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^19*x + (r^104*x + r^78)/(x^2 + x + 1)
z^2 + z = r^19*x + (r^104*x + r^61)/(x^2 + x + 1) + 1
has genus 5 and has 232 rational points.