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The elliptic curves over number fields other than $\Q$ come from several sources.

Imaginary quadratic fields

The curves defined over the five Euclidean Imaginary quadratic fields consist of the curves in John Cremona's 1981 thesis, extended by him with Warren Moore in 2014 to conductor norm $10000$. In almost all cases, curves were found to match cuspidal Bianchi newforms, using custom code by Cremona and Moore, supplemented by Magma's EllipticCurveSearch function written by Steve Donnelly. Additionally, curves with CM by the field in question, which are not associated to cuspidal Bianchi newforms, were found from their $j$-invariants by Cremona.

Totally real fields

Over $\Q(\sqrt{5})$ the curves of conductor norm up to about $5000$ were provided by Alyson Deines from joint work of Jonathan Bober, Alyson Deines, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana, Ashwath Rabindranath, Paul Sharaba and William Stein (see http://arxiv.org/abs/1202.6612). All the other curves were found from their associated Hilbert newforms using Magma's EllipticCurveSearch function, using a script written by John Cremona. This process is still ongoing, and in particular the data for fields of degree $6$ is limited. Otherwise the extent of the data matches that of the Hilbert Modular Form data for totally real fields of degrees 2, 3, 4, 5 and 6.

Other fields

In some cases the same curve occurs in more than one of the above sources, in which case efforts have been made not to include it more than once in the database.