1README file for programs mwrank, findinf, ratpoint
2
3There are three main programs for elliptic curves in this distribution: mwrank,
4point_search and quartic_points. The first two work on elliptic curves over Q
5given by Weierstrass equations with standard coefficients
6[a1,a2,a3,a4,a6]. If the input curve is non-minimal or non-standard
7(where standard means a1,a3 in {0,1} and a2 in {-1,0,1}) the input
8curve is replaced by a standard minimal model, and points found are
9transferred back to the original input model. The programs tell you
10when the model is changed.
11
12In all cases there are 2 possible input formats for curves:
13
14[a1,a2,a3,a4,a6]
15a1 a2 a3 a4 a6
16
17So if the first character read is '[' it expects a single non-space
18delimiter between coefficients and one at the end; otherwise it
19expects 5 integers separated by space. If the input is invalid the
20programs stop, so you can use 0 0 0 0 0 to end the input stream.
21
22
23point_search searches for points on the curves up to a given bound on the
24logarithmic height, using sieving. One can opt for either seeing all
25points as they are found (verbose=1) or nothing until a final report
26(verbose=0); also, if verbose=1 then one can opt for a simple list of
27the points found, with no post-processing, or a full processing of the
28points to find linear dependencies between them using the height
29pairing. The "regulator" output is just the regulator of the subgroup
30of points found. Currently the sieve is based on Michael Stoll's
31programme (which originated with a program of Elkies, with
32developments by Stoll and Colin Stahlke). Added in Jan 2005: the
33subgroup of points found is (by default) automatically saturated at
34the end.
35
36mwrank is described more fully in the file mwrank.info. Briefly,
37mwrank does a 2-descent to determine the rank and gives points
38covering the cosets of 2E(Q) in E(Q); it then goes on to complete to a
39Mordell-Weil basis via a sieve-based saturation process after
40determining an upper bound for the index.
41
42quartic_points works with quartics g(x) with integer coefficients,
43g(x)=a*x^4+b*x^3+c*x^2+d*x+e. It finds the Jacobian E of the curve
44C:y^2=g(x), searches C for rational points and maps any point found to
45E using the 2-covering map. Current version quits as soon as one
46point on C is found.
47
48Other programs: tmrank is a version of mwrank designed for testing
49with the input file tmrank.in. Each line of tmrank.in contains a
50curve and its correct rank, and tmrank uses mwrank to compute the rank
51and checks the result.
52
53 tate: Runs Tate's algorithm, outputs detail of
54 reduction types etc.
55 conductor: just output conductor
56 torsion: outputs number (and optionally structure) of
57 torsion points
58 allisog: find the complete isogeny class of the input
59 curve by computing a succession of isogenies
60 of prime degree. WARNING: this is unreliable
61 and causes lots of warning messages if run in
62 standard precision. With higher precision
63 (under LiDIA) it is reliable, though a lot
64 slower. I also have a Pari/GP program to do
65 the same thing, available on request.
66
67These four can be run interactively (by entering "tty" for the input
68filename at the prompt) or in batch mode by entering the filename.
69The file should contain a list of curves in the standard format, with
70[0,0,0,0,0] to terminate cleanly.
71
72 twist: computes quadratic twists of a given curve,
73 giving their minimal models etc. Interactive.
74
75 indep: tests whether given points on a curve are
76 independent, using 2-descent NOT heights. Test with
77 "indep < e23".
78
79Systems supported: no binaries are now provided, only source.
80
81Sources: Available from https://github.com/JohnCremona/eclib/ It
82requires pre-installation of Pari and NTL.
83
84 John Cremona
85 john.cremona@gmail.com
86
87 original: 6/2/95
88
89 updated: 7/8/95, 12/1/98, 20/4/00, 16/8/00, 5/7/01, 7/1/05, 10/5/05,
90 31/12/11, 2/1/16
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93