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
