Entry 76

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

$$j_{6A}(\tau) = \left(\left(\frac{d_2\,d_3}{d_1\,d_6}\right)^6-\left(\frac{d_1\,d_6}{d_2\,d_3}\right)^6\right)^2$$ Examples. 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 well-behaved integers $$\begin{align}j_{6A}\big(\tfrac12\sqrt{-10/3}\big) &= (8\sqrt5)^2\\ j_{6A}\big(\tfrac12\sqrt{-14/3}\big) &= (8\sqrt{14})^2\\ j_{6A}\big(\tfrac12\sqrt{-26/3}\big) &= (20\sqrt{26})^2\\ j_{6A}\big(\tfrac12\sqrt{-34/3}\big) &= (140\sqrt2)^2 \end{align}$$ as well as $$\begin{align}j_{6A}\Big(\tfrac{1+\sqrt{-7/3}}{2}\Big) &= -(4\sqrt7)^2\\ j_{6A}\Big(\tfrac{1+\sqrt{-11/3}}{2}\Big) &= -20^2\\ j_{6A}\Big(\tfrac{1+\sqrt{-19/3}}{2}\Big) &= -52^2\\ j_{6A}\Big(\tfrac{1+\sqrt{-31/3}}{2}\Big) &= -(28\sqrt{31})^2\\ j_{6A}\Big(\tfrac{1+\sqrt{-59/3}}{2}\Big) &= -1060^2\end{align}$$ Note that the prime-generating polynomials $$F(n)=n^2-n+41\\ F(n)=6n^2-6n+31$$ where the latter is prime for \(30\) consecutive values \(n=0 - 29\). Solving \(F(n)=0\) yields \(n=\frac{1+\sqrt{-163}}{2}\) and \(n=\frac{1+\sqrt{-59/3}}{2}\), respectively hence $$j_{1A}\Big(\tfrac{1+\sqrt{-163}}2\Big) = -640320^3\\ j_{6A}\Big(\tfrac{1+\sqrt{-59/3}}2\Big) = -1060^2\quad$$