An Interesting Family of Curves of Genus 1

We study the family of elliptic curvesy2 = x3−t2x+1, both overQ(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.