# A database of elliptic curves over Q(sqrt 5): a first report

@article{Bober2013ADO,
title={A database of elliptic curves over Q(sqrt 5): a first report},
author={Jonathan W. Bober and Alyson Deines and Ariah Klages-Mundt and Benjamin L{\'e}v{\^e}que and R. Andrew Ohana and Ashwath Rabindranath and Paul Sharaba and William A. Stein},
journal={arXiv: Number Theory},
year={2013}
}
• Published 29 February 2012
• Mathematics, Computer Science
• arXiv: Number Theory
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.
8 Citations

## Tables from this paper

On non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals
The study of rational points on an abelian variety A over a number field K gives rise to its Tate-Shafarevich group X(A/K), which plays an important role in understanding the arithmetic of A/K. The
Computing Power Series Expansions of Modular Forms
• Mathematics
• 2014
We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation of a fundamental domain and linear algebra. As
• Mathematics, Computer Science
IACR Cryptol. ePrint Arch.
• 2019
This paper considers four aspects of supersingular isogeny graphs, study each thoroughly and, where appropriate, discuss how they relate to one another, and provides an analysis of the distances of connected components of $\mathcal{S}$.
On some implementations of modular forms and related topics(Survey, Activity Group "Algorithmic Number Theory and Its Applications")
• Mathematics
• 2015
A recent progress of the implementation of Siegel modular forms that is an important generalization of elliptic modular forms is focused on.
The L-Functions and Modular Forms Database Project
In the lecture, I gave a very brief introduction to L-functions for non-experts and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through the website www.lmfdb.org.
A database of Hilbert modular forms
• Mathematics
• 2016
We describe the computation of tables of Hilbert modular forms of parallel weight 2 over totally real fields.

## References

SHOWING 1-10 OF 32 REFERENCES
On the modularity of elliptic curves over Q
• Mathematics
• 1999
In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular.
On families of n-congruent elliptic curves
We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of
Torsion groups of elliptic curves over quadratic fields
• Mathematics
• 2011
We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear
A Database of Elliptic Curves - First Report
• Computer Science
ANTS
• 2002
The end goal is to find as many curves with conductor less than 108 as possible, and a first stage of processing is started (computation of analytic rank data), with point searching to be carried out in a later second stage of computation.
Periods and Points Attached to Quadratic Algebras
• Mathematics
• 2004
Φ : H/Γ0(N) −→ E(C), (0–1) where H is the Poincare upper half-plane and Γ0(N) is Hecke’s congruence group of level N . Fix a quadratic field K; when it is imaginary, the theory of complex
Computing Special Values of Motivic L-Functions
An algorithm to compute values L(s) and derivatives L (k) (S) of L-functions of motivic origin numerically to required accuracy to apply to any L-series whose Γ-factor is of the form AS with d arbitrary and complex λ j.
Rational isogenies of prime degree
• Mathematics
• 1978
In this table, g is the genus of Xo(N), and v the number of noncuspidal rational points of Xo(N) (which is, in effect, the number of rational N-isogenies classified up to "twist"). For an excellent
The arithmetic of elliptic curves
• J. Silverman
• Mathematics, Computer Science