I would like to construct moduli space of elliptic curves with cyclic group $C_n$-action. In other words, I want to classify a pair $(C,\phi)$, where $C$ is an elliptic curve and $\phi:C_n\rightarrow Aut(C)$ is an action, up to $C_n$ equivariant isomorphisms. Is there any good description of this moduli space?
Since endowing $C$ with a $C_n$-action is the same as specifying an $n$-torsion, this may be related to level structure (which unfortunately I don't know anything about).
Note I mean by $Aut(C)$ simply isomorphism group of $C$, not respecting the group structure on $C$.
Note2 I forgot to say an important point; my $C_n$-action is always translation.