Entry 75

V. Level 5. The McKay-Thompson series of class 5B for the Monster.

$$j_{5B}(\tau) =\left(\frac{\eta(\tau)}{\eta{(5\tau)}}\right)^{6}$$

Examples. Surprisingly, the following "cubes" are powers of the golden ratio $\phi=\frac{1+\sqrt5}2$  $$\begin{align}j_{5B}\Big(\tfrac{1+\sqrt{-7/5}}{2}\Big) &= -(\sqrt5)^3\,\phi^3\\ j_{5B}\Big(\tfrac{1+\sqrt{-23/5}}{2}\Big) &=  -(\sqrt5)^3\,\phi^9\\ j_{5B}\Big(\tfrac{1+\sqrt{-47/5}}{2}\Big) &=  -(\sqrt5)^3\,\phi^{15}\end{align}$$ These discriminants $d=5m$ have class number $h(-d)=2$.