= Solution
Let $\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2$ and $T=\mathbb C/\Lambda$. Its fundamental parallelogram with opposite sides identified is a <torus>, so $\chi(T)=0$. The <Weierstrass elliptic function> is even and doubly periodic. One way to verify the latter from its series is to differentiate: the locally normally convergent derivative series is
$$
\wp'(z)=-2\sum_{\omega\in\Lambda}(z-\omega)^{-3},
$$
which is absolutely convergent away from the <lattice> and periodic by reindexing. Thus $\wp(z+\omega_j)-\wp(z)$ is constant. Evaluating at $z=-\omega_j/2$ and using evenness shows that this constant is zero for each generating period.
Therefore $[z]\mapsto\wp(z)$ descends to a <meromorphic function> on $T$, with one double pole at $[0]$. Viewed as a map to the <Riemann sphere>, a pole is a holomorphic point in the target coordinate $1/w$. The map is consequently holomorphic everywhere and
$$
\boxed{T\longrightarrow\mathbb{CP}^1,\qquad[z]\longmapsto\wp(z),\qquad\deg\wp=2.}
$$
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