A Database of Elliptic Curves — First Report

Abstract

In the late 1980s, Brumer and McGuinness [2] undertook the construction of a database of elliptic curves whose absolute discriminant |Δ| was both prime and satisfied |Δ| ≤ 10⁸. While the restriction to primality was nice for many reasons, there are still many curves of interest lacking this property. As ten years have passed since the original experiment, we decided to undertake an extension of it, simultaneously extending the range for the type of curves they considered, and also including curves with composite discriminant. Our database can be crudely described as being the curves with |Δ| ≤ 10¹² which either have conductor smaller than 10⁸ or have prime conductor less than 10¹⁰—but there are a few caveats concerning issues like quadratic twists and isogenous curves. For each curve in our database, we have undertaken to compute various invariants (as did Brumer and McGuinness), such as the Birch—Swinnerton-Dyer L-ratio, generators, and the modular degree. We did not compute the latter two of these for every curve. The database currently contains about 44 million curves; the end goal is find as many curves with conductor less than 10⁸ as possible, and we comment below on this direction of growth of the database. Of these 44 million curves, we have started a first stage of processing (computation of analytic rank data), with point searching to be carried out in a later second stage of computation.