Entry 142

Define the McKay-Thompson series of Class 3A for the Monster $$j_3 = j_{3}(\tau) =\left(\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{6}+3^3 \left(\frac{\eta(3\tau)}{\eta(\tau)}\right)^{6}\right)^2$$ and the Euler-Jerrard quintic $$x^5+5\sqrt{\alpha}\, x^2 -\sqrt{\alpha} = 0$$ Alternatively $$y^3(y-5)^2 =j_3$$

Conjecture: "If \(\tau\) is a complex quadratic such that \(j_3=j_{3}(\tau)\) is an algebraic number, then the quintic above has a solvable Galois group."

Example: Let \(j_3\big(\tfrac{1+\sqrt{-89/3}}2\big)=-300^3\), then $$y^3(y-5)^2 = -300^3$$ is solvable in radicals.