The Maximal Rank of Elliptic Delsarte Surfaces

Shioda described in his article from 1986 a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over k(t). In this article we find all elliptic curves over k(t) for which his method is applicable. For each of these curves we also compute the Mordell-Weil rank.