Let s be an element of F_8 such that s^3 + s + 1 = 0. Let D be the plane quartic defined by
(s*x^2 + s*y^2 + s^3*z^2 + s^3*x*y + s^3*y*z + s^3*x*z)^2 + x*y*z*(x + y + z) = 0,
and let C be the double cover of D defined by
w^2 + w*z = s*x*z + s^2*y*z.
Then the real Weil polynomial of C over F_8 is (t + 1)^5, and C has 140 points over F_64.

The Magma routines in the file IsogenyClasses.magma (available at the URL above) show that there are no genus-5 curves over F_64 that have 141, 142, 143, or 144 points. Thus, N_64(5) is equal to either 140 or 145.