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.