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

Updated 2016-10-17 (last major update 2016-10-17)


This site is a front-end to the ecdata repository, hosted at GitHub, which contains data files for (modular) elliptic curves over Q, 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. As of 2000 the book is out of print, and CUP have no plans to reprint it.

For a more sophisticated web interface to this data and much more, use the LMFDB.

The files correspond to tables 1-5 in the book (Table 5 is not in the First Edition), with additional tables:

From September 2005, a new labelling scheme was 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 and 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 tables currently contain data for conductors up to 400000.

Acknowledgements


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

TABLE TEN: IMAGES OF TWO-ADIC GALOIS REPRESENTATION


Recent update notes17 October 2016


john dot cremona at gmail dot com