I stumbled upon the fact that the Bolza surface can be obtained as the locus of the equation,

$y^2 = x^5-x$

Its automorphism group has the highest order for genus 2, namely 48. I recognized $x^5-x$ as a polynomial invariant of the octahedron. If we use the analogous polynomial invariant of the icosahedron, then does the genus 5 surface,

$y^2 = x(x^{10}+11x-1)$?

have special properties? How close does the order of its automorphism group get to the bound 84(g-1)? (For g = 5, this would be 336.)