 Fermat cubic curve

The Fermat cubic $x^3+y^3+z^3=0$ is a smooth projective curve: its three first partial derivatives vanish simultaneously only at the forbidden zero vector. The genus of a smooth plane curve formula gives genus one.

This is the degree-three case of a Fermat equation; in general $x^n+y^n+z^n=0$ defines a Fermat curve.