Entry 184

Using the \(j_2(\tau)\) from the previous entries, then the following quintics have a solvable Galois group

$$\begin{align}y(y-5)^4 &= -(4\sqrt2)^4\\ y(y-5)^4 &=  -(12\sqrt2)^4\\ y(y-5)^4 &=  -(84\sqrt2)^4\quad\end{align}$$

as well as,

$$\begin{align}y(y-5)^4 &= (4\sqrt{3})^4\\ y(y-5)^4 &= 12^4\\ y(y-5)^4 &= (12\sqrt{11})^4\\ y(y-5)^4 &= 396^4\qquad\end{align}$$

and an example with class number \(4\)

$$\quad y(y-5)^4 = -2^{9}\cdot3^4\big(111+13\sqrt{73}\big)^3$$

though for all radical \(j_2(\tau)\) as also discussed in Entry 141.