Let $\Lambda$ be the lattice $\mathbb{Z}+\mathbb{Z} i, X$ the torus $\mathbb{C} / \Lambda$, and $\wp$ the Weierstrass elliptic function with respect to $\Lambda$.

(i) Let $x \in X$ be the point given by $0 \in \Lambda$. Determine the group

$$G=\{f \in \operatorname{Aut}(X) \mid f(x)=x\}$$

(ii) Show that $\wp^{2}$ defines a degree 4 holomorphic map $h: X \rightarrow \mathbb{C} \cup\{\infty\}$, which is invariant under the action of $G$, that is, $h(f(y))=h(y)$ for any $y \in X$ and any $f \in G$. Identify a ramification point of $h$ distinct from $x$ which is fixed by every element of $G$.

[If you use the Monodromy theorem, then you should state it correctly. You may use the fact that $\operatorname{Aut}(\mathbb{C})=\{a z+b \mid a \in \mathbb{C} \backslash\{0\}, b \in \mathbb{C}\}$, and may assume without proof standard facts about $\wp$.]