Entry 83

Level 13. Define \(d_k = \eta(k\tau)\) with Dedekind eta function \(\eta(k\tau)\) and the McKay-Thompson series of class 13B $$j_{13B}(\tau) = \left(\frac{d_1}{d_{13}}\right)^2$$

Examples. Let \(d=13m\) with class number \(h(-d)=2\) for \(m=7, 31\) and we find a phenomenon similar to Level 5 $$\begin{align}j_{13B}\Big(\tfrac{1+\sqrt{-7/13}}{2}\Big) & = -\sqrt{13}\left(\tfrac{3+\sqrt{13}}2\right)\\ j_{13B}\Big(\tfrac{1+\sqrt{-31/13}}{2}\Big) & = -\sqrt{13}\left(\tfrac{3+\sqrt{13}}2\right)^3 \end{align}$$

But where Class 5B involves the fundamental unit \(U_5\) or the golden ratio, Class 13B involves \(U_{13} = \tfrac{3+\sqrt{13}}2\) also known as the bronze ratio.