Let s in F_8 satisfy s^3 + s + 1 = 0. Let E be the elliptic curve over F_8 defined by y^2 + y = x^3, and let C be the degree-3 Kummer cover of E defined by z^3 = ((s*x^2 + s^6*x + s^3)*y + (s^6*x^3 + x^2 + s^6*x))/(x^2 + x + s^5). Then C is a curve of genus 5 with real Weil polynomial equal to x * (x^4 - 6*x^2 + 3), and C has 133 points over F_64.

This curve was found by a computer search of degree-3 Kummer extensions of E of genus 5.