Elliptic Curve Data

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

Updated 2020-11-27 (last major update 2019-07-22)

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:

Table 6 gives the isogeny matrices between curves in each isogeny class;
Table 7 lists the integral points on each curve;
Table 8 gives information on which curve is optimal, and the Manin constant;
Table 9 gives information on the mod-p Galois representations attached to each curve;
Table 10 gives information on the 2-adic Galois representations attached to each curve.
Table 11 gives information on the growth of torsion in number fields of small degree, for each curve.

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 400000, 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). 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 500000.

Acknowledgements

The curves were computed via modular symbols, using the C++ library eclib (see https://github.com/JohnCremona/eclib), run on machines in a variety of places, including clusters at Nottingham (1999-2007, conductors up to 130000); Warwick (2011-2016, conductors 130000 to 400000); and the Google Computing Platform (15-22 July 2019, conductors 400000-500000). Thanks to the universities of Nttingham and Warwick, and to the Simons Foundation for the GCP runs. Additional data is computed using a SageMath script to interface a variety of software, including the following:
The modular degrees for conductors over 12000 were computed using Mark Watkins's programs ec and sympow, via SageMath.
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 first computed using Sage in an implementation due to Michael Mardaus, Tobias Nagel and JEC. For all curves we checked on 2018-12-19 whether these agree with Magma (version V2.24-1): in 207 cases Magma found more integral points (never fewer), and the lists here are now hoped to be complete.
The images of the mod p Galois representations were computed by Andrew Sutherland.
The images of the 2-adic Galois representations were computed using a Magma program provided by Jeremy Rouse.
The torsion growth data was computed by Enrique Gonzalez-Jimenez and Filip Najman.

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 370000 the optimal Γ₀(N) curve is the one labelled 1 (except for class 990h when it is the curve labelled 3). For N>370000, 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 recognizable. 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 410000) is 5625=75² for 165066d3.

shas.html
A summary table of large Shas. Note that up to conductor 500000 there are 243527 elliptic curves with non-trivial Sha, with ranks 0 (222922 curves), 1 (20563 curves), 2 (42 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 parametrizations 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 parametrizations 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 all integral points on all curves.
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 400000, 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 can (using our modular symbols programs) determine the optimal curve in any individual case, this takes a long time to do for all remaining cases; this is ongoing. 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 detailed notes with full results for all conductors to 500000. These updated results include the proof 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 always the first curve listed.

Table of results known regarding optimality and Manin constant in all isogeny classes. For conductors greater than 400000, the values of the Manin constant are conditional on the first curve in the class being optimal.
Data format with sample lines:

N	class	#	[a1,a2,a3,a4,a6]	Optimality code	Manin constant
11	a	1	[0,-1,1,-10,-20]	1	1
11	a	2	[0,-1,1,-7820,-263580]	0	1
11	a	3	[0,-1,1,0,0]	0	5
499992	a	1	[0,-1,0,4481,148204]	3	1
499992	a	2	[0,-1,0,-29964,1526004]	3	1
499992	a	3	[0,-1,0,-446624,115024188]	3	1
499992	a	4	[0,-1,0,-164424,-24344100]	0	1

The optimality code is 0 for "not optimal", 1 for "optimal" and n for "one of n possibly optimal curves in this isogeny class". In the case of isogeny class 11a above, out of the three curves in the class, the optimal curve is 11a1 (which is $X_0(11)$), the first two curves have Manin constant 1, while the curve 11a3 (which is $X_1(11)$) has Manin constant equal to 5. In the class 499992a, out of four curves, the optimal curve is certainly one of the first 3; and if the optimal curve is indeed 499992a1 then all the Manin constants are equal to 1.

TABLE NINE: IMAGES OF GALOIS REPRESENTATIONS

A table giving the for each elliptic curve the primes p for which the mod-p Galois representation is not maximal, 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 and the image is shown when it is not maximal, where maximal means as large as possible given the constraints imposed by the endomorphism ring of E. 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:

Curve list of non-surjective images

11a1 5Cs.1.1

27a1 3Cs.1.1

37a1

Curve	list of non-surjective images
11a1	5Cs.1.1
27a1	3Cs.1.1
37a1

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

TABLE ELEVEN: TORSION GROWTH

Torsion growth data has been computed so far for curves of conductor up to 400000, for extensions of degree up to 23. (In degrees 11, 13, 17, 19, 22 and 23 there are no cases of torsion growth).

For 2 ≤ d ≤ 23 we give for every curve E a list of the number fields K of degree d (if any) such that E(K)_tors is strictly larger than E(Q)_tors (and in case d is composite, strictly larger than E(K')_tors for all subfields K'⊂ K).

Fields are specified by the coefficients of a canonical defining polynomial. Torsion structure is shown as [n] or [m,n] with m dividing n.

Data format with sample lines (one in degree 2, one in degree 12):

Curve label [Torsion][Field] [Torsion][Field] [Torsion][Field]

130014e1 [2,2][1806,-1,1] [4][-23,-1,1] [4][175,-1,1]

130032e1 [4][-10346,8862,-13965,14728,-4689,-1362,387,156,129,-86,9,0,1]

Recent update notes: 27 November 2020
john dot cremona at gmail dot com

Curve label	[Torsion][Field]	[Torsion][Field]	[Torsion][Field]
130014e1	[2,2][1806,-1,1]	[4][-23,-1,1]	[4][175,-1,1]
130032e1	[4][-10346,8862,-13965,14728,-4689,-1362,387,156,129,-86,9,0,1]