2013 AMC 12B Problem 25

Let G be the set of polynomials of the form P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50, where c_1,c_2,\cdots, c_{n-1} are integers and P(z) has distinct roots of the form a+ib with a and b integers. How many polynomials are in G?

\textbf{(A)}\ 288\qquad\textbf{(B)}\ 528\qquad\textbf{(C)}\ 576\qquad\textbf{(D)}\ 992\qquad\textbf{(E)}\ 1056

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