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We study the intersection of two particular Fermat hypersurfaces in$\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we studyarithmetic properties of this curve in terms of its quotients. Explicitcomputation of the quotients is done using a Gr\"obner basis algorithm. We alsostudy 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 quotientis $(4,4)$-split.

The Intersection of Two Fermat Hypersurfaces in P^3 via Computation of Quotient Curves