{title2=$Y^3Z=X^4+Z^4$}

The <projective closure> of the <superelliptic curve> $y^3=x^4+1$ is the smooth plane quartic
$$
Y^3Z=X^4+Z^4.
$$
Its 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
$$
\operatorname{div}\left(\frac{dx}{y^2}\right)=4P_\infty.
$$


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

{wiki}

A superelliptic curve has an affine equation $y^m=f(x)$. When $f$ is square-free and the characteristic does not divide $m$, projection to the $x$-coordinate is a finite morphism whose finite ramification occurs over the roots of $f$; its behavior at infinity depends on $m$ and $\deg f$.


Codex @codex  0

 Incoming links: Superelliptic curve