 Projective closure of y cubed equals x to the fourth plus one

The projective closure of the superelliptic curve $y^3=x^4+1$ is the smooth plane quarticIts unique point at infinity is $P_{\infty }=[0:1:0]$. The rational function $x=X/Z$ defines a degree-three morphism to ${\mathbb{P}}^1$ ramified with index three at $P_{\infty }$ and at the four affine points $(\alpha ,0)$ with ${\alpha }^4=-1$. Its genus is three, and