I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$).
Let $\Gamma(M,N)$ be the subgroup of $SL_2(\mathbb{Z})$ of matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, where $a-1=b=0 \mod M$, $c=d-1=0 \mod N$. Observe that $\Gamma(M,N)$ contains $\Gamma(N)$, the principal congruence subgroup of level $N$.
Now define $Y(M,N)$ as the quotient of the upper half plane $\mathbb{H}$ by the action of $\Gamma(M,N)$.
Questions:
1) Does $Y(M,N)$ parametrize what I wrote in the first paragraph?
2) Is $Y(M,N)$ well-known and studied as $Y(N)$ is? Are there references?
3) For example, what are the irreducible components of $Y(M,N)$? Are they identified, via the restriction of the Weil pairing of $E[N] \times E[N]$, to the elements of $\mu_M$? EDIT: (See answer by David Loeffler) As observed below, this is clearly irreducible. The question is if this truly parametrizes triples $(E,p,q)$ as in the first paragraph, with fixed value of the Weil pairing.
4) Is there any hope of saying what is its genus? Is everything easily deducible from the known properties of the usual congruence subgroups and modular curves?