Entry 77

Level 6. Define \(d_k = \eta(k\tau)\) with Dedekind eta function \(\eta(k\tau)\) and the McKay-Thompson series of class 6B for the Monster.

$$j_{6B}(\tau) = \left(\frac{d_2\,d_3}{d_1\,d_6}\right)^{12}$$

Example. We select \(d=12m\) with class number \(h(-d)=4\) and find  

$$m=10, 14, 26, 34\\ m=7, 11, 19, 31, 59$$ such that the following are special quadratic irrationals $$\begin{align}j_{6B}\big(\tfrac12\sqrt{-10/3}\big) &= U_5^{12}=\big(\tfrac{1+\sqrt5}2\big)^{12}\\ j_{6B}\big(\tfrac12\sqrt{-14/3}\big) &= U_{14}^2 =\big(15+4\sqrt{14}\big)^2 \\ j_{6B}\big(\tfrac12\sqrt{-26/3}\big) &= U_{26}^4 =\big(5+\sqrt{26}\big)^4\\ j_{6B}\big(\tfrac12\sqrt{-34/3}\big) &= U_2^{12}=\big(1+\sqrt{2}\big)^{12}\end{align}$$

$$\quad \begin{align}j_{6B}\Big(\tfrac{1+\sqrt{-7/3}}{2}\Big) &= -U_{21}^{3}= -\big(\tfrac{5+\sqrt{21}}2\big)^{3}\\ j_{6B}\Big(\tfrac{1+\sqrt{-11/3}}{2}\Big) &= -U_{11}^2 = -\big(10+3\sqrt{11}\big)^2\\ j_{6B}\Big(\tfrac{1+\sqrt{-19/3}}{2}\Big) &= -U_{3}^6 = -\big(2+\sqrt{3}\big)^6\\ j_{6B}\Big(\tfrac{1+\sqrt{-31/3}}{2}\Big) &= -U_{93}^{3}= -\big(\tfrac{29+3\sqrt{93}}2\big)^{3}\\ j_{6B}\Big(\tfrac{1+\sqrt{-59/3}}{2}\Big) &= -U_{59}^2 = -\big(530+69\sqrt{59}\big)^2 \end{align}$$

since they are fundamental units \(U_n\). Note the integer $$\left(\sqrt{-\big(530+69\sqrt{59}\big)^2}-1/\sqrt{-\big(530+69\sqrt{59}\big)^2}\right)^2 = -1060^2$$ and similarly for the others as discussed in the previous entry.