2004 AMC 12B 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 $f(z) = z$?

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.