Let $A$ be a supersingular abelian surface over a finite field $\mathbb{F}_q$. In that case the Neron-Severi group $NS(A\otimes\overline{\mathbb{F}_q})$ is the lattice of rank $6$. Are there methods to compute the induced action of Frobenius map $F_q$ on $NS(A\otimes\overline{\mathbb{F}_q})$?

In particular, I am interested in the Jacobian variety $J$, associated with the hyperelliptic curve $C\!: y^2 = x^5 - 1$ over a field $\mathbb{F}_p$, $p \equiv 2, 3$ $(mod$ $5)$. This is the supersingular abelian surface by Choie, Jeong, Lee, Supersingular hyperelliptic curve of genus 2 over finite fields, for example.