  We study the intersection of two particular Fermat hypersurfaces in
$\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we study
arithmetic properties of this curve in terms of its quotients. Explicit
computation of the quotients is done using a Gr\"obner basis algorithm. We also
study the $p$-rank, zeta function, and number of rational points, of the modulo
$p$ reduction of the curve. We show that the Jacobian of the genus 2 quotient
is $(4,4)$-split.
