Your curve is hyperelliptic.

If $X_g$ is an hyperelliptic curve of genus $g$, then $\textrm{Aut}(X_g)$ is a central extension of degree $2$ of one of the groups $$\mathbb{Z}_n, D_n, A_4, S_4, A_5,$$

see [this paper][1].

In the case of Bolza curve the polynomial $x^5-x$ is invariant by the automorphism group of the octahedron, which is $S_4$. In fact, the automorphism group of the Bolza curve is a central extension of $S_4$ by the group of order $2$ generated by the hyperelliptic involution, hence it has order $2 \cdot |S_4|=48$.

Regarding your curve, the polynomial at the right hand side is invariant by the automorphism group of the icosahedron, which is $A_5$. Then the automorphism group is a central extension of $A_5$ by the hyperelliptic involution, hence it has order $2 \cdot |A_5|= 120$.


  [1]: http://www.google.com/url?sa=t&rct=j&q=hurwitz%2520curvge%2520is%2520never%2520hyperelliptic&source=web&cd=3&ved=0CDMQFjAC&url=http%253A%252F%252Fciteseerx.ist.psu.edu%252Fviewdoc%252Fdownload%253Fdoi%253D10.1.1.89.7871%2526rep%253Drep1%2526type%253Dpdf&ei=6_5NT_qOLsbS4QTtsKjhAg&usg=AFQjCNFV9des8_dziECG87F8kVgVwzvv2g&cad=rja