How could I find the constant term in the expansion
$$\frac{1}{(1-z_1z_2^{1/5}z_3^{-5}z_4)(1-z_1^{-5}z_2^{-1/5}z_3z_4)(1-z_1z_2z_3z_4)(1-z_4)(z_4^{6t})} $$
in terms of $t$? The reason I'm asking is because this results from an attempt to find the discrete volume of a polytope. I won't go into the details of how I arrived here, but I used Euler's Generating Function, whose corollary relates it to Ehrhart polynomials, and attained this monstrosity of an expression. If I were to try to solve the system of equations arising from setting the exponents of each of the variables $z_1,z_2,z_3,z_4$ equal to $0$, I would come back to a certain system of equations that I was trying to solve in the first place, so this isn't a viable option. I'm really struggling with this and I've spent some time on it, but I can't make meaningful progress. I would sincerely appreciate any help.
This has no constant term, as it contains a factor $z_4^{-6 t}$.