Icosahedral Equation

Hunt (1996) gives the ``dehomogenized'' icosahedral equation as

$$[(z^{20}+1)-228(z^{15}-z^5)+494z^{10})^3+1728uz^5(z^{10}+11z^5-1)^5=0.$$

Other forms include

$I(u, v, Z) = u^5 v^5 (u^{10} + 11 u^5 v^5 - v^{10})^5$
$- [u^{30} + v^{30} - 10005 (u^{20} v^{10} + u^{10} v^{20})+ 522 (u^25 v^5 - u^5 v^25)]^2 Z=0$

and

$$I(z, 1, Z)=z^5(-1+11z^5+z^{10})^5-[1+z^{30}-10005(z^{10}+z^{20})+522(-z^5+z^{25})]^2Z=0.$$

References

Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, p. 146, 1996.