#11346: major bug in the conductor function for elliptic curves over number
fields
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Reporter: was | Owner: cremona
Type: defect | Status: new
Priority: critical | Milestone: sage-4.7.1
Component: elliptic curves | Keywords:
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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Joanna Gaski found a serious bug in the function for computing conductors
of elliptic curves over number fields, when the input curve is not
integral. Witness:
{{{
sage: K.<g> = NumberField(x^2 - x - 1) #, embedding=1.6)
sage: E1 = EllipticCurve(K,[0,0,0,-1/48,-161/864]); E1
Elliptic Curve defined by y^2 = x^3 + (-1/48)*x + (-161/864) over Number
Field in g with defining polynomial x^2 - x - 1
sage: factor(E1.conductor())
(Fractional ideal (3)) * (Fractional ideal (-2*g + 1))
sage: factor(E1.integral_model().conductor())
(Fractional ideal (2))^4 * (Fractional ideal (3)) * (Fractional ideal
(-2*g + 1))
}}}
The bug is actually in the local_data() function, which computes the
possible primes of bad reduction by taking the support of the
discriminant. However, this is simply wrong if the input curve is not
integral.
{{{
sage: E1.discriminant().support()
[Fractional ideal (-2*g + 1), Fractional ideal (3)]
sage: E1.integral_model().discriminant().support()
[Fractional ideal (-2*g + 1), Fractional ideal (2), Fractional ideal (3)]
}}}
The one-line fix is to first compute an integral model, then ask for the
discriminant of that model in the local_data function.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11346>
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