Let r be an element of F_128 that satisfies r^7 + r + 1 = 0. Then the curve C over F_128 defined by the two equations
y^2 + y = r^23*x + r^75/(x^2 + x) + 1
z^2 + z = r^75*x + r^101/(x^2 + x) + 1
has genus 6 and has 251 rational points.