Bring's curve
In mathematics, Bring's curve is the curve in the projective space 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 is the direct product, which has order 240.
The curve may also be modelled by the singular plane curve in
This form was used by Hulek and Craig to study modular properties of the curve.
Fundamental domain and systole
Bring's curve can be obtained as a Riemann surface by associating sides of a hyperbolic icosagon. The identification pattern is given in the adjoining diagram. The icosagon can be tessellated by 240 triangles. The actions that transport one of these triangles to another give the full group of automorphisms of the surface. 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 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, and we have
as expected.
The systole of the surface has length
and multiplicity 20, a geodesic loop of that length consisting of the concatenated altitudes of twelve of the 240 triangles.
Similarly to the Klein quartic, Bring's surface does not maximize the systole length among compact Riemann surfaces in its topological category despite maximizing the size of the automorphism group. The systole is presumably maximized by the surface referred to a M4 in. The systole length of M4 is
and has multiplicity 36.