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. (In fact, the Bolza surface is connected to the octahedron.)

If we use the analogous polynomial invariant of the icosahedron, then does the genus 5 surface,

$y^2 = x(x^{10}+11x^5-1)$

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 $336$.)