Klein quartic

Description

The Klein quartic is the plane curve $$X = \set{ [x : y : z] \in \mathbb{P}^2(\C) : x^3 y + y^3 z + z^3 x = 0 }.$$ It is a smooth curve of degree $4$, so it is a compact Riemann surface of genus $\frac{(4-1)(4-2)}{2} = 3$ by the genus–degree formula.

Two automorphisms can be checked by hand. Let $\zeta = e^{2\pi i / 7}$. The map $[x : y : z] \mapsto [\zeta^4 x : \zeta^2 y : \zeta z]$ preserves $X$, because $x^3 y$ is multiplied by $\zeta^{12+2} = 1$, $y^3 z$ by $\zeta^{6+1} = 1$, and $z^3 x$ by $\zeta^{3+4} = 1$. The cyclic shift $[x : y : z] \mapsto [y : z : x]$ permutes the three monomials, so it also preserves $X$.

The full group of holomorphic automorphisms of $X$ is isomorphic to $\mathrm{PSL}(2, \F_7)$, of order $168$. This equals the bound $84(g - 1) = 168$ of Hurwitz's automorphisms theorem for $g = 3$. Up to isomorphism, $X$ is the only compact Riemann surface of genus $3$ with $168$ automorphisms.