Bring's curve

Bring's curve is related to the small stellated dodecahedron and the dodecadodecahedron.^[1]

In mathematics, Bring's curve (also called Bring's surface and, by analogy with the Klein quartic, the Bring sextic) is the curve in the projective space ${\displaystyle \mathbb {P} ^{4}}$ cut out by the homogeneous equations

${\displaystyle v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.}$

It was named by Klein (2003, p.157) after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund. Note that the roots x_i of the Bring quintic ${\displaystyle x^{5}+ax+b=0}$ satisfies Bring's curve since ${\displaystyle \sum _{i=1}^{5}x_{i}^{k}=0}$ for ${\displaystyle k=1,2,3.}$

The automorphism group of the curve is the symmetric group S₅ of order 120, given by permutations of the 5 coordinates. This is the largest possible automorphism group of a genus 4 complex curve.

The curve can be realized as a triple cover of the sphere branched in 12 points, and is the Riemann surface associated to the small stellated dodecahedron. It has genus 4. The full group of symmetries (including reflections) is the direct product ${\displaystyle S_{5}\times \mathbb {Z} _{2}}$, which has order 240.