Quotient of Punctured Disc

Question

Let $D=\{z\in\mathbb{C}:0<|z|<1\}$. So $D$ is a punctured disc. If I put the group action generated by $z\mapsto ze^{2\pi i/n}$, then consider the quotient. As at every point, there is an invariant neighbourhood around it, we ge get a Riemann surface. For different $n$, do we get different Reimann surface or not? I think the answer is that they are the same, as we take the holomorphic map between the one given by $n_1$ and $n_2$ by doing scaling along the angle, and this map should be biholomorphic. Is this correct?