Entry 186

Using the \(j_3(\tau)\) from the previous entry, then the following quintics have a solvable Galois group

$$\begin{align}\qquad y^3(y-5)^2 &= -3\cdot 4^3\\ y^3(y-5)^2 &= -5\cdot 12^3\\ y^3(y-5)^2 &= -7\cdot 36^3\\ y^3(y-5)^2 &= -\sqrt{11}\,(150\sqrt3+78\sqrt{11})^3\end{align}$$

as well as

$$\begin{align}y^3(y-5)^2 &= -(2\sqrt3)^6\\ y^3(y-5)^2 &= -(4\sqrt3)^6\\ y^3(y-5)^2 &= -(10\sqrt3)^6\qquad \qquad\end{align}$$

as discussed in Entry 142.