','66a3','102b','420c2']: 

....: assert(lmfdb_to_cremona(cremona_to_lmfdb(label)) == label) 

sage: for label in ['438.c2','306.b','462.f3']: 

....: assert(cremona_to_lmfdb(lmfdb_to_cremona(label)) == label) 

""" 

from sage.libs.pari.all import pari 

m = cremona_label_regex.match(cremona_label) 

if m is None: 

raise ValueError("Invalid Cremona label") 

N, cremona_iso, cremona_number = m.groups() 

if CDB is None: 

CDB = CremonaDatabase() 

classes = CDB.isogeny_classes(N) 

ft = int(53) 

tff = int(255) # This should be enough to distinguish between curves (using heuristics from Sato-Tate for example) 

isos = [] 

for i, iso in enumerate(classes): 

alist = iso[0][0] 

E = pari(alist).ellinit(precision=ft) 

isos.append((E.ellan(tff, python_ints=True), cremona_letter_code(i))) 

isos.sort() 

sorted_letters = [iso[1] for iso in isos] 

lmfdb_iso = cremona_letter_code(sorted_letters.index(cremona_iso)) 

if len(cremona_number) > 0: 

iso_class = sorted([(curve[0],str(i+1)) for i,curve in enumerate(classes[class_to_int(cremona_iso)])]) 

sorted_numbers = [curve[1] for curve in iso_class] 

lmfdb_number = str(sorted_numbers.index(cremona_number)+1) 

return N + '.' + lmfdb_iso + lmfdb_number 

else: 

return N + '.' + lmfdb_iso 

 

def lmfdb_to_cremona(lmfdb_label, CDB=None): 

""" 

Converts an LMFDB labe into a Cremona label. 

 

See :func:`parse_lmfdb_label` for an explanation of LMFDB labels. 

 

INPUT: 

 

- ``lmfdb_label`` -- a string, the LMFDB label of a curve. 

This can be the label of a curve (e.g. '990.j1') or of an isogeny 

class (e.g. '990.j') 

- ``CDB`` -- the Cremona database in which to look up the isogeny 

classes of the same conductor. 

 

OUTPUT: 

 

- ``cremona_label`` -- a string, the corresponding Cremona label. 

 

EXAMPLES:: 

 

sage: from sage.databases.cremona import cremona_to_lmfdb, lmfdb_to_cremona 

sage: lmfdb_to_cremona('990.h3') 

'990j1' 

sage: cremona_to_lmfdb('990j1') 

'990.h3' 

""" 

from sage.libs.pari.all import pari 

m = lmfdb_label_regex.match(lmfdb_label) 

if m is None: 

raise ValueError("Invalid LMFDB label") 

N, lmfdb_iso, lmfdb_number = m.groups() 

if CDB is None: 

CDB = CremonaDatabase() 

classes = CDB.isogeny_classes(N) 

ft = int(53) 

tff = int(255) # This should be enough to distinguish between curves (using heuristics from Sato-Tate for example) 

isos = [] 

for i, iso in enumerate(classes): 

alist = iso[0][0] 

E = pari(alist).ellinit(precision=ft) 

isos.append((E.ellan(tff, python_ints=True), cremona_letter_code(i))) 

isos.sort() 

cremona_iso = isos[class_to_int(lmfdb_iso)][1] 

if len(lmfdb_number) > 0: 

iso_class = sorted([(curve[0],i+1) for i,curve in enumerate(classes[class_to_int(cremona_iso)])]) 

cremona_number = str(iso_class[int(lmfdb_number)-1][1]) 

return N + cremona_iso + cremona_number 

else: 

return N + cremona_iso 

 

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], 

'a2': [[0, -1, 1, -7820, -263580], 0, 1], 

'a3': [[0, -1, 1, 0, 0], 0, 5]} 

""" 

def __init__(self, name, read_only=True, build=False): 

""" 

Initialize the database. 

 

TESTS:: 

 

sage: c = CremonaDatabase('cremona mini') 

sage: c.name 

'cremona mini' 

""" 

self.name = name 

name = name.replace(' ','_') 

db_path = os.path.join(SAGE_SHARE, 'cremona', name+'.db') 

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.') 

SQLDatabase.__init__(self, db_path, read_only=read_only, \ 

skeleton=_miniCremonaSkeleton) 

return 

if not os.path.isfile(db_path): 

raise ValueError("Desired database (='%s') does not "%self.name \ 

+ "exist") 

SQLDatabase.__init__(self, db_path, 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: next(it).label() 

'11a1' 

sage: next(it).label() 

'11a2' 

sage: next(it).label() 

'11a3' 

sage: next(it).label() 

'14a1' 

sage: skip = [next(it) for _ in range(100)] 

sage: next(it).label() 

'45a3' 

""" 

query = "SELECT curve FROM t_curve,t_class USING(class) ORDER BY conductor" 

for c in self.__connection__.cursor().execute(query): 

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: sorted(CremonaDatabase().curves(990)) 

['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 coefficients_and_data(self, label): 

""" 

Return the Weierstrass coefficients and other data for the 

curve with given label. 

 

EXAMPLES:: 

 

sage: c, d = CremonaDatabase().coefficients_and_data('144b1') 

sage: c 

[0, 0, 0, 6, 7] 

sage: d['conductor'] 

144 

sage: d['cremona_label'] 

'144b1' 

sage: d['rank'] 

0 

sage: d['torsion_order'] 

2 

 

Check that :trac:`17904` is fixed:: 

 

sage: 'gens' in CremonaDatabase().coefficients_and_data('100467a2')[1] # optional - database_cremona_ellcurve 

True 

 

 

""" 

# There are two possible strings: the Cremona label and the LMFDB label. 

# They are distinguished by the presence of a period. 

if label.find('.') == -1: 

cremona_label = label 

lmfdb_label = None 

else: 

cremona_label = lmfdb_to_cremona(label) 

lmfdb_label = label 

 

N, iso, num = parse_cremona_label(cremona_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 = next(q) 

except StopIteration: 

if N < self.largest_conductor(): 

message = "There is no elliptic curve with label " + label \ 

+ " in the database" 

elif is_package_installed('database_cremona_ellcurve'): 

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) 

ainvs = eval(c[0]) 

data = {'cremona_label': label, 

'rank': c[1], 

'torsion_order': c[2], 

'conductor': N} 

if lmfdb_label: 

data['lmfdb_label'] = lmfdb_label 

if len(c) > 3: 

data['modular_degree'] = (c[3]) 

data['gens'] = eval(c[4]) 

data['db_extra'] = list(c[5:]) 

return ainvs, data 

 

def data_from_coefficients(self, ainvs): 

""" 

Return elliptic curve data for the curve with given 

Weierstrass coefficients. 

 

EXAMPLES:: 

 

sage: d = CremonaDatabase().data_from_coefficients([1, -1, 1, 31, 128]) 

sage: d['conductor'] 

1953 

sage: d['cremona_label'] 

'1953c1' 

sage: d['rank'] 

1 

sage: d['torsion_order'] 

2 

 

Check that :trac:`17904` is fixed:: 

 

sage: ai = EllipticCurve('100467a2').ainvs() # optional - database_cremona_ellcurve 

sage: 'gens' in CremonaDatabase().data_from_coefficients(ai) # optional - database_cremona_ellcurve 

True 

""" 

ainvs = str(list(ainvs)) 

if self.get_skeleton() == _miniCremonaSkeleton: 

q = self.__connection__.cursor().execute("SELECT curve,rank,tors " 

+ 'FROM t_curve,t_class USING(class) WHERE eqn=?', 

(ainvs.replace(' ', ''),)) 

else: 

q = self.__connection__.cursor().execute("SELECT curve,rank,tors," 

+ "deg,gens,cp,om,L,reg,sha FROM t_curve,t_class " 

+ "USING(class) WHERE eqn=?", 

(ainvs.replace(' ', ''),)) 

try: 

c = next(q) 

except StopIteration: 

raise RuntimeError("There is no elliptic curve with coefficients " 

+ ainvs + " in the database") 

label = str(c[0]) 

N, iso, num = parse_cremona_label(label) 

data = {'cremona_label': label, 

'rank': c[1], 

'torsion_order': c[2], 

'conductor': N} 

if len(c) > 3: 

data['modular_degree'] = (c[3]) 

data['gens'] = eval(c[4]) 

data['db_extra'] = list(c[5:]) 

return data 

 

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 

 

Old (pre-2006) Cremona labels are also allowed:: 

 

sage: c.elliptic_curve('9450KKKK1') 

Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 5*x + 7 over Rational Field 

 

Make sure :trac:`12565` is fixed:: 

 

sage: c.elliptic_curve('10a1') 

Traceback (most recent call last): 

... 

ValueError: There is no elliptic curve with label 10a1 in the database 

""" 

data = self.data_from_coefficients(ainvs) 

return elliptic.EllipticCurve(ainvs, **data) 

 

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 or LMFDB label) 

 

OUTPUT: 

 

- an :class:`sage.schemes.elliptic_curves.ell_rational_field.EllipticCurve_rational_field` 

 

.. note:: 

 

For more details on LMFDB labels see :func:`parse_lmfdb_label`. 

 

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 

 

You can also use LMFDB labels:: 

 

sage: c.elliptic_curve('462.f3') 

Elliptic Curve defined by y^2 + x*y = x^3 - 363*x + 1305 over Rational Field 

""" 

ainvs, data = self.coefficients_and_data(label) 

return elliptic.EllipticCurve(ainvs, **data) 

 

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(key=sort_key) 

for k in K: 

v = A[k] 

# test if not first curve in class 

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__ 

#print "Computing largest conductor." 

q = self.__connection__.cursor().execute('SELECT conductor FROM ' \ 

+ 't_class ORDER BY conductor DESC LIMIT 1') 

self.__largest_conductor__ = next(q)[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__ = next(q)[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 next(q)[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 next(q)[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__ = next(q)[0] 

return self.__number_of_isogeny_classes__ 

q = self.__connection__.cursor().execute('SELECT COUNT(class) FROM ' \ 

+ 't_class WHERE conductor=?',(int(N),)) 

return next(q)[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 = next(q)[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)) 

 

############################################################################### 

# Functions for loading data from Cremona's ftpdata directory. 

############################################################################### 

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 

 

# Since July 2014 the data files have been arranged in 

# subdirectories (see trac #16903). 

allcurves_dir = os.path.join(ftpdata,'allcurves') 

allbsd_dir = os.path.join(ftpdata,'allbsd') 

allgens_dir = os.path.join(ftpdata,'allgens') 

degphi_dir = os.path.join(ftpdata,'degphi') 

num_curves, num_iso_classes = self._init_allcurves(allcurves_dir, largest_conductor) 

self.__number_of_curves__ = num_curves 

self.__number_of_isogeny_classes__ = num_iso_classes 

if hasattr(self, 'degphi'): 

self._init_degphi(degphi_dir, largest_conductor) 

if hasattr(self, 'allbsd'): 

self._init_allbsd(allbsd_dir, largest_conductor) 

if hasattr(self, 'allgens'): 

self._init_allgens(allgens_dir, 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: 

 

- `ftpdata` (string) -- the name of the directory in which the data is 

 

- ``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 = sorted(os.listdir(ftpdata)) 

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 open(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], 

'a2': [[0, -1, 1, -7820, -263580], 0, 1], 

'a3': [[0, -1, 1, 0, 0], 0, 5]} 

""" 

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' 

""" 

self.name = name 

name = name.replace(' ','_') 

db_path = os.path.join(SAGE_SHARE, 'cremona', name+'.db') 

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.') 

SQLDatabase.__init__(self, db_path, read_only=read_only, \ 

skeleton=_cremonaSkeleton) 

return 

if not os.path.isfile(db_path): 

raise ValueError("Desired database (='%s') does not "%self.name \ 

+ "exist") 

SQLDatabase.__init__(self, db_path, 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, 4.07927920046493, 0.453253244496104, 1.0, 1] 

sage: c.allbsd(12001)['a1'] # optional - database_cremona_ellcurve 

[2, 3.27608135248722, 1.54910143090506, 0.236425971187952, 1.0] 

""" 

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 = sorted(os.listdir(ftpdata)) 

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 open(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 = sorted(os.listdir(ftpdata)) 

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 open(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 = sorted(os.listdir(ftpdata)) 

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 open(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 

 

Verify that :trac:`12341` has been resolved:: 

 

sage: c = CremonaDatabase('should not exist',mini=True) 

Traceback (most recent call last): 

... 

ValueError: Desired database (='should not exist') does not exist 

sage: c = CremonaDatabase('should not exist',mini=False) 

Traceback (most recent call last): 

... 

ValueError: Desired database (='should not exist') does not exist 

sage: from sage.env import SAGE_SHARE 

sage: os.path.isfile(os.path.join(SAGE_SHARE,'cremona','should_not_exist.db')) 

False 

""" 

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 is_package_installed('database_cremona_ellcurve'): 

name = 'cremona' 

else: 

name = 'cremona mini' 

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)