The conversion from E to E.abelian_group() does not perserve group order for elliptic curves over finite fields.

[koffie]

One would expect that the following code is deterministic and prints 10 times the same:

sub_gens=[(542, 488, 1)]
p=709
for i in xrange(10):
    E2=EllipticCurve(GF(p),[606,486])
    ab=E2.abelian_group()
    sub_gens=[E2(i) for i in sub_gens]
    index=ab.order()/ab.submodule([ab(i) for i in sub_gens]).order()
    sub=ab.submodule([ab(i) for i in sub_gens])
    print index,[(ab(i),i.order(),ab(i).order()) for i in sub_gens]

However the output is:

6 [((542 : 488 : 1), 57, 114)]
228 [((542 : 488 : 1), 57, 3)]
6 [((542 : 488 : 1), 57, 114)]
3 [((542 : 488 : 1), 57, 228)]
3 [((542 : 488 : 1), 57, 228)]
6 [((542 : 488 : 1), 57, 114)]
6 [((542 : 488 : 1), 57, 114)]
3 [((542 : 488 : 1), 57, 228)]
228 [((542 : 488 : 1), 57, 3)]
6 [((542 : 488 : 1), 57, 114)]

Component: group theory

Reviewer: Jeroen Demeyer

Issue created by migration from https://trac.sagemath.org/ticket/13757

[koffie]

comment:1

This weird problem seems to be caused by the following

vec=(0,2)
ab_elt=ab(vec)
for i in xrange(10):
    print vec,ab_elt
    vec=ab_elt.vector()
    ab_elt=ab(vec)

[koffie]

comment:2

This outputs:

(0, 2) (665 : 37 : 1)
(0, 2) (11 : 229 : 1)
(2, 0) (665 : 37 : 1)
(0, 2) (11 : 229 : 1)
(2, 0) (665 : 37 : 1)
(0, 2) (11 : 229 : 1)
(2, 0) (665 : 37 : 1)
(0, 2) (11 : 229 : 1)
(2, 0) (665 : 37 : 1)
(0, 2) (11 : 229 : 1)

[koffie]

comment:4

I've made this critical since this bug affect the initialisation of all things inheriting from FGP_Module and silently produces completely wrong answers.

[fchapoton]

comment:5

trying to look at the problem, here is part of the problem maybe:

sage: p=709
sage: E2=EllipticCurve(GF(p),[606,486])
sage: ab=E2.abelian_group()
sage: ab((2,0))
(11 : 229 : 1)
sage: ab(vector((2,0)))
(409 : 94 : 1)
sage: ab((0,2))
(409 : 94 : 1)
sage: ab(vector((0,2)))
(11 : 229 : 1)

so there is a strange transposition of the arguments.

But there is also something random happening in the choice of the embedding:

sage: ab=EllipticCurve(GF(p),[606,486]).abelian_group()
sage: ab(vector((2,0)))
(534 : 340 : 1)
sage: ab=EllipticCurve(GF(p),[606,486]).abelian_group()
sage: ab(vector((2,0)))
(350 : 542 : 1)

[fchapoton]

comment:10

This seems to have been solve by #16261

[jdemeyer]

Reviewer: Jeroen Demeyer