The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing an elliptic curve $E$, together with two cyclic subgroups $G$ and $H$, of order respectively $M$ and $N$, of the group of degree $0$ line bundles modulo linear equivalence on $E$.

(note that the subgroup generated by $G$ and $H$ need not be of rank $2$, although it will be so generically)

When $N=1$, $Y_0(M,N)$ is just the well known modular curve $Y_0(M)$.

Questions:

1) Are there references for this moduli space/modular curve? Is this well studied/well known?

2) I think of this as the fiber product $Y_0(M) \times_{\mathcal{M}_{1,1}} Y_0(N)$ (the fiber product over the moduli space of elliptic curves). Is this correct?

3) What can we say about the curve $Y_0(M,N)$? Is it irreducible? Are there some kind of formulas for its genus as there are for $Y_0(M)$?