# 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} }

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

On non-square order Tate-Shafarevich groups of non-simple abelian surfaces over the rationals

- Mathematics
- 2014

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…

Adventures in Supersingularland

- Mathematics, Computer ScienceIACR 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

- MathematicsFound. Comput. Math.
- 2016

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

- Mathematics
- 2011

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 ScienceANTS
- 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

- MathematicsExp. Math.
- 2004

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

- Mathematics, Computer ScienceGraduate texts in mathematics
- 1986

It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.

Explicit Computations of Hilbert Modular Forms on ℚ(√5)

- MathematicsExp. Math.
- 2005

This article presents an algorithm to compute Hilbert modular forms on the quadratic field ℚ(√5) with prime level of norm less than 100 (up toℚ-isogeny).

Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes

- MathematicsExp. Math.
- 2007

An algorithm for determining elliptic curves defined over a given number field with a given set of primes of bad reduction over various quadratic fields is described.