Vol. 1, No. 1, 2013

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A database of elliptic curves over ${\mathbb Q}(\sqrt{5})$: a first report

Jonathan Bober, Alyson Deines, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana, Ashwath Rabindranath, Paul Sharaba and William Stein

Vol. 1 (2013), No. 1, 145–166
Abstract

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

Keywords
elliptic curves, totally real number fields, Hilbert modular forms, tables, sage
Mathematical Subject Classification 2010
Primary: 11-04
Secondary: 11G05
Milestones
Accepted: 13 May 2013
Published: 14 November 2013
Authors
Jonathan Bober
Department of Mathematics
University of Washington
Seattle, WA 98195-4350
United States
Howard House
University of Bristol
Queens Avenue
Bristol
BS8 1SN
United Kingdom
http://sage.math.washington.edu/home/bober/www/
Alyson Deines
Department of Mathematics
University of Washington
Amherst, MA 01002
United States
http://www.math.washington.edu/~adeines/Alyson_Deines/Welcome.html
Ariah Klages-Mundt
Department of Mathematics
Amherst College
1555 Keefe Campus Center
Amherst, MA 01002
United States
http://https://www.amherst.edu/users/K/aklagesmundt12
Benjamin LeVeque
Mathematics Department
Brown University
Providence, RI 02906
United States
R. Andrew Ohana
Department of Mathematics
University of Washington
Seattle, WA 98195
United States
Ashwath Rabindranath
Department of Mathematics
University of Michigan
2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States
Paul Sharaba
Department of Mathematics
Cleveland State University
Cleveland, OH 44115
United States
William Stein
Department of Mathematics
University of Washington
423 Padelford Hall
Seattle, WA 98195-4361
United States
http://www.williamstein.org