J. Silverman described an algorithm for computing rational points on
rank-1 elliptic curves via L-series and canonical heights, in a paper
cleverly entitled, "Computing Rational Points on
Rank-1 Elliptic Curves via L-Series and Canonical Heights",
Math Comp 68 (1999) 835-858.
- Allan MacLeod has implemented this in
UBASIC (an extended-precision arithmetic language which runs on DOS and
Windows machines. Available from ftp://rkmath.rikkyo.ac.jp/pub).
Input the curve's coordinates early in the code
and type 'run'. Works OK if the height and conductor are not too large.
- Silverman's method requires an extremely accurate computation of the
expected height (which may be computed from L-series, assuming some
standard conjectures). Tom Womack has code for computing
the rank, which runs under PARI (see http://www.parigp-home.de)
- Silverman's method includes careful consideration of the use of
precision, the search for points on the non-identity component (when present),
and so on. Throwing all that hard work to the wind I have cobbled together
a rough-and ready implementation of the main technique. This runs under
Maple but is probably easily ported to any CAS because, hey, it ain't deep.
Here are implementation notes, a tiny
users' guide, and a bunch of very long strings of
polynomials, described in the implementation notes:
S-FILE0 etc. (this page still under construction)
Last modified 2003/09/01 by Dave Rusin. Mail: