Opened 11 years ago

Closed 11 years ago

# major bug in the conductor function for elliptic curves over number fields

Reported by: Owned by: William Stein John Cremona critical sage-4.7.1 elliptic curves sage-4.7.1.alpha2 William Stein Robert Bradshaw N/A

### Description (last modified by William Stein)

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.

### comment:1 Changed 11 years ago by William Stein

Description: modified (diff)

### comment:2 Changed 11 years ago by William Stein

Status: new → needs_review

### comment:3 Changed 11 years ago by Robert Bradshaw

Status: needs_review → positive_review

### comment:4 Changed 11 years ago by John Cremona

Authors: → William Stein → Robert Bradshaw

Additional comment on this and #11347. I think a better fix would have been to add one line in the constructor for `EllipticCurveLocalData` in sage.schemes.elliptic_curves.ell_local_data, namely

```    E = E.integral_model()
```

rather than having a lot of separate functions have to remember to do this. The fixes here and at #11347 are fine by themselves, but it remains the case that

```sage: K.<g> = NumberField(x^2 - x - 1)
sage: E = EllipticCurve(K,[0,0,0,-1/48,161/864])
sage: E.local_data()
```

fails.

I don't have time to rework this right now, so have not tagged this "needs work"...