Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are,
$$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 + 1$$
$$H=z(z^{10} + 11z^5 - 1)$$
$$T=z^{30} + 522 z^{25} - 10005 z^{20} - 10005 z^{10} - 522 z^5 + 1$$
These obey the nice relationship,
$$f^3+1728H^5 = T^2$$
Question: What are the three corresponding equations for the snub cube? Do they obey a similar relation?