Elliptic Curve Data
by J. E. Cremona
University of Warwick, U.K.

Updated 2014-08-29 (last major update 2014-05-12)


UK site
US mirror (hosted by William Stein)

This site contains various data files concerning modular elliptic curves, in a standard format to make them easily readable by other programs. For a typeset version of the same data (with some extra data about local reduction data) for conductors up to 1000, you can refer to the book Algorithms for modular elliptic curves , CUP 1992, second revised edition 1997. See the book's web site for more information, including errata for the current (2nd) edition, and errata to the first edition (not maintained since the appearance of the second edition). The errata lists include errors and omissions in the tables. The files here have the corrected data in them.

Note: As of 2000 the book is out of print, and CUP have no plans to reprint it.

The files correspond to tables 1-5 in the book (Table 5 is not in the First Edition), with additional tables: Table 6 gives the isogeny matrices between curves in each isogeny class, and Table 7 lists the integral points on each curve. They are compressed with gzip, which adds the suffix ".gz" to the filename.; You may need to uncompress after transfer using gunzip, or your browser might uncompress the files automatically for you to view them.

The tables currently contain data for conductors up to 350000.

From September 2005: New labelling scheme introduced for isogeny classes. The old scheme started A,B,...,Z,AA,BB,...,ZZ,AAA,BBB,... and had become unwieldy. The new scheme is a straight base 26 encoding with a=0, b=1 etc., with the classes numbered from 0 amd leading a's deleted: a,b,...,z,ba,bb,...bz,ca,cb,... . The change to lower case is to make codes such as bb unambiguous between the old and new systems. For conductors less than 1728 the number of isogeny classes is at most 25 and the only change is from upper to lower case.

We give all curves in each isogeny class. For all classes of curves of conductor less than 60000, and many others, the first one listed in each class is proved to be the so-called "optimal" or "strong Weil" curve attached to each newform (referred to as optimal curves from now on). For conductors above 60000, see the section "Optimality and the Manin constant" below.

Some of the data is common to all curves in the isogeny class.

The modular degrees for conductors over 12000 were computed using Mark Watkins's programs ec and sympow.

Generators for many rank 1 curves were computed using either Magma's HeegnerPoint function (written by Mark Watkins) or GP's ellheegner function written by Christophe Delaunay and Bill Allombert, based on the same ideas of Delaunay and Watkins.

The integral points for all curves were computed using Sage in a new implementation due to Michael Mardaus, Tobias Nagel and JEC. For conductors up to 1000 we have checked that these agree with Magma (version V2.14-14). N.B. It is known that these lists are in some cases incomplete, due to bugs in the program mentioned; they need to be corrected.

The images of the mod p Galois representations were computed by Andrew Sutherland.


SUMMARY TABLES

TABLE ONE: CURVES

TABLE TWO: GENERATORS

TABLE THREE: HECKE EIGENVALUES

TABLE FOUR: BSD DATA and ANALYTIC ORDERS OF SHA

TABLE FIVE: PARAMETRIZATION DEGREES

TABLE SIX: ISOGENY MATRICES

TABLE SEVEN: INTEGRAL POINTS

TABLE EIGHT: OPTIMALITY AND THE MANIN CONSTANT

TABLE NINE: IMAGES OF GALOIS REPRESENTATIONS


Recent update notes29 August 2014


john dot cremona at gmail dot com