def simon_two_descent(E, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30):
"""
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 : -55/27 : 1)]]
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)])
TESTS::
sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x'))
sage: E.simon_two_descent()
(1, 1, [(-1 : 0 : 1)])
"""
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'.
if not K is QQ:
K = K.change_names('a')
E_orig = E
E = EllipticCurve(K,[K(list(a)) for a in E.ainvs()])
F = E.integral_model()
if K != QQ:
# 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()
if K == QQ:
cmd = 'ellrank([%s,%s,%s,%s,%s]);' % F.ainvs()
else:
cmd = 'bnfellrank(K, [%s,%s,%s,%s,%s]);' % F.ainvs()
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
return 'fail'
if verbose >= 2:
print "v = ", 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)})
inv_transform = F.isomorphism_to(E)
ans[2] = [inv_transform(F(P)) for P in ans[2]]
ans[2] = [E_orig([K_orig(list(c)) for c in list(P)]) for P in ans[2]]
return ans
