AMC 12 2019b Problem 17

How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

$\textbf{(A) }0 \textbf{(B) }1 \textbf{(C) }2 \textbf{(D) }4 \textbf{(E) }\text{infinitely many}$