Coverage for local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/gp_simon.py : 80%

""" 

Denis Simon's PARI scripts 

""" 

from __future__ import absolute_import 

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

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

# 

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

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

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

from sage.structure.parent_gens import localvars 

from sage.interfaces.gp import Gp 

from sage.misc.sage_eval import sage_eval 

from sage.misc.randstate import current_randstate 

from sage.rings.all import QQ, ZZ 

from .constructor import EllipticCurve 

gp = None 

def init(): 

""" 

Function to initialize the gp process 

""" 

global gp 

if gp is None: 

import os 

from sage.env import DOT_SAGE 

logfile = os.path.join(DOT_SAGE, 'gp-simon.log') 

gp = Gp(script_subdirectory='simon', logfile=logfile) 

gp.read("ellQ.gp") 

gp.read("ell.gp") 

gp.read("qfsolve.gp") 

gp.read("resultant3.gp") 

def simon_two_descent(E, verbose=0, lim1=None, lim3=None, limtriv=None, 

maxprob=20, limbigprime=30, known_points=[]): 

""" 

Interface to Simon's gp script for two-descent. 

.. NOTE:: 

Users should instead run E.simon_two_descent() 

EXAMPLES:: 

sage: import sage.schemes.elliptic_curves.gp_simon 

sage: E=EllipticCurve('389a1') 

sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E) 

(2, 2, [(1 : 0 : 1), (-11/9 : 28/27 : 1)]) 

TESTS:: 

sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x')) 

sage: E.simon_two_descent() 

(1, 1, [(0 : 0 : 1)]) 

An example with an elliptic curve defined over a relative number field:: 

sage: F.<a> = QuadraticField(29) 

sage: x = QQ['x'].gen() 

sage: K.<b> = F.extension(x^2-1/2*a+1/2) 

sage: E = EllipticCurve(K,[1, 0, 5/2*a + 27/2, 0, 0]) # long time (about 3 s) 

sage: E.simon_two_descent(lim1=2, limtriv=3) 

(1, 1, ...) 

Check that :trac:`16022` is fixed:: 

sage: K.<y> = NumberField(x^4 + x^2 - 7) 

sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0]) 

sage: E.simon_two_descent(lim1=2, limtriv=3) # long time (about 3 s) 

(1, 1, ...) 

An example that checks that :trac:`9322` is fixed (it should take less than a second to run):: 

sage: K.<w> = NumberField(x^2-x-232) 

sage: E = EllipticCurve([2-w,18+3*w,209+9*w,2581+175*w,852-55*w]) 

sage: E.simon_two_descent() 

(0, 2, []) 

""" 

init() 

current_randstate().set_seed_gp(gp) 

K = E.base_ring() 

K_orig = K 

# The following is to correct the bug at #5204: the gp script 

# fails when K is a number field whose generator is called 'x'. 

# It also deals with relative number fields. 

E_orig = E 

if not K is QQ: 

K = K_orig.absolute_field('a') 

from_K,to_K = K.structure() 

E = E_orig.change_ring(to_K) 

known_points = [P.change_ring(to_K) for P in known_points] 

# Simon's program requires that this name be y. 

with localvars(K.polynomial().parent(), 'y'): 

gp.eval("K = bnfinit(%s);" % K.polynomial()) 

if verbose >= 2: 

print("K = bnfinit(%s);" % K.polynomial()) 

gp.eval("%s = Mod(y,K.pol);" % K.gen()) 

if verbose >= 2: 

print("%s = Mod(y,K.pol);" % K.gen()) 

else: 

from_K = lambda x: x 

to_K = lambda x: x 

# The block below mimicks the defaults in Simon's scripts, and needs to be changed 

# when these are updated. 

if K is QQ: 

cmd = 'ellrank(%s, %s);' % (list(E.ainvs()), [P.__pari__() for P in known_points]) 

if lim1 is None: 

lim1 = 5 

if lim3 is None: 

lim3 = 50 

if limtriv is None: 

limtriv = 3 

else: 

cmd = 'bnfellrank(K, %s, %s);' % (list(E.ainvs()), [P.__pari__() for P in known_points]) 

if lim1 is None: 

lim1 = 2 

if lim3 is None: 

lim3 = 4 

if limtriv is None: 

limtriv = 2 

gp('DEBUGLEVEL_ell=%s; LIM1=%s; LIM3=%s; LIMTRIV=%s; MAXPROB=%s; LIMBIGPRIME=%s;'%( 

verbose, lim1, lim3, limtriv, maxprob, limbigprime)) 

if verbose >= 2: 

print(cmd) 

s = gp.eval('ans=%s;'%cmd) 

if s.find(" *** ") != -1: 

raise RuntimeError("\n%s\nAn error occurred while running Simon's 2-descent program"%s) 

if verbose > 0: 

print(s) 

v = gp.eval('ans') 

if v=='ans': # then the call to ellrank() or bnfellrank() failed 

raise RuntimeError("An error occurred while running Simon's 2-descent program") 

if verbose >= 2: 

print("v = %s" % v) 

# pari represents field elements as Mod(poly, defining-poly) 

# so this function will return the respective elements of K 

def _gp_mod(*args): 

return args[0] 

ans = sage_eval(v, {'Mod': _gp_mod, 'y': K.gen(0)}) 

lower = ZZ(ans[0]) 

upper = ZZ(ans[1]) 

points = [E_orig([from_K(c) for c in list(P)]) for P in ans[2]] 

points = [P for P in points if P.has_infinite_order()] 

return lower, upper, points