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

Abstract

Let F be the cubic field of discriminant −23 and ${\mathcal{O}}_F$ its ring of integers. Let Γ be the arithmetic group ${GL}_2\left({\mathcal{O}}_F\right)$, and for every ideal $\mathfrak{n}\subset {\mathcal{O}}_F$, let ${\Gamma }_0\left(\mathfrak{n}\right)$ be the congruence subgroup of level $\mathfrak{n}$. In [Gunnells and Yasaki 13], the cohomology of various ${\Gamma }_0\left(\mathfrak{n}\right)$ 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.

2000 AMS Subject Classification::

• Primary 11F75
• Secondary 11F67, 11G05, 11Y99

Keywords:

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

Notes

¹We note that recent remarkable work in [Scholze 13] explains how to attach Galois representations to Hecke eigenclasses in the mod p and characteristic 0 cohomology of certain locally symmetric spaces. At present, the example we consider falls outside the scope of that work, since our field F is neither totally real nor CM.

²Available at http://www.lmfdb.org.

³We take this opportunity to point out an error in [Gunnells and Yasaki 13], in which we neglected to include the orientation module in our coefficients. None of the results there or here are affected by this oversight.

⁴The Hecke computations became impractical at these levels because of our implementation. With better code, we could undoubtedly treat some levels above norm 911, but even then, we do not expect to handle level norms above 5000.

⁵The given equation of the second curve is the canonical model, which is a global minimal model. The curve actually found using this method had the coefficients [a₁, a₂, a₃, a₄, a₆] = [16a² + 24a + 10, −1872a² − 152a + 952, −1872a² − 152a + 952, 0, 0].