A Table of Elliptic Curves over the Cubic Field of Discriminant-23

作者：Donnelly Steve; Gunnells Paul E^*; Klages Mundt Ariah; Yasaki Dan

来源：Experimental Mathematics, 2015, 24(4): 375-390.

DOI：10.1080/10586458.2014.1002141

• Elliptic curves over number fields
• modular symbols
• cohomology of arithmetic groups
• Cremona-Lingham algorithm

Let F be the cubic field of discriminant -23 and O-F its ring of integers. Let Gamma be the arithmetic group GL(2)(O-F), and for every ideal n subset of O-F, let Gamma(0)(n) be the congruence subgroup of level n. In [Gunnells and Yasaki 13], the cohomology of various Gamma(0)(n) was computed, along with the action of the Hecke operators. The goal of that work was to test the modularity of elliptic curves over F. In the present article, we complement and extend those results in two ways. First, we tabulate more elliptic curves than were found earlier using various heuristics ("old and new" cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then 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.

• 出版日期2015