International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 6, Pages 431-434
doi:10.1155/S0161171200002210

An interesting family of curves of genus 1

Abstract

We study the family of elliptic curves y2=x3−t2x+1, both over ℚ(t) and over ℚ. In the former case, all integral solutions are determined; in the latter case, computation in the range 1≤t≤999 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.