2024 AMC 12B 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}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\frac{4}{3}\qquad\textbf{(D)}~\frac{3}{2}\qquad\textbf{(E)}~\frac{5}{3}

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