From: elkies@math.harvard.edu (Noam Elkies) Subject: Re: looking for rational point of infinite order on an elliptic Date: 11 Aug 00 21:45:50 GMT Newsgroups: sci.math.numberthy Summary: [missing] >Can anyone find a Mordell-Weil rational generator for a point of infinite >order on the elliptic curve [ 0, 0, 0, -29929, 0 ]? This is the "congruent-number curve" equivalent to 173y^2=x^3-x. The Heegner point method, as described in my ANTS-I paper, efficiently finds generators of such curves which have rank 1. [Actually in this case one has to use a variant, called "mock-Heegner points" if I remember right, because the conductor 32 of the initial curve y^2=x^3-x contains a prime ramified in the twisting field Q(sqrt(-173)).] Looking up tables I made some year back I find the rational point of infinite order with x-coordinate 4772605191075675626409/36682809607700857441 multiplying this by 173 yields the x-coordinate of the corresponding point on your curve [0,0,0,-29929,0]. --Noam D. Elkies ============================================================================== From: john.cremona@maths.nottingham.ac.uk (Prof. J. E. Cremona) Subject: Re: looking for rational point of infinite order on an elliptic Date: 13 Aug 00 00:11:38 GMT Newsgroups: sci.math.numberthy I ran mwrank with the congruent number curve -29929 and with paramters -b 13 it found the generator in 97s. The first descent quartic is (-1,0,519,0,-59858), the relevant 2nd descent quartic is (173,0,0,0,692) (which you will see if you set -v 2 ) with rational point x=177521/137059. One can do a third descent in this case (not implemented generally in mwrank) which leads to the third descent quartic (13,-476,6456,-39132,88727) which has the "easy" rational point with x=341/69. So now you have two independent methods. For anoter challenge, I remind readers that the Mordell curve y^2=x^3+7823 has analytic rank 1 but (as far as I know) no known rational points. It is the one missing case from the tables of Gebel, petho and Zimmer. John Cremona