On decompositions of singular abelian surfaces

Inspired by a work of Ma, in which he counts the number of decompositions of abelian surfaces by lattice-theoretical tools, we explicitly find all such decompositions in the case of singular abelian surfaces. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group on the factors of a given decomposition. Also, we provide an alternative and simpler proof of Ma's formula.