Elliptic Curve Data

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

Updated 2015-05-19 (last major update 2015-05-19)

UK site (the US mirror hosted by William Stein is no longer in operation)

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 360000.

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

Lists of number of curves (up to isogeny) of each individual conductor (each file is 10000 lines long).
Table showing number of curves of each range of 10000 conductors, sorted by rank.

TABLE ONE: CURVES

One entry for each isomorphism class of curves, giving conductor N, letter id for isogeny class, number of the curve in the class, coefficients of minimal Weierstrass equation, rank r, order of torsion subgroup |T|. For all N up to 60000 the optimal Γ₀(N) curve is the one labelled 1 (except for class 990h when it is the curve labelled 3). For N>60000, this is probably also true, but in some cases remains conditional on Stevens' Conjecture (see the section "Optimality and the Manin constant" below).

Data format with sample line:

N C # curve r t

2730 bd 1 [1,0,0,-25725,1577457] 0 12

where:
- N = conductor
- C = isogeny class (letter(s))
- # = number of curve in class = 1 (except for 990h3)
- curve = curve coefficients in format [a1,a2,a3,a4,a6]
- r = rank
- t = order of torsion
Simple searches may be carried out with the unix/linux utility awk. For example:
- All curves with torsion of order 12:
  awk '$6==12' allcurves.* | sort -n -k 1
- All curves with torsion of order 16:
  awk '$6==16' allcurves.*
- All curves of rank 3:
  awk '$5==3' allcurves.* | sort -n -k 1
If it is desired to have the curve coefficients in five separate fields with spaces as field separators, this can be achieved using scripts such as this:
sed 's/[]\[,]/ /g' allcurves.00000-10000

N	C	#	curve	r	t
2730	bd	1	[1,0,0,-25725,1577457]	0	12

TABLE TWO: GENERATORS

For every curve, generators are given for the Mordell group, in projective coordinates. N.B. In all cases I have checked that the point(s) given are indeed generators. Each entry consists of conductor N, isogeny class code, number of curve in class, curve coefficients, rank r, torsion structure (as a list of t structure constants for t=0,1 or 2, i.e. in the form [] or [t] or [t1,t2]) and r+t points in projective coordinates (torsion last). For example, the entry

389

[0,1,1,-2,0]

[]

[0:0:1]

[1:0:1]

means that curve 389a1 = [0,1,1,-2,0] has rank 2 and trivial torsion, with generators [0:0:1]=(0,0) and [1:0:1]=(1,0), while the entry

4602

[1,1,0,-37746035,-89296920339]

[2]

[175781888357266265777015693706802984972253428834450486976370 : 19575260230015313702261379022151675961965157108920263594545223 : 11451799510178287699130942513632433218384249076487302907]

[7094:-3547:1]

means that curve 4602a1 = [1,1,0,-37746035,-89296920339] has rank 1 with generator

 77985922458974949246858229195945103471590      19575260230015313702261379022151675961965157108920263594545223
[----------------------------------------- ,    -------------------------------------------------------------- ]
    2254020761884782243^2                               2254020761884782243^3

together with torsion of order 2 generated by [7094:-3547:1] = (7094,-3547).

N.B. From April 2011 the format of these files was changed to include information about the torsion; there is therefore now a line in the allgens files for every curve, not just those of positive rank. The files for N<130000 were updated accordingly on 15/4/11.

TABLE THREE: HECKE EIGENVALUES

Hecke eigenvalues for p<100 for each of the corresponding newforms for Γ₀(N). When p|N the entry is simply "+" or "-" and is a W-eigenvalue, as in Antwerp IV. When there are primes p|n with p>100 the corresponding eigenvalue(s) are in extra column(s), as in

101 a 0 -2 -1 -2 -2 1 3 -5 1 -4 -9 -2 8 -8 7 -2 -14 4 2 13 8 -9 -4 14 2 +(101)

10201 a 0 2 -1 2 2 1 3 -5 1 4 -9 -2 -8 -8 7 2 14 -4 -2 13 -8 -9 4 -14 2 +(101)

19153 a 2 0 -1 0 -4 7 -3 -3 -6 3 8 -2 0 1 1 0 15 6 -13 12 -2 2 9 -9 -10 +(107) -(179)

so the total number of fields is 27, 28 or 29 on each line (assuming N<1113121=101*103*107)

TABLE FOUR: BSD DATA and ANALYTIC ORDERS OF SHA

1-1000
Birch--Swinnerton-Dyer data for the optimal curve in each class, exactly as in the book. Column headings: Conductor, class id letter, rank, real period w, L^(r)(1)/r!, regulator R, rational factor, S. Here the rational factor is L^(r)(1)/wRr!; when r=0 this is exact and given as a pair of integers (numerator denominator); when r>0 it is approximate, but easily recognisable. Lastly, S is the value of the order of the Tate-Shafarevich group as predicted by B-SD (the "analytic order of Sha"), given the previous data and also the local factors and torsion. When r=0 this is exact; when r>0 it is approximate, and was computed to several places but to save space is just entered as 1.0. (S>1 in only 4 cases, where S=4 or 9).
Same as previous but with data for all the curves (not only the optimal ones) up to the current bound.
Data format with sample lines:

N C # curve r t cp om L R S

11 a 1 [0,-1,1,-10,-20] 0 5 5 1.269209304 0.25384186 1 1

5077 a 1 [0,0,1,-7,6] 3 1 1 4.151687983 1.73184990 0.41714355 1.00000000

where:
- N = conductor
- C = isogeny class (letter(s))
- # = number of curve in class
- curve = curve coefficients in format [a1,a2,a3,a4,a6]
- r = rank
- t = order of torsion
- cp = product of Tamagawa factors c_p
- om = real period
- L = L^(r)(E,1)/r!.
- R = Regulator
- S = (Analytic) order of Sha.

Lists of the curves with non-trivial Tate-Shafarevich group, according to the BSD conjecture; i.e., curves whose "analytic order of Sha" is greater than 1. The record (to conductor 350000) is 5625=75² for 165066d3.

shas.html
A summary table of large Shas. Note that (up to conductor 360000) there are 171596 elliptic curves with non-trivial Sha, with ranks 0 (158043 curves), 1 (13540 curves) and 2 (13 curves, all with 2-torsion).

N	C	#	curve	r	t	cp	om	L	R	S
11	a	1	[0,-1,1,-10,-20]	0	5	5	1.269209304	0.25384186	1	1
5077	a	1	[0,0,1,-7,6]	3	1	1	4.151687983	1.73184990	0.41714355	1.00000000

TABLE FIVE: PARAMETRIZATION DEGREES

Optimal curves:

A table of the degree of the modular parameterizations of each optimal curve.
Data format with sample line:

N id degree primes curve

5077 a 1 1984 {2,31} [0,0,1,-7,6]
where "primes" is the set of primes dividing the degree.

All curves:

A table of the degree of the modular parameterizations of every curve.
Data format with sample line:

N id # curve degree

11 a 1 [0,-1,1,-10,-20] 1

11 a 2 [0,-1,1,-7820,-263580] 5

11 a 3 [0,-1,1,0,0] 5

N	id	degree	primes	curve
5077	a 1	1984	{2,31}	[0,0,1,-7,6]

N	id	#	curve	degree
11	a	1	[0,-1,1,-10,-20]	1
11	a	2	[0,-1,1,-7820,-263580]	5
11	a	3	[0,-1,1,0,0]	5

TABLE SIX: ISOGENY MATRICES

A table giving the degrees of isogenies within each isogeny class. One row for each isogeny class.
Data format with sample line:

N	class	#	[a1,a2,a3,a4,a6]	curves in the class	isogeny matrix
14	a	1	[1,0,1,4,-6]	[[1,0,1,4,-6],[1,0,1,-36,-70],[1,0,1,-171,-874],[1,0,1,-1,0],[1,0,1,-2731,-55146],[1,0,1,-11,12]]	[[1,2,3,3,6,6],[2,1,6,6,3,3],[3,6,1,9,2,18],[3,6,9,1,18,2],[6,3,2,18,1,9],[6,3,18,2,9,1]]

where the isogeny matrix has (i,j) entry d when the there is a cyclic isogeny of degree d from curve i to curve j.

TABLE SEVEN: INTEGRAL POINTS

A table giving the x-coordinates of (almost) all integral points on all curves. Corrected 2009-01-01, but still incomplete.
Data format with sample line:

Curve	[a1,a2,a3,a4,a6]	x-coordinates of integral points
114114bz1	[1,0,0,-858375,380956041]	[-1098,-1042,-990,-954,-756,-522,-426,-72,36,102,270,354,414,498,596,630,726,918,960,1334,1590,1818,1974,2702,3006,3690,5250,6966,8352,9702,18054,24438,31848,48150,119988,295254,913014]

TABLE EIGHT: OPTIMALITY AND THE MANIN CONSTANT

For isogeny classes of curves of conductor greater than 60000, we have not yet determined in all cases which curve in each class is optimal. However, in all cases we have verified that the Manin constant of the optimal curve is equal to 1 (as it is conjectured to be for every optimal curve), even in cases where we do not know for sure which curve is optimal.

While we could (using our modular symbols programs) determine the optimal curve in any individual case, this would take a long time to do for all remaining cases. For more details on this, see my Appendix to the paper "The Manin Constant" by Amod Agashe, Ken Ribet and William Stein [Pure and Applied Mathematics Quarterly, Vol. 2 no.2 (2006), pp. 617-636.] and these notes updating the results for all conductors to 360000. These updated results include the verification that Manin's constant is 1 in all cases, together with a list of which curves in the class might be optimal, given the incomplete modular symbol computations carried out to date. Note, however, that it follows from computation of the modular degrees of all curves in the class (which computation is conditional on Stevens's conjecture) that the optimal curve is indeed the first curve listed.

Summary of results known regarding optimality of all classes with conductor greater than 60000 and size at least 2.
Sample lines:

Class Manin's constant possible optimal curves

60003a c=1 optimal curve is E1

60005a c=1 2 possible optimal curves: E1 E2

61710cz c=1 6 possible optimal curves: E1 E2 E3 E5 E6 E8

Class	Manin's constant	possible optimal curves
60003a	c=1	optimal curve is E1
60005a	c=1	2 possible optimal curves: E1 E2
61710cz	c=1	6 possible optimal curves: E1 E2 E3 E5 E6 E8

TABLE NINE: IMAGES OF GALOIS REPRESENTATIONS

A table giving the for each elliptic curve the primes p≤37 for which the mod-p Galois representation is not surjective, as computed by Andrew Sutherland, together with a code identifying the image (as a subgroup of GL(2,p), up to conjugation). For curves with CM the representation is never surjective but the image is only shown for p≤37. For curves without CM, the representation is surjective for all but finitely many primes (Serre) and is conjectured to be surjective for p>37, but this was only checked for 37<p<80.
Data format with sample lines:

N	class	#	[a1,a2,a3,a4,a6]	rank	torsion	list of non-surjective images
11	a	1	[0,-1,1,-10,-20]	0	5	5Cs.1.1
27	a	1	[0,0,1,0,-7]	0	3	3Cs.1.1 5Nn 7Ns 11Nn 13Ns 17Nn 19Ns 23Nn 29Nn 31Ns 37Ns
37	a	1	[0,0,1,-1,0]	1	1

TABLE TEN: IMAGES OF TWO-ADIC GALOIS REPRESENTATION

A table giving, for each elliptic curve without CM, the image of the 2-adic Galois representation, as computed using a Magma program provided by Jeremy Rouse and David Zureick Brown. There are 1208 possible images. The index is the index of the image in GL(2,Z_2). The level is the largest n such that the image contains the kernel of reduction modulo 2^n. The generators are generating matrices for the image modulo the level. Note that the action of Galois is on the right, so points in E[2^r] are represented as row vectors v and if M is in GL_2(Z_2) the action of M is v |-> v*M. The label is a label for the associated modular curve. See Rouses's web page for details.
Data format with sample lines:

N class # [a1,a2,a3,a4,a6] index level matrix generators label

11 a 1 [0,-1,1,-10,-20] 1 1 [] X1

15 a 1 [1,1,1,-10,-10] 96 8 [[5,4,2,3],[1,0,0,5],[1,4,0,5],[1,0,4,5]] X187d

N	class	#	[a1,a2,a3,a4,a6]	index	level	matrix generators	label
11	a	1	[0,-1,1,-10,-20]	1	1	[]	X1
15	a	1	[1,1,1,-10,-10]	96	8	[[5,4,2,3],[1,0,0,5],[1,4,0,5],[1,0,4,5]]	X187d

Recent update notes: 19 May 2015

john dot cremona at gmail dot com