Entry 141

Define the McKay-Thompson series of Class 2A for the Monster $$j_2 = j_{2}(\tau) =\left(\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{12}+2^6 \left(\frac{\eta(2\tau)}{\eta(\tau)}\right)^{12}\right)^2$$ and the Bring-Jerrard quintic $$x^5-5\alpha x -\alpha=0$$ Alternatively $$y(y-5)^4 =j_2$$

Conjecture: "If \(\tau\) is a complex quadratic such that \(j_2=j_{2}(\tau)\) is an algebraic number, then the quintic above has a solvable Galois group."

Example: Let \(j_2\big(\tfrac12\sqrt{-10}\big)=12^4\), then $$\quad y(y-5)^4=12^4\\ z^5-5z-12=0$$ are solvable in radicals.