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11347 global_minimal_model function is sometimes wrong over number fields, when input model isn't integral. William Stein John Cremona " 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. = 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!" defect closed critical sage-4.7.1 elliptic curves fixed sage-4.7.1.alpha2 William Stein John Cremona N/A