Bring's curve

cut out by the homogeneous equations 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.

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.

The full group of symmetries (including reflections) is the direct product

Bring's curve can be obtained as a Riemann surface by associating sides of a hyperbolic icosagon (see fundamental polygon).

The actions that transport one of these triangles to another give the full group of automorphisms of the surface (including reflections).

This also tells us that there does not exist a Hurwitz surface of genus 4.

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

From this presentation, information about the linear representation theory of the symmetry group of Bring's surface can be computed using GAP.

Little is known about the spectral theory of Bring's surface, however, it could potentially be of interest in this field.

The Bolza surface and Klein quartic have the largest symmetry groups among compact Riemann surfaces of constant negative curvature in genera 2 and 3 respectively, and thus it has been conjectured that they maximize the first positive eigenvalue in the Laplace spectrum.

There is strong numerical evidence to support this hypothesis, particularly in the case of the Bolza surface, although providing a rigorous proof is still an open problem.