  In this paper the authors study quotients of the product of elliptic curves
by a rigid diagonal action of a finite group $G$. It is shown that only for $G
= \operatorname{He(3)}, \mathbb Z_3^2$, and only for dimension $\geq 4$ such an
action can be free. A complete classification of the singular quotients in
dimension 3 and the smooth quotients in dimension $4$ is given. For the other
finite groups a strong structure theorem for rigid quotients is proven.
