From:	nfrogers@math.harvard.edu 
Reply-To: nfrogers@math.harvard.edu
To: was@math.harvard.edu
Subject: RE: Neron
Date: Thu, 31 Jul 2003 15:46:24 -0400

Basically, Noam and I have been looking at ranks in the family x^3 +
y^3 = k.  I used Neron to eliminate the last few cases to prove that
21691 and 489489 are minimal for ranks 4 and 5, respectively.  Perhaps
more interestingly, computations on Neron have also led to some of the
first examples of rank 8 in this family, and the first example of a
rank 9 curve.  The rank 9 example is particularly interesting because
the 3-isogenous curve xy(x+y) = k has rational 3-torsion, making this
the first known example of an elliptic curve over Q with rational
3-torsion and rank at least 9.

Nick