page_title: Bring's curve
section_title: Fundamental domain and systole
hierarchical_section_title: Bring's curve / Fundamental domain and systole
context_page_description: In mathematics, Bring's curve is the curve given by the equations
  It was named by Klein after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund.
  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 is the direct product, which has order 240.
context_section_description: 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 that 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. In particular, the group has four 1 dimensional, four 4 dimensional, four 5 dimensional, and two 6 dimensional irreducible representations - it can be seen that these quantities satisfy  Burnside's lemma:
  as expected.
  The systole of the surface has length
  Similarly to the Klein quartic, Bring's surface does not maximize the systole length among compact Riemann surfaces in its topological category (that is, surfaces having the same genus) despite maximizing the size of the automorphism group. The systole is presumably maximized by the surface referred to a M4 in (Schmutz 1993). The systole length of M4 is
  and has multiplicity 36.
caption: The fundamental icosagon for Bring's surface, complete with side identifications.