A Database of Elliptic Curves---First Report

William A. Stein

Mark Watkins

This paper appeared in the ANTS V proceedings.
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In the late 1980s, Brumer and McGuinness cite{BMCG} undertook the construction of a database of elliptic curves whose discriminant Delta was both prime and satisfied |Delta|<= 10^8. 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 |Delta|<= 10^{12} which either have conductor smaller than 10^8 or have prime conductor less than 10^10---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 10^8 as possible, and we comment on this direction of growth of the database below. 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.

Our general frame of mind is that computation of many of the invariants is rather trivial, for instance, the discriminant, conductor, and even the isogeny structure. We do not even save these data, expecting them to be recomputable quite easily in real time. For instance, for each isogeny class, we store only one representative (the one of minimal Faltings height), as we view the construction of isogenous curves as a ``fast'' process. It is only information like analytic ranks, modular degrees (both of which use computation of the Frobenius traces a_p), and coordinates of generators that we save; saving the a_p would take too much storage space. It might be seen that our database could be used a ``seed'' for other more specialised databases, as we can quickly calculate the less time-consuming information and append it to the saved data.