Entry 143

Define the McKay-Thompson series of Class 4A for the Monster $$j_4 = j_{4}(\tau) = \left(\left(\frac{\eta(\tau)}{\eta(4\tau)}\right)^{4} + 4^2 \left(\frac{\eta(4\tau)}{\eta(\tau)}\right)^{4}\right)^2 = \left(\frac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \right)^{24}$$ and the Bring-Jerrard quintic $$y^5+5y=\left(\frac{64}{\sqrt{j_4}}-\sqrt{j_4}\right)^{1/2}$$

Conjecture: "If \(\tau\) is a complex quadratic such that \(j_4=j_{4}(\tau)\) is an algebraic number, then the quintic above has a solvable Galois group."

Example: Let \(j_4\big(\tfrac12\sqrt{-7}\big)=2^{12}\), then $$y^5+5y=\sqrt{-63}$$ is solvable in radicals.