Elliptic curve data by J. E. Cremona, University of Exeter, U.K. Updated 16 July 1998 (email to cremona@maths.exeter.ac.uk) This directory contains various data files concerning modular elliptic curves, in a standard format to make them easily readable by other programs. For a beautifully typeset version of the same data (with some extra data about local reduction data) for conductors up to 1000, you will have to buy the book ("Algorithms for modular elliptic curves", CUP 1992, second revised edition 1997). See the Web site http://www.maths.ex.ac.uk/~cremona/amec for more information. See the directory (ditto)/book for errata to the First Edition of the book, including errors and omissions in the tables. The files here have the corrected data in them. 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. If using ftp to fetch the files, use binary mode and uncompress after transfer using gunzip. If accessing via the Web, your browser might uncompress the files automatically for you to view then. At present (July 1998) most tables contain data for conductors up to 5300, while some only go up to 1000. These bounds will increase in time! TABLE ONE: CURVES ================= curves.1-5300 ------------- One entry for isogeny class of curves, giving conductor N, letter id for isogeny class, coefficients of minimal Weierstrass equation, rank r, order of torsion subgroup |T|. For all N up to 5300 this is the strong Weil curve. Format of each line: N C # curve r t where: N = conductor, C = isogeny class (letter(s)), # = number of curve in class = 1, "curve" = curve coefficients in format [a1,a2,a3,a4,a6], r = rank, t = order of torsion. e.g. 2730 DD 1 [1,0,0,-25725,1577457] 0 12 Simple searches may be carried out with the unix utility awk or its gnu version gawk. For example, to find the only curve in the list with 12 torsion points (the one just given) awk '$6==12' curves.* and the only curve of rank 3: awk '$5==3' curves.* allcurves.1-5300 ---------------- Similar to previous file, but for all curves in the isogeny class, with the curve number in the third field. e.g. 30 A 2 [1,0,1,-19,26] 0 12 The 18 curves in the table with torsion of order 12: gawk '$10==12' allcurves.* The only curve of rank 3: gawk '$9==3' allcurves.* Note: If it is desired to have the curve coefficients in five separate fields with spaces as field separators, this can be achieved using the following script. sed 's/[]\[,]/ /g' curves.1-5300 TABLE TWO: GENERATORS ===================== gens.1-5300 ----------- 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. NOTE In many BUT NOT ALL cases I have checked that the point(s) given are indeed generators, but in some cases it is possible that they generate a subgroup of finite index. This can easily (in principle) be checked for any given curve if required. Each entry consists of conductor N, isogeny class number (not letter), number of curve in class, curve coefficients, rank r, 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 the rather large generator [175781888357266265777015693706802984972253428834450486976370 : 19575260230015313702261379022151675961965157108920263594545223 : 11451799510178287699130942513632433218384249076487302907] = 77985922458974949246858229195945103471590 [----------------------------------------- , 2254020761884782243 ^2 19575260230015313702261379022151675961965157108920263594545223 -------------------------------------------------------------- ] 2254020761884782243 ^3 allgenerators.1-1000 -------------------- Generators for ALL curves of positive rank up to conductor 1000. Shorter, plainer format than gens files (will be standardized when this file is complete to 5300). Again, I have checked in most but not quite all cases that the points do not in fact generate a subgroup of finite index >1. TABLE THREE: HECKE EIGENVALUES ============================== aplist.1-5300 ------------- 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) so the total number of fields is 27 or 28 on each line. TABLE FOUR: BSD DATA ==================== bsd.1-1000 ---------- Birch--Swinnerton-Dyer data for the first (strong Weil) curve in each class. Column headings: Conductor, class id, 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, 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). lf1.1-5300 ---------- A partial version of the previous table, up to level 5300. Sample entry: 5077 1: Rank = 3 L^(r)(f,1)/r! = 1.73184990011834 allbsd.1-1000 ------------- Same as previous but with a line for all the curves, not just the strong Weil curves. An extra column gives the number of the curve in each isogeny class. bigshas.1-1000 -------------- A list of the 106 curves of conductor up to 1000 with non-trivial Tate-Shafarevich group, according to the BSD conjecture. Similar tables for N>1000 will be prepared once all the generators have been found. Again, partial tables are available on request. TABLE FIVE: PARAMETRIZATION DEGREES =================================== degphi.1-5300 ------------- A table of the degree of the modular parametrizations of each strong Weil curve of conductor up to 5300.