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