Coverage for local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ec_database.py : 76%

r""" 

Tables of elliptic curves of given rank 

The default database of curves contains the following data: 

+------+------------------+--------------------+ 

| Rank | Number of curves | Maximal conductor | 

+======+==================+====================+ 

| 0 | 30427 | 9999 | 

+------+------------------+--------------------+ 

| 1 | 31871 | 9999 | 

+------+------------------+--------------------+ 

| 2 | 2388 | 9999 | 

+------+------------------+--------------------+ 

| 3 | 836 | 119888 | 

+------+------------------+--------------------+ 

| 4 | 10 | 1175648 | 

+------+------------------+--------------------+ 

| 5 | 5 | 37396136 | 

+------+------------------+--------------------+ 

| 6 | 5 | 6663562874 | 

+------+------------------+--------------------+ 

| 7 | 5 | 896913586322 | 

+------+------------------+--------------------+ 

| 8 | 6 | 457532830151317 | 

+------+------------------+--------------------+ 

| 9 | 7 | ~9.612839e+21 | 

+------+------------------+--------------------+ 

| 10 | 6 | ~1.971057e+21 | 

+------+------------------+--------------------+ 

| 11 | 6 | ~1.803406e+24 | 

+------+------------------+--------------------+ 

| 12 | 1 | ~2.696017e+29 | 

+------+------------------+--------------------+ 

| 14 | 1 | ~3.627533e+37 | 

+------+------------------+--------------------+ 

| 15 | 1 | ~1.640078e+56 | 

+------+------------------+--------------------+ 

| 17 | 1 | ~2.750021e+56 | 

+------+------------------+--------------------+ 

| 19 | 1 | ~1.373776e+65 | 

+------+------------------+--------------------+ 

| 20 | 1 | ~7.381324e+73 | 

+------+------------------+--------------------+ 

| 21 | 1 | ~2.611208e+85 | 

+------+------------------+--------------------+ 

| 22 | 1 | ~2.272064e+79 | 

+------+------------------+--------------------+ 

| 23 | 1 | ~1.139647e+89 | 

+------+------------------+--------------------+ 

| 24 | 1 | ~3.257638e+95 | 

+------+------------------+--------------------+ 

| 28 | 1 | ~3.455601e+141 | 

+------+------------------+--------------------+ 

Note that lists for r>=4 are not exhaustive; there may well be curves 

of the given rank with conductor less than the listed maximal conductor, 

which are not included in the tables. 

AUTHORS: 

- William Stein (2007-10-07): initial version 

- Simon Spicer (2014-10-24): Added examples of more high-rank curves 

See also the functions cremona_curves() and cremona_optimal_curves() 

which enable easy looping through the Cremona elliptic curve database. 

""" 

from __future__ import absolute_import 

import os 

from ast import literal_eval 

from .constructor import EllipticCurve 

class EllipticCurves: 

def rank(self, rank, tors=0, n=10, labels=False): 

r""" 

Return a list of at most `n` non-isogenous curves with given 

rank and torsion order. 

INPUT: 

- ``rank`` (int) -- the desired rank 

- ``tors`` (int, default 0) -- the desired torsion order (ignored if 0) 

- ``n`` (int, default 10) -- the maximum number of curves returned. 

- ``labels`` (bool, default False) -- if True, return Cremona 

labels instead of curves. 

OUTPUT: 

(list) A list at most `n` of elliptic curves of required rank. 

EXAMPLES:: 

sage: elliptic_curves.rank(n=5, rank=3, tors=2, labels=True) 

['59450i1', '59450i2', '61376c1', '61376c2', '65481c1'] 

:: 

sage: elliptic_curves.rank(n=5, rank=0, tors=5, labels=True) 

['11a1', '11a3', '38b1', '50b1', '50b2'] 

:: 

sage: elliptic_curves.rank(n=5, rank=1, tors=7, labels=True) 

['574i1', '4730k1', '6378c1'] 

:: 

sage: e = elliptic_curves.rank(6)[0]; e.ainvs(), e.conductor() 

((1, 1, 0, -2582, 48720), 5187563742) 

sage: e = elliptic_curves.rank(7)[0]; e.ainvs(), e.conductor() 

((0, 0, 0, -10012, 346900), 382623908456) 

sage: e = elliptic_curves.rank(8)[0]; e.ainvs(), e.conductor() 

((1, -1, 0, -106384, 13075804), 249649566346838) 

For large conductors, the labels are not known:: 

sage: L = elliptic_curves.rank(6, n=3); L 

[Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2582*x + 48720 over Rational Field, 

Elliptic Curve defined by y^2 + y = x^3 - 7077*x + 235516 over Rational Field, 

Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2326*x + 43456 over Rational Field] 

sage: L[0].cremona_label() 

Traceback (most recent call last): 

... 

LookupError: Cremona database does not contain entry for Elliptic Curve defined by y^2 + x*y = x^3 + x^2 - 2582*x + 48720 over Rational Field 

sage: elliptic_curves.rank(6, n=3, labels=True) 

[] 

""" 

from sage.env import ELLCURVE_DATA_DIR 

data = os.path.join(ELLCURVE_DATA_DIR, 'rank%s'%rank) 

try: 

f = open(data) 

except IOError: 

return [] 

v = [] 

tors = int(tors) 

for w in f.readlines(): 

N, iso, num, ainvs, r, t = w.split() 

N = int(N) 

t = int(t) 

if tors and tors != t: 

continue 

# Labels are only known to be correct for small conductors. 

# NOTE: only change this bound below after checking/fixing 

# the Cremona labels in the elliptic_curves package! 

if N <= 400000: 

label = '%s%s%s'%(N, iso, num) 

else: 

label = None 

if labels: 

if label is not None: 

v.append(label) 

else: 

E = EllipticCurve(literal_eval(ainvs)) 

E._set_rank(r) 

E._set_torsion_order(t) 

E._set_conductor(N) 

if label is not None: 

E._set_cremona_label(label) 

v.append(E) 

if len(v) >= n: 

break 

return v 

elliptic_curves = EllipticCurves()