A Database of Elliptic Curves---First Report
This paper appeared in the ANTS V proceedings.
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Introduction
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.