The Klein quartic has the smooth projective plane modelIts affine chart is . If is a primitive seventh root of unity, the projective transformationis an automorphism of order seven.
The coordinate function has degree three on the Klein quartic. It has ramification index three at , index two at the seven affine points satisfying and , and index two at . The total ramification is ten, so the Riemann-Hurwitz formula gives genus three.
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The Klein quartic is a notable and interesting example of a mathematical object in the field of topology and algebraic geometry. Specifically, it is a compact Riemann surface of genus 3, which can be represented as a complex algebraic curve of degree 4.