APECS says it has rank 3, based on B-SD conjecture and possibly R.H. as well. It has Weierstrass form Y^2+Y*X = X^3-X^2-4317*X+116441 and in that form has at least 56 integer points. On Wed, Mar 7, 2012 at 11:56 AM, ed pegg <ed@mathpuzzle.com> wrote:
A while back, I made a list of elliptic curves that hit a lot of simple integer coordinates. I made a demo of it at http://demonstrations.wolfram.com/EllipticCurvesOnASmallLattice/ A good info page - http://en.wikipedia.org/wiki/Elliptic_curve .
Here's one of the curves, it goes through 21 points in that space. Outside of (-9,-9) to (9,9),it also goes through the points {-42, 7}, {-20, 1}, and {121, -38}. eq = x^3 + 3 x^2 y + 16 x^2 - 2 x y^2 - 13 x y - 17 x + y^3 + 6 y^2 - 37 y - 210; pts = Select[Tuples[Range[-10, 10], {2}], Evaluate[eq /. {x -> #[[1]], y -> #[[2]]} ] == 0 &] ; Graphics[{Point[pts], First[ContourPlot[eq == 0, {x, -10, 10}, {y, -10, 10}]]}] Is there a convenient way to find the rank of this curve? A a glance, it looks like it might be rank 7-9 or so. Not a record setter by any means, but I'm curious. The way I found these curves with lots of lattice points was to just pick 9 points at random and fed them into the http://demonstrations.wolfram.com/NinePointCubic/ equation, and kept the resulting curve if it went through a lot of other lattice points. With more careful selection of the random points, it seems like it would be easy to generate a number of rank 9 elliptic curves. Or am I just being naive? Ed Pegg Jr
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