Let $C_1,C_2$ be smooth, projective curves of genera $g_1,g_2 \geq 2$. Assume that a group $G$ of order $(g_1 - 1)(g_2 - 1)$ acts on $C_1$ and $C_2$ such that $C_1/G \cong \mathbb{P}^1$ and $C_2/G \cong \mathbb{P}^1$. Assume that $G$ acts freely on $C_1 \times C_2$, then the quotient $S = (C_1 \times C_2)/G$ is smooth.

I know that $q(S) = p_g(S)$: we have $\chi(C_1 \times C_2) = (g_1 - 1)(g_2 - 1)$, and therefore $\chi(S) = 1$, which implies that $q(S) = p_g(S)$. Is it also true that both these numbers are equal to $0$? If so, why?