A table of elliptic curves over the cubic field of discriminant -23

TL;DR

This paper extends previous work by systematically tabulating elliptic curves over the cubic field with discriminant -23, using advanced heuristics and algorithms to find new curves and compute their invariants, including ranks and isogeny classes.

Contribution

It provides the first comprehensive table of elliptic curves over this specific cubic field, including conjectural ranks 1 and 2, and introduces improved search techniques for such curves.

Findings

Discovered elliptic curves of ranks 1 and 2 over the field.

Extended the database of elliptic curves beyond previous work.

Computed invariants like rank and isogeny classes for these curves.

Abstract

Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics ("old and new" cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then by using more sophisticated search techniques (for instance, torsion subgroups, twisting, and the Cremona-Lingham algorithm) to find them. We then compute further invariants of these curves, such as their rank and representatives of all isogeny classes. Our enumeration includes conjecturally the first elliptic curves of ranks 1 and 2 over this field, which occur at levels of norm 719 and 9173 respectively.