I apologize in advance if my question is rather trivial, but i have trouble understanding a basic fact about elliptic curves.. I have always wrote an elliptic curve $E$ as $\mathbb{C}/\Lambda$, where $\Lambda$ is a lattice of rank 2, i.e. $\Lambda=\mathbb{Z}+\mathbb{Z}\tau$, $\tau\in\mathbb{H}$ the upper half plane.
Now I have read the equality $E=\mathbb{C}/\Lambda=\mathbb{C}^*/\mathbb{Z}$ where the last quotient is given by multiplication with $t^n$, $t=e^{i2\pi\tau}$.
My question is: how can $\mathbb{C}/\Lambda$ and $\mathbb{C}^*/\mathbb{Z}$ be the same thing? the two actions seem really different to me..