Updated 2005-06-20
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). 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.
At present the tables contain data for conductors up to 70000.
We give all curves in each isogeny class. For levels (or conductors) up to 20000, the first one listed in each class is the so-called "optimal" or "strong Weil" curve attached to each newform (referred to as optimal curves from now on). For levels above 20000 we have not yet determined which curve in each class is optimal (except, of course, where the class has only one curve); but thanks to Mark Watkins's program, we can say that the first curve listed is optimal conditional if the Stephens' conjecture is valid. The optimal curve can be determined in any individual case, but this would take a long time to do for all remaining cases. Hence the numbering of the curves within each class may change for conductors over 20000 --but only if Stevens's conjecture is false. 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 program.
One entry for isogeny class of curves, giving conductor N, letter id for isogeny class, number of the curve in the class (=1 almost always) coefficients of minimal Weierstrass equation, rank r, order of torsion subgroup |T|, degree of modular parametrization. For all N up to 20000 this is the optimal curve. For N>20000 it may or may not be. For modular degrees for N>20000, see below.
Data format with sample line:
| N | C | # | curve | r | t | deg |
|---|---|---|---|---|---|---|
| 2730 | DD | 1 | [1,0,0,-25725,1577457] | 0 | 12 | 9216 |
Similar to previous files, but for all curves in the isogeny class, with the curve number in the third field, and no degree field. For N>12000 the ordering of the curves in each class should be regarded as provisional, since we do not (in most cases) yet know the optimal curve which by convention is numbered 1.
Data format with sample line:| N | C | # | curve | r | t |
|---|---|---|---|---|---|
| 30 | A | 2 | [1,0,1,-19,26] | 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' curves.00000-10000
For the first curve in each isogeny class, generators are given for the Mordell group modulo torsion, in projective coordinates, when the rank is positive. N.B. In all but a very few cases I have checked that the point(s) given are indeed generators (modulo torsion). Each entry consists of conductor N, isogeny class number (not letter), number of curve in class (=1), curve coefficients, rank r (>0), and r sets of three projective coordinates. For example, the entry
| 389 | 1 | 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 with generators [0:0:1]=(0,0) and [1:0:1]=(1,0), while the entry
| 4602 | 1 | 1 | [1,1,0,-37746035,-89296920339] | 1 | [175781888357266265777015693706802984972253428834450486976370 : 19575260230015313702261379022151675961965157108920263594545223 : 11451799510178287699130942513632433218384249076487302907] |
means that curve 4602A1 = [1,1,0,-37746035,-89296920339] has rank 1 with a rather large generator,
77985922458974949246858229195945103471590 19575260230015313702261379022151675961965157108920263594545223 [----------------------------------------- , -------------------------------------------------------------- ] 2254020761884782243^2 2254020761884782243^3
Generators for all curves of positive rank, in the same format as above. In all but a very few cases I have shown that these points do generate the full Mordell-Weil group modulo torsion.
Hecke eigenvalues for p<100 for each of the corresponding newforms for Gamma_0(N). When p|N the entry is simply "+" or "-" and is a W-eigenvalue, as in Antwerp IV. When there is a prime p|n with p>100 the corresponding eigenvalue is in a final column, 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) |
Birch--Swinnerton-Dyer data for the first (strong Weil or 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 50000.
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 |
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 level 70000) is 441 for 42565B1-2.
A summary table of large Shas.
A table of the degree of the modular parametrizations of each "optimal" or "strong Weil" curve. For conductors greater than 20000 the optimality is conditional on Stephens' conjecture, and for conductors greater than 12000 the degree was computed using Mark Watkins's program ec. Optimality and degrees for conductors greater than 20001 may become unconditional in due course.
Data format with sample line:
| N | id | degree | primes | curve |
|---|---|---|---|---|
| 5077 | A1 | 1984 | {2,31} | [0,0,1,-7,6] |
Recent update notes: 20 June 2005