$X_1(M,MN)$ (awaiting review)

$X_1(M,MN)$ is the modular curve $X_H$ for $H\le \GL_2(\widehat\Z)$ the inverse image of $\begin{pmatrix} 1 & M* \\ 0 & 1+M* \end{pmatrix} \subset \GL_2(\Z/MN\Z)$. As a moduli space it parameterizes triples $(E,P,Q)$, where $E$ is an elliptic curve over $k$, $P \in E[MN](k)$ is a point of order $MN$, and $Q\in E[M](k)$ is a point of order $M$ such that $E[M]=\langle NP,Q\rangle$.

The canonical field of definition of $X_1(M,MN)$ is $\Q(\zeta_M)$, which means that the database of modular curves $X_H/\Q$ only includes $X_1(M,MN)$ for $M\le 2$.