Abstract

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.