All complex $5\times 5$ Hadamard matrices are equivalent to the Fourier matrix $$F_5=F_5^{(0)}=\left[\begin{array}{lllll} 1& 1 & 1 & 1 & 1 \\ 1& w & w^2& w^3& w^4\\ 1& w^2& w^4& w & w^3\\ 1& w^3& w & w^4& w^2\\ 1& w^4& w^3& w^2& w \end{array}\right]:\quad w=\exp(2\pi i/5)$$ which is an isolated point [62].