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Concept
Bring's curve
Summary
In mathematics, Bring's curve (also called Bring's surface and, by analogy with the Klein quartic, the Bring sextic) is the curve in cut out by the homogeneous equations
It was named by after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund. Note that the roots xi of the Bring quintic satisfies Bring's curve since for
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 , which has order 240.
Bring's curve can be obtained as a Riemann surface by associating sides of a hyperbolic icosagon (see fundamental polygon). The identification pattern is given in the adjoining diagram. The icosagon (of area , by the Gauss-Bonnet theorem) can be tessellated by 240 (2,4,5) triangles. The actions that transport one of these triangles to another give the full group of automorphisms of the surface (including reflections). Discounting reflections, we get the 120 automorphisms mentioned in the introduction. Note that 120 is less than 252, the maximum number of orientation preserving automorphisms allowed for a genus 4 surface, by Hurwitz's automorphism theorem. Therefore, Bring's surface is not a Hurwitz surface. This also tells us that there does not exist a Hurwitz surface of genus 4.
The full group of symmetries has the following presentation:
where is the identity action, is a rotation of order 5 about the centre of the fundamental polygon, is a rotation of order 2 at the vertex where 4 (2,4,5) triangles meet in the tessellation, and is reflection in the real line. From this presentation, information about the linear representation theory of the symmetry group of Bring's surface can be computed using GAP.
Official source
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