This question was prompted by the post here, and I asked this earlier, deleted it, and due to pressure exerted by Ilya Nikokoshev, I am asking it again. Apologies to Pavel Etingof.

Q1. Let $\Lambda$ be a lattice in $\mathbb{C}$. We look at the behavior of the zero set of the Weierstrass $\wp$-function for this lattice. By integration around a unit cell for the lattice, we see that the number of poles and zeros are the same. So there has to be two zeros. We position the fundamental domain to be symmetric about $0$, and from the expression for $\wp$, we see that the zero should be $z$ and $-z$ in case they are distinct. Otherwise, it is a double zero, which is one of the $2$-torsion points.

Now, in the case that the zero is not a double zero, can anything be said about its location from the knowledge of $\Lambda$?

Q2. This is stupid. . But, what is the degree of the branched covering $\wp: \mathbb{C}/\Lambda \rightarrow \mathbb{P}^1(\mathbb{C})$? I must confess that I am not good in this stuff.