Your curve is hyperelliptic.

If $X_g$ is a 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 [Shaska - Determining the automorphism group of a hyperelliptic curve](http://arxiv.org/abs/math.AG/0312284).

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://scholar.google.com/scholar_url?hl=en&q=http://arxiv.org/pdf/math.AG/0312284&sa=X&scisig=AAGBfm1ulu1cuyVqpwFOf4s7OzHqIHVD8w&oi=scholarr