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  "body": "\n## Description\n\nThe *Klein quartic* is the plane curve\n$$X = \\set{ [x : y : z] \\in \\mathbb{P}^2(\\C) : x^3 y + y^3 z + z^3 x = 0 }.$$\nIt is a smooth curve of degree $4$, so it is a compact [[Riemann surface|5K0N8QMV6Q]] of genus $\\frac{(4-1)(4-2)}{2} = 3$ by the genus–degree formula.\n\nTwo automorphisms can be checked by hand.\nLet $\\zeta = e^{2\\pi i / 7}$.\nThe 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 $\\zeta^{3+4} = 1$.\nThe cyclic shift $[x : y : z] \\mapsto [y : z : x]$ permutes the three monomials, so it also preserves $X$.\n\nThe full group of holomorphic automorphisms of $X$ is isomorphic to $\\mathrm{PSL}(2, \\F_7)$, of order $168$.\nThis equals the bound $84(g - 1) = 168$ of Hurwitz's automorphisms theorem for $g = 3$.\nUp to isomorphism, $X$ is the only compact Riemann surface of genus $3$ with $168$ automorphisms.\n\n## Explanations\n\n### L1 human\n\nThe Klein quartic is a surface with three holes that is as symmetric as a surface of its genus can be.\nNo compact Riemann surface of genus $3$ has more than $168$ symmetries, and this one has exactly $168$.\nIt can be built by gluing $24$ regular heptagons, with three at each corner, in the hyperbolic plane.\nIts equation $x^3 y + y^3 z + z^3 x = 0$ is short, and one can see some symmetries directly: a cyclic shift of the variables, and a scaling by seventh roots of unity.\n",
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