P4
v+w+x+y+z=v2+w2+x2+y2+z2=v3+w3+x3+y3+z3=0.
x5+ax+b=0
5 | |
\sum | |
i=1 |
k | |
x | |
i |
=0
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
S5 x Z2
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
12\pi
\langler,s,t|r5=s2=t2=rtrt=stst=(rs)4=(sr3sr2)2=e\rangle
where
e
r
s
t
4(12)+4(42)+4(52)+2(62)=4+64+100+72=240
as expected.
The systole of the surface has length
12\sinh-1\left(\tfrac{1}{2}\sqrt{\tfrac{1}{2}(\sqrt{5}-1)}\right) ≈ 4.60318
and multiplicity 20, a geodesic loop of that length consisting of the concatenated altitudes of twelve of the 240 (2,4,5) triangles.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 . The systole length of M4 is
2\cosh-1\left(\tfrac{1}{2}(5+3\sqrt{3})\right) ≈ 4.6245,
and has multiplicity 36.
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. Following this pattern, one may reasonably conjecture that Bring's surface maximizes the first positive eigenvalue of the Laplacian (among surfaces in its topological class).