389a sd22 - modified

699 days ago by daniel.disegni

#The INPUTS are an elliptic curve over Q of rank 2 and a (small, if possible) odd prime p; optional: disc_bound=50; #koly_bound=500 #We choose a Heegner discriminant D, the corresponding quadratic field will be K, and we'll list Kolyvagin primes for D and ell #E(K) is isomorphic (up to 2-torsion) to the direct sum of E(Q) and ED(Q) #Then we need to find whether the reduction mod ell from E(K)/pE(K) to E(GF(ell^2))/pE(GF(ell^2) is surjective #and in case it is surjective, find a generator for its kernel (this is related to the Kolyvagin class \tau_ell) #This map is the direct sum (modulo some possible problems at 2) of the reductions of E and ED mod ell #First we study the part coming from the twist (a map of 1-dimensional F_p -vector spaces) #Then the part coming from E/QQ #WHAT ARE WE DOING (assuming Sha(E/K)[p]=0): the reduction map is surjective if and only if the modified Selmer group #H(ell)=H^1_{F(\ell)}(K,E[p]) has dimension 1. #(Otherwise, the dimension would be bigger as the dimension of the Selmer group H(1)=Sel(K,E[p]) is 3[=rank E/K]). # This is a necessary condition for having a nonvanishing Kolyvagin class \tau_ell #In this case, H(ell) is a line in Sel(K,E[p])^+=E(Q)/p (a Z/p-module of rank 2), and we identify this line. #FIXME: we still should check that the Galois action on E[p] has full image in GL(2,Z/p) (this should fail only rarely) E=EllipticCurve('389a'); p=3 disc_bound=50 koly_bound=500 gens=E.gens(); if len(gens)!=2: print 'The elliptic curve you chose does not have rank 2' P1=gens[0]; P2=gens[1]; heegner_discriminants_good=[D for D in E.heegner_discriminants(disc_bound) if GF(p)(D)!=0 and GF(p)(E.base_extend(QuadraticField(D, 'a')).torsion_order())!=0] for D in heegner_discriminants_good: K.<a>=QuadraticField(D); print 'Discriminant =', D ED=E.quadratic_twist(D) if len(ED.gens())>1: print 'Bad luck: the twist of E by your discriminant has too large rank (>1)' else: PD=ED.gens()[0] kolyprimes=[ell for ell in prime_range(koly_bound) if GF(E.conductor())(ell)!=0 and E.heegner_point(D,ell).satisfies_kolyvagin_hypothesis(p)] print 'List of Kolyvagin primes for',D, ':', kolyprimes for ell in kolyprimes: print ell EDred=ED.change_ring(GF(ell)) PDred=EDred(PD) if PDred.is_divisible_by(p): print 'The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_', ell,'=0' else: Ered=E.change_ring(GF(ell)) Ker=[] for i,j in GF(p)^2: if Ered(ZZ(i)*P1+ZZ(j)*P2).is_divisible_by(p): Ker.append((i,j)) if len(Ker)>p : print 'The part of the map coming from E/QQ is not surjective. Therefore tau_', ell,'=0' else: print 'The reduction mod', ell, 'map is surjective. The Kolyvagin class tau_', ell,'is a multiple of',Ker[1][0],'*',P1,'+',Ker[1][1],'*', P2, 'in E(Q)/',p,'E(Q).' print '' 
        
WARNING: Output truncated!  
full_output.txt



Discriminant = -4
List of Kolyvagin primes for -4 : [59, 83, 107, 227, 263, 347, 479, 491]
59
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 59 =0

83
The reduction mod 83 map is surjective. The Kolyvagin class tau_ 83 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

107
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 107 =0

227
The reduction mod 227 map is surjective. The Kolyvagin class tau_ 227 is
a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

263
The reduction mod 263 map is surjective. The Kolyvagin class tau_ 263 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

347
The reduction mod 347 map is surjective. The Kolyvagin class tau_ 347 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

479
The reduction mod 479 map is surjective. The Kolyvagin class tau_ 479 is
a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

491
The reduction mod 491 map is surjective. The Kolyvagin class tau_ 491 is
a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

Discriminant = -7
List of Kolyvagin primes for -7 : [5, 17, 41, 59, 83, 173, 227, 269,
479]
5
The reduction mod 5 map is surjective. The Kolyvagin class tau_ 5 is a
multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

17
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 17 =0

41
The reduction mod 41 map is surjective. The Kolyvagin class tau_ 41 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

59
The reduction mod 59 map is surjective. The Kolyvagin class tau_ 59 is a
multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

83
The reduction mod 83 map is surjective. The Kolyvagin class tau_ 83 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

173
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 173 =0

227
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 227 =0

269
The part of the map coming from E/QQ is not surjective. Therefore tau_
269 =0

479
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 479 =0

Discriminant = -11
List of Kolyvagin primes for -11 : [17, 29, 41, 83, 107, 173, 227, 233,
263, 281, 347, 479, 491]
17
The reduction mod 17 map is surjective. The Kolyvagin class tau_ 17 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

...

The reduction mod 449 map is surjective. The Kolyvagin class tau_ 449 is
a multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

Discriminant = -20
List of Kolyvagin primes for -20 : [17, 53, 59, 113, 173, 233, 479, 491]
17
The reduction mod 17 map is surjective. The Kolyvagin class tau_ 17 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

53
The reduction mod 53 map is surjective. The Kolyvagin class tau_ 53 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

59
The reduction mod 59 map is surjective. The Kolyvagin class tau_ 59 is a
multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

113
The reduction mod 113 map is surjective. The Kolyvagin class tau_ 113 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

173
The reduction mod 173 map is surjective. The Kolyvagin class tau_ 173 is
a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

233
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 233 =0

479
The reduction mod 479 map is surjective. The Kolyvagin class tau_ 479 is
a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

491
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 491 =0

Discriminant = -35
List of Kolyvagin primes for -35 : [41, 53, 59, 107, 113, 233, 263, 269,
347, 479]
41
The reduction mod 41 map is surjective. The Kolyvagin class tau_ 41 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

53
The reduction mod 53 map is surjective. The Kolyvagin class tau_ 53 is a
multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

59
The reduction mod 59 map is surjective. The Kolyvagin class tau_ 59 is a
multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

107
The reduction mod 107 map is surjective. The Kolyvagin class tau_ 107 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

113
The reduction mod 113 map is surjective. The Kolyvagin class tau_ 113 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

233
The reduction mod 233 map is surjective. The Kolyvagin class tau_ 233 is
a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

263
The reduction mod 263 map is surjective. The Kolyvagin class tau_ 263 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

269
The part of the map coming from E/QQ is not surjective. Therefore tau_
269 =0

347
The reduction mod 347 map is surjective. The Kolyvagin class tau_ 347 is
a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

479
The part of the reduction map coming from the twist is not surjective
(i.e., it is zero). Therefore tau_ 479 =0
WARNING: Output truncated!  
full_output.txt



Discriminant = -4
List of Kolyvagin primes for -4 : [59, 83, 107, 227, 263, 347, 479, 491]
59
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 59 =0

83
The reduction mod 83 map is surjective. The Kolyvagin class tau_ 83 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

107
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 107 =0

227
The reduction mod 227 map is surjective. The Kolyvagin class tau_ 227 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

263
The reduction mod 263 map is surjective. The Kolyvagin class tau_ 263 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

347
The reduction mod 347 map is surjective. The Kolyvagin class tau_ 347 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

479
The reduction mod 479 map is surjective. The Kolyvagin class tau_ 479 is a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

491
The reduction mod 491 map is surjective. The Kolyvagin class tau_ 491 is a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

Discriminant = -7
List of Kolyvagin primes for -7 : [5, 17, 41, 59, 83, 173, 227, 269, 479]
5
The reduction mod 5 map is surjective. The Kolyvagin class tau_ 5 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

17
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 17 =0

41
The reduction mod 41 map is surjective. The Kolyvagin class tau_ 41 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

59
The reduction mod 59 map is surjective. The Kolyvagin class tau_ 59 is a multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

83
The reduction mod 83 map is surjective. The Kolyvagin class tau_ 83 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

173
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 173 =0

227
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 227 =0

269
The part of the map coming from E/QQ is not surjective. Therefore tau_ 269 =0

479
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 479 =0

Discriminant = -11
List of Kolyvagin primes for -11 : [17, 29, 41, 83, 107, 173, 227, 233, 263, 281, 347, 479, 491]
17
The reduction mod 17 map is surjective. The Kolyvagin class tau_ 17 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

...

The reduction mod 449 map is surjective. The Kolyvagin class tau_ 449 is a multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

Discriminant = -20
List of Kolyvagin primes for -20 : [17, 53, 59, 113, 173, 233, 479, 491]
17
The reduction mod 17 map is surjective. The Kolyvagin class tau_ 17 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

53
The reduction mod 53 map is surjective. The Kolyvagin class tau_ 53 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

59
The reduction mod 59 map is surjective. The Kolyvagin class tau_ 59 is a multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

113
The reduction mod 113 map is surjective. The Kolyvagin class tau_ 113 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

173
The reduction mod 173 map is surjective. The Kolyvagin class tau_ 173 is a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

233
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 233 =0

479
The reduction mod 479 map is surjective. The Kolyvagin class tau_ 479 is a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

491
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 491 =0

Discriminant = -35
List of Kolyvagin primes for -35 : [41, 53, 59, 107, 113, 233, 263, 269, 347, 479]
41
The reduction mod 41 map is surjective. The Kolyvagin class tau_ 41 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

53
The reduction mod 53 map is surjective. The Kolyvagin class tau_ 53 is a multiple of 1 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

59
The reduction mod 59 map is surjective. The Kolyvagin class tau_ 59 is a multiple of 0 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

107
The reduction mod 107 map is surjective. The Kolyvagin class tau_ 107 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

113
The reduction mod 113 map is surjective. The Kolyvagin class tau_ 113 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

233
The reduction mod 233 map is surjective. The Kolyvagin class tau_ 233 is a multiple of 1 * (-1 : 1 : 1) + 0 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

263
The reduction mod 263 map is surjective. The Kolyvagin class tau_ 263 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

269
The part of the map coming from E/QQ is not surjective. Therefore tau_ 269 =0

347
The reduction mod 347 map is surjective. The Kolyvagin class tau_ 347 is a multiple of 2 * (-1 : 1 : 1) + 1 * (0 : -1 : 1) in E(Q)/ 3 E(Q).

479
The part of the reduction map coming from the twist is not surjective (i.e., it is zero). Therefore tau_ 479 =0