International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 6, Pages 431-434

An interesting family of curves of genus 1

Andrew Bremner

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 y2=x3t2x+1, both over (t) and over . In the former case, all integral solutions are determined; in the latter case, computation in the range 1t999 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.