Elliptic curve data by J. E. Cremona, University of Exeter, U.K. (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, revised edition in preparation 1996). See the directory ../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 (mine does). TABLE ONE: CURVES ================= curves.1-5077 ------------- 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|, degree of modular parametrization X_0(N)->E (for N<=3200 only; an entry of 0 means degree not yet computed). For N<=3200 this is the strong Weil curve; for larger N it may not be. The file has a 2-line header which should be stripped before inputting into your program. For input into e.g. apecs you will have to use a text editor first. Simple searches may be carried out with the unix utility awk or its gnu version gawk. For example, to find the two curves in the list with 12 torsion points: awk '$9==12' curves.1-5077 allcurves.1-1000 ---------------- As previous file but for all curves in the isogeny class, with the curve number as a new third field; no deg(phi) field. allcurves.1001-5077 ------------------- As previous file but for the larger range 1001-5077 of conductors. NOTE that for N>3200 the numbering of the curves in the class should be regarded as provisional, as we have not yet computed which is the strong Weil curve. TABLE TWO: GENERATORS ===================== generators.1-1000 ----------------- for the FIRST (=strong Weil) curve in each isogeny class for N up to 1000, generators are given for the Mordell group modulo torsion, in projective coordinates, when the rank is positive. Each entry consists of conductor N, isogeny class letter, rank r, and r sets of three projective coordinates. For example, the entry 389 A 2 0 0 1 1 0 1 means that curve 389A1 has rank 2 with generators [0:0:1]=(0,0) and [1:0:1]=(1,0), while the entry 873 C 1 3639568 106817593 4096 means that curve 873C1 has rank 1 with generator [3639568:106817593:4096] = (227473/256, 106817593/4096). allgenerators.1-1000 -------------------- Generators for ALL curves of positive rank up to conductor 1000. Same format as "generators", with one line per curve, with an extra field for the curve-number in the class. I have checked in most but not quite all cases that the points do not in fact generate a subgroup of finite index >1. Generators for the curves of conductor > 1000 are on the way. I now have all generators for the first curve in each class, and can provide a list to anyone who asks. TABLE THREE: HECKE EIGENVALUES ============================== aplist.1-1000 ------------- 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. aplist.1001-5077 ---------------- Same as previous for the range 1001-5077. 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). 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.1-3000 ------------- A table of the degree of the modular parametrizations of each strong Weil curve of conductor up to 3000.