Entry 180

Using the \(j\)-functions from the previous entries, then the following quintics have a solvable galois group $$\begin{align}z^3(z^2+5z+40) &= -960^3\\ z^3(z^2+5z+40) &= -5280^3\\ z^3(z^2+5z+40) &= -640320^3\qquad\end{align}$$ and all other radical \(j(\tau)\). Examples from class number \(2\)$$\begin{align}z^3(z^2+5z+40) &= 2^3(25-13\sqrt{5})^3\\ z^3(z^2+5z+40) &= 30^3(31-9\sqrt{13})^3\\ z^3(z^2+5z+40) &= 60^3(2837-468\sqrt{37})^3\end{align}$$

a form related to the Brioschi quintic and discussed in Entry 140.