Let $C$ be the projective curve (over an algebraically closed field $k$ of characteristic zero) defined by the affine equation

$$x^{5}+y^{5}=1$$

Determine the points at infinity of $C$ and show that $C$ is smooth.

Determine the divisors of the rational functions $x, y \in k(C)$.

Show that $\omega=d x / y^{4}$ is a regular differential on $C$.

Compute the divisor of $\omega$. What is the genus of $C$ ?