  Given an abelian surface, the number of its distinct decompositions into a
product of elliptic curves has been described by Ma. Moreover, Ma himself
classified the possible decompositions for abelian surfaces of Picard number $1
\leq \rho \leq 3$. We explicitly find all such decompositions in the case of
abelian surfaces of Picard number $\rho= 4$. 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. We
also provide an alternative and simpler proof of Ma's formula, and an
application to singular K3 surfaces.
