Abstract. Let F be the cubic field of discriminant −23 and OF its ring of integers. Let 􀀀 be the arithmetic group GL2(OF ), and for any ideal n ⊂ OF let 􀀀0(n) be the congruence subgroup of level n. In [16], two of us (PG and DY) computed the cohomology of various 􀀀0(n), along with the action of the Hecke operators. The goal of [16] was to test the modularity of elliptic curves over F. In the present paper, we complement and extend the results of [16] in two ways. First, we tabulate more elliptic curves than were found in [16] 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.