We know that an elliptic curve over $\mathbb{C}$ is a torus, and I've read that an abelian surface over $\mathbb{C}$ is isomorphic to $\mathbb{C}^2/\Lambda$, where $\Lambda \cong \mathbb{Z}^4$. So I suppose that an abelian variety of dimension $n$ over $\mathbb{C}$ is isomorphic to $\mathbb{C}^n/\Lambda$, where $\Lambda \cong \mathbb{Z}^{2n}$.
Question 1. I wonder to what extent is the above true over the algebraic closure of a number field $k$? For example, is an elliptic curve over $\bar{k}$ isomorphic to a torus over $\bar{k}$?
Question 2. In algebraic geometry we also have the definition of a torus to be a product of multiplicative groups $\mathbb{G}_m$. For an elliptic curve over $\mathbb{C}$, would it then be isomorphic to $\mathbb{G}_m^n$ for some $n$? If so, how do we determine $n$?