A Database of Elliptic Curves over Q(sqrt(5)) - First Report

TL;DR

This paper presents a comprehensive tabulation of modular elliptic curves over Q(sqrt(5)), including computational methods and conjectural classifications up to a certain conductor norm, advancing the understanding of elliptic curves over this field.

Contribution

It introduces an efficient algorithm implementation and provides the first extensive database of elliptic curves over Q(sqrt(5)) with conjectural completeness up to a specified conductor.

Findings

Computed tables of Hilbert modular forms over Q(sqrt(5))

Constructed elliptic curves corresponding to these forms

Conjecturally classified all elliptic curves with conductor norm ≤ 1831

Abstract

We describe a tabulation of (conjecturally) modular elliptic curves over the field Q(sqrt(5)) up to the first curve of rank 2. Using an efficient implementation of an algorithm of Lassina Dembele, we computed tables of Hilbert modular forms of weight (2,2) over Q(sqrt(5)), and via a variety of methods we constructed corresponding elliptic curves, including (again, conjecturally) all elliptic curves over Q(sqrt(5)) that have conductor with norm less than or equal to 1831.