Let s in F_9 satisfy s^2 = s + 1, and let E be the elliptic curve y^2 = x^3 + s*x^2 + s^5. Let C be the degree-5 Kummer cover of E defined by z^5 = ((s^3*x + s^5)*y + (s^5*x^3 + s^3*x - 1))/(x + s^3). Then C is a genus-5 curve with real Weil polynomial equal to (x - 1)^2 * (x + 1)^3, and C has 167 points over F_81.

This curve was found by a computer search of degree-5 Kummer extensions of E of genus 5.