AMC 12 2024b Problem 12

Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?

\textbf{(A)}\frac{3}{4} \textbf{(B)}1 \textbf{(C)}\frac{4}{3} \textbf{(D)}\frac{3}{2} \textbf{(E)}~\frac{5}{3}