Entry 79

Level 7. Define \(d_k = \eta(k\tau)\) with Dedekind eta function \(\eta(k\tau)\) and the McKay-Thompson series of class 7B for the Monster $$j_{7B}(\tau) = \left(\frac{d_1}{d_7}\right)^4$$ Examples. We select \(d=7m\) with class number \(h(-d)=2\) and find \(m=5,13,61\) such that the following are special quadratic irrationals $$\begin{align}j_{7B}\Big(\tfrac{1+\sqrt{-5/7}}{2}\Big) &= -7\,U_{5}^{2} = -7\left(\tfrac{1+\sqrt5}2\right)^{2}\\ j_{7B}\Big(\tfrac{1+\sqrt{-13/7}}{2}\Big) &= -7\,U_{13}^{2} = -7\left(\tfrac{3+\sqrt{13}}2\right)^{2}\\ j_{7B}\Big(\tfrac{1+\sqrt{-61/7}}{2}\Big) &= -7\,U_{61}^{2} = -7\left(\tfrac{39+5\sqrt{61}}2\right)^{2}\end{align}$$

as they involve fundamental units \(U_n\). These are analogous to the examples in Level 3B which have the form \(-3^3\,U_n^2\). The last implies the integer $$\left(\sqrt{-7\,U_{61}}+7/\sqrt{-7\,U_{61}}\right)^2 = -7\times39^2=-22^3+1$$ and solutions to the curve \(x^3-1 = 7y^2\) as discussed in the previous entry.