major bug in the conductor function for elliptic curves over number fields
|Reported by:||was||Owned by:||cremona|
|Authors:||William Stein||Reviewers:||Robert Bradshaw|
|Report Upstream:||N/A||Work issues:|
Description (last modified by was)
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) 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.