Let $G$ be the set of polynomials of the form $P(z)=z^n+c_{n-1}{z}^{n-1}+\dots +c_2{z}^2+c_{1}z+10$, where $c_1,c_2,\dots ,c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?