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