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R1.<x> = PolynomialRing(QQ)
R2.<x,y> = PolynomialRing(QQ)
def extended_field( f, A, B, deg_mult ):
# f: a polynomial whose roots are the x-coordinates of some points of an
# elliptic curve E: y^2 = x^3 + Ax + B.
# return value: a field containing the x and y coordinates of those points
# deg_mult: a positive integer known to be a multiple of the degree of the
# extended field.
#
# This function uses the properties of resultants (I can provide a pdf
# explaining how it works).
#
# It is useful to compute, e.g., the fields obtained by adjoining the
# coordinates of the n-division points of a point (using the n-uplication
# formulas to get the required polynomials).
#
# When used to compute the 2-division fields, it gives the same output as
# E.division_field(2).
g = y^2 - x^3 - A*x - B
res = f.resultant(g,x)
res = res.subs(y=x)
#K.<b> = f.splitting_field()
#print aux
pol = pari(R1(res*f)).nfsplitting(deg_mult)
K.<b> = NumberField(g)
return K
L = []
for A in range(1,9):
for B in range(1,9):
print "Current list of examples:", len(L), "elements. List:"
print L
E = EllipticCurve([0,0,0,A,B])
print "*************************"
print E
print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
print "CM:", E.has_cm()
rep = E.galois_representation()
print "mod 2 rep is surjective:", rep.is_surjective(2)
if E.rank() == 0:
print "Stopping because rank 0"
print ""
continue
if len(E.gens()) == 0:
print "Stopping because no points of infinite order found"
print ""
continue
if E.has_cm():
print "Stopping because CM"
print ""
print ""
continue
if not rep.is_surjective(2):
print "Stopping because mod 2 rep is not surjective"
print ""
continue
K_2.<a> = E.division_field(2)
print "2-division field:", K_2
P = E(0)
flag = False
for P in E.gens():
if len(P.division_points(2)) == 0:
flag = True
break
if not flag:
print "Stopping because the points found are 2-divisible"
print ""
continue
print "Taking the 2-division of P =", P
# The following polynomial is derived from the duplication formula
# (Silverman, p.54) using the x-coordinate of the 2-division point as
# an indeterminate.x
f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
#print f_P.roots()
M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
print "2-division field of P:", M
if M.degree() != 24:
print "Stopping because 2-division of P is too small"
print ""
continue
div_pol_4 = E.division_polynomial(4)
#if div_pol_4.splitting_field('zz').degree() < 48:
# print "Stopping because splitting field of div_pol_4 < 48"
# exit()
K_4 = extended_field( div_pol_4, A, B, 96 )
print "4-division field:", K_4
if K_4.degree() != 96:
print "Stopping because mod 4 representation not surjective"
print ""
continue
if len(f_P.roots(ring=K_4)) == 0:
print "Stopping because M is not contained in K_4"
print ""
continue
print "----------------"
print "|Example Found!|"
print "----------------"
L.append((A,B))
print "*************************"
print ""
|
