Define $X^{\prime}:=\left\{(x, y) \in \mathbb{C}^{2}: x^{3} y+y^{3}+x=0\right\}$.

(a) Prove by defining an atlas that $X^{\prime}$ is a Riemann surface.

(b) Now assume that by adding finitely many points, it is possible to compactify $X^{\prime}$ to a Riemann surface $X$ so that the coordinate projections extend to holomorphic maps $\pi_{x}$ and $\pi_{y}$ from $X$ to $\mathbb{C}_{\infty}$. Compute the genus of $X$.

(c) Assume that any holomorphic automorphism of $X^{\prime}$ extends to a holomorphic automorphism of $X$. Prove that the group Aut $(\mathrm{X})$ of holomorphic automorphisms of $X$ contains an element $\phi$ of order 7 . Prove further that there exists a holomorphic map $\pi: X \rightarrow \mathbb{C}_{\infty}$ which satisfies $\pi \circ \phi=\pi$.