global_minimal_model function is sometimes wrong over number fields, when input model isn't integral.

[williamstein]

The discriminant and conductor of a global minimal model must be divisible by the same primes. However the following code (extracted from examples computed by Joanna Gaski), illustrates the Sage global_minimal_model function producing a model that can't possibly be a global minimal model (since the conductor and discriminant are divisible by different primes).

sage: K.<g> = NumberField(x^2 - x - 1)
sage: E = EllipticCurve(K,[0,0,0,-1/48,161/864]).global_minimal_model(); E
Elliptic Curve defined by y^2 = x^3 + (-1)*x^2 + 12 over Number Field in g with defining polynomial x^2 - x - 1
sage: E.conductor().factor()
(Fractional ideal (3)) * (Fractional ideal (-2*g + 1))
sage: E.discriminant().factor()
(-1) * 2^12 * 3 * (-2*g + 1)^2

Again, the bug is that the global_minimal_model function is assuming that its input is integral, and the fix is easy, probably.

sage: E = EllipticCurve(K,[0,0,0,-1/48,161/864]).integral_model().global_minimal_model(); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 over Number Field in g with defining polynomial x^2 - x - 1
sage: E.conductor().factor()
(Fractional ideal (3)) * (Fractional ideal (-2*g + 1))
sage: E.discriminant().factor()
(-1) * 3 * (-2*g + 1)^2

Yes, inspecting the source code shows a typo related to this, i.e., somebody defines E to be a global integral model, then forgets to actually use E!

Component: elliptic curves

Author: William Stein

Reviewer: John Cremona

Merged: sage-4.7.1.alpha2

Issue created by migration from https://trac.sagemath.org/ticket/11347

[williamstein]

comment:3

Attachment: trac_11347.patch.gz

[JohnCremona]

comment:4

Patch looks good to me -- the typo could well have been mine...

Testing now.

[JohnCremona]

Reviewer: John Cremona

[JohnCremona]

comment:5

Applies fine to 4.7.rc1, tests pass.

[JohnCremona]

Author: William Stein

[JohnCremona]

comment:6

See the comment (#4) at #11346.

[jdemeyer]

Merged: sage-4.7.1.alpha2