International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 6, Pages 431-434
An interesting family of curves of genus
Department of Mathematics, Arizona State University, Tempe 85287-1804, AZ, USA
Received 22 October 1998
Copyright © 2000 Andrew Bremner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the family of elliptic curves , both over
and over . In the former case, all
integral solutions are determined; in the latter case, computation
in the range shows large ranks are common, giving
a particularly simple example of curves which (admittedly over a
small range) apparently contradict the once held belief that the
rank under specialization will tend to have minimal rank consistent
with the parity predicted by the Selmer conjecture.