Updated 2017-11-23 (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.
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 Γ_{0}(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 |
Simple searches may be carried out with the unix/linux utility awk. For example:
awk '$6==12' allcurves.* | sort -n -k 1
awk '$6==16' allcurves.*
awk '$5==3' allcurves.* | sort -n -k 1
sed 's/[]\[,]/ /g' allcurves.00000-10000
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 | a | 1 | [0,1,1,-2,0] | 2 | [] | [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 | a | 1 | [1,1,0,-37746035,-89296920339] | 1 | [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 2254020761884782243^3together 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.
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) |
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).
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 |
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 |
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]] |
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] |
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 400000. 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 indeed the first curve listed.
Summary of results known regarding optimality of all classes with conductor greater than 60000 and size at least 2.
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 |
Curve | list of non-surjective images |
---|---|
11a1 | 5Cs.1.1 |
27a1 | 3Cs.1.1 |
37a1 |
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 |
For 2≤ d≤7 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 indicated either by their LMFDB label, or (for fields not currently in the LMFDB) by the coefficients of a defining polynomial. Torsion structure is shown as [n] or [m,n] with m dividing n.
Data format with sample lines:
Curve label | Torsion:field | Torsion:field | Torsion:field |
---|---|---|---|
15a2 | [2,4]:2.0.4.1 | [2,4]:2.0.20.1 | [2,4]:2.2.5.1 |
651e3 | [3]:-651,0,0,1 | [2]:3.1.2604.1 | |
26b2 | [7]:7.1.52706752.1 |
Recent update notes: 23 November 2017