Source of the curves

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 $10000$. Curves were found to match almost all the cuspidal Bianchi newforms (with trivial character, weight \(2\) and rational coefficients) in the database, using custom code by Cremona and Moore, supplemented by Magma's EllipticCurveSearch function written by Steve Donnelly; in the remaining cases it has been proved that there is no matching curve. 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: it is complete for fields of degree \({}<6\), but a few curves are still missing over 13 of the 55 sextic fields represented in the database. 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

• Steve Donnelly, Paul E. Gunnells, Ariah Klages-Mundt, and Dan Yasaki provided the curves over the mixed cubic field 3.1.23.1.
• Marc Masdeu provided $\Q$-curves over quadratic fields.
• Haluk Sengun provided curves with everywhere good reduction over imaginary quadratic fields.
• S. Yokoyama and Masdeu provided curves with everywhere good reduction over real quadratic fields (see here).

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.

Source of the data for each curve

Models

For each curve, the model in the database is a global minimal model where one exists (for example when the base field has class number one), and otherwise a semi-minimal model which is nonminimal at precisely one prime. These (semi-)minimal models we computed in SageMath using code written by John Cremona. Among all (semi-)minimal models we scale by units in order to minimize the archimedean embeddings of \(c_4\) and \(c_4\) simultaneously using a lattice basis reduction method, also implemented in SageMath by John Cremona.

Local data

Conductors and local reduction data were computed using Tate's Algorithm, as implemented in SageMath by John Cremona, except for the local root numbers which were computed using Magma

Isogenies

Complete isogeny classes were computed using implementations in SageMath by John Cremona and Ciaran Schembri of Billerey's algorithm (to determine the reducible primes) and Kohel-Vélu formulas.

Mordell-Weil groups and generators

Where these exist in the database they were computed by John Cremona using Magma's MordellWeilShaInformation function, implemented by Steve Donnelly

Galois representations

The images of mod-\(\ell\) Galois representations were computed by Andrew Sutherland using the methods described in [10.1017/fms.2015.33, arXiv:1504.0761] (including a generalization to handle curves with complex multiplication).

Base change and \(\mathbb{Q}\)-curves

The property of being the base change of an elliptic curve over \(\mathbb{Q}\) and of being a \(\mathbb{Q}\)-curve were computed by John Cremona using his implementation in SageMath.