Evans, Pulham, and Sheenan computed the number of complete 4-subgraphs of Paley graphs by counting the number of edges of
the subgraph containing only those nodes x for which x and
x−1 are quadratic residues. Here we obtain formulae for the number of edges of generalizations of these subgraphs using Gaussian hypergeometric series and elliptic curves. Such
formulae are simple in several infinite families, including those studied by Evans, Pulham, and Sheenan.