Explicit equations of the fake projective plane $(a=7,p=2,\emptyset,D_3 X_7)$

Abstract

We find explicit equations of the fake projective plane $(a=7,p=2,\emptyset,D_3 X_7)$, which lies in the same class as the fake projective plane $(a=7,p=2,\emptyset,D_3 2_7)$ with $21$ automorphisms whose equations were previously found by Borisov and Keum. The method involves finding a birational model of a common Galois cover of these two surfaces.