AMC 12 2013b 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 \textbf{(B)}\ 528 \textbf{(C)}\ 576 \textbf{(D)}\ 992 \textbf{(E)}\ 1056