AMC 12 2004b Problem 16

A function $f$ is defined by $f(z) = i\overline{z}$, where $i=\sqrt{-1}$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$?

$\mathrm{(A)}\ 0 \mathrm{(B)}\ 1 \mathrm{(C)}\ 2 \mathrm{(D)}\ 4 \mathrm{(E)}\ 8$