Extent of elliptic curve data over number fields (reviewed)

The database currently contains 725,187 elliptic curves in 350,509 isogeny classes, over 423 number fields of degree 2 to 6. Elliptic curves defined over $\mathbb{Q}$ are contained in a separate database.

Imaginary quadratic fields

Most of the elliptic curves in the database defined over number fields are defined over the five Euclidean imaginary quadratic fields fields, with discriminants $-3$, $-4$, $-7$, $-8$, $-11$. Over each of these fields the database contains elliptic curves of conductor norm bounded by (respectively) 150000, 100000, 50000, 50000, 50000. Within these bounds the database contains all modular elliptic curves; however, modularity has not yet been proved for elliptic curves over imaginary quadratic fields in general, and non-modular curves (if any exist) are not in the database. Assuming modularity, the database is complete within these bounds, by comparison with the database of Bianchi modular forms.

In addition the database contains a small number of elliptic curves with everywhere good reduction defined over some other imaginary quadratic fields.

Totally real fields

The database contains elliptic curves defined over totally real fields of degree \(2, 3, 4, 5\) and \(6\). For each field the database contains curves of conductor norm up to a bound which depends on the field. Over totally real fields of degree \(2\) and \(3\), it is known that all elliptic curves are modular, and hence the database is complete for these fields, by comparison with the database of Hilbert Modular Forms. In higher degrees the database contains elliptic curves matching each of the Hilbert Modular Forms (of parallel weight \(2\), trivial character and rational coefficients) over the field in question, for fields of degree \(2,3,4\) and \(5\), and hence contains all modular elliptic curves of conductor norm up to the bound for that field. Over 21 of the 34 fields of degree 6, the database contains curves matching all of the relevant Hilbert Modular Forms; over the other 13 fields there are a total of 55 isogeny classes missing.

Other fields

The only other field for which the database contains elliptic curves is the mixed signature cubic field 3.1.23.1. The elliptic curves here have conductor norm up to \(20000\) and match automorphic forms over this field. No results on modularity are known for this field, but the curves in the database include all modular curves within these bounds.

For the complete list of fields over which the database contains elliptic curves, the numbers of curves and isogeny classes, and the bounds on the conductor norms in each case, see this page.