# Moduli space of elliptic curves with $C_n$ action

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.

-
It's not entirely clear what exactly you want. Adding a fixed $n$-torsion point $P$ does not give you an automorphism of the curve, since if $\phi$ is the map $Q\mapsto Q+P$, then $\phi(Q_1+Q_2)\neq \phi(Q_1)+\phi(Q_2)$. In fact, automorphism groups of elliptic curves over number fields are either $C_2$ (generic case - no CM, or CM by a ring with no exciting units), or $C_4$ or $C_6$ (if you have CM by $\mathbb{Q}(i)$, respectively by $\mathbb{Q}(\zeta_3)$). – Alex B. Dec 14 '12 at 15:00
I meant by $Aut(C)$ simply $isomorphism$ group of $C$, not respecting the group structure on $C$. (it is typical to write $Aut(X)$ for the automorphism group of a general variety $X$). Sorry for the confusion. – M. K. Dec 14 '12 at 20:38
Dear M.K., Why not read something about level structures; there are many sources? Also, do your elliptic curves have a fixed point on them? Regards, – Matt E Dec 14 '12 at 20:44
Dear Matt E, I briefly go through some lecture notes on level structure of elliptic curve, but it seems to me that it parametrizes a "pair" of $n$-torsion points and I am not convinced that it may be modified to help my case. As to your second question, elliptic curve always has an origin by definition. – M. K. Dec 14 '12 at 22:56
I forgot to say an important point; my $C_n$-action is always translation. Thank you all for comments clarifying ambiguous points. – M. K. Dec 14 '12 at 22:58