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.