A Database of Elliptic Curves over Q(sqrt(5)) - First Report
Jonathan Bober, Alyson Deines, Ariah Klages-Mundt, Benjamin LeVeque,, R. Andrew Ohana, Ashwath Rabindranath, Paul Sharaba, William Stein

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.
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Taxonomy
TopicsCryptography and Residue Arithmetic · Coding theory and cryptography · Algebraic Geometry and Number Theory
