Entry 80

Level 10. Define $d_k=\eta(k\tau)$ with the Dedekind eta function $\eta(\tau)$ and the McKay-Thompson series of class 10A for the Monster. 

$$j_{10A}(\tau) = \left(\left(\frac{d_2\,d_5}{d_1\,d_{10}}\right)^3-\left(\frac{d_1\,d_{10}}{d_2\,d_5}\right)^3\right)^2$$ Examples. We select $d=20m$ with class number $h(-d)=4$ and find odd $m=17$ as well as $$m=6, 14, 26, 38$$ such that the following are well-behaved integers $$\begin{align}j_{10A}\Big(\tfrac{1+\sqrt{-17/5}}{2}\Big) &= -18^2\\ j_{10A}\big(\tfrac12\sqrt{-6/5}\big) &= 6^2\\ j_{10A}\big(\tfrac12\sqrt{-14/5}\big) &= 14^2\\ j_{10A}\big(\tfrac12\sqrt{-26/5}\big) &= 36^2\\ j_{10A}\big(\tfrac12\sqrt{-38/5}\big) &= 76^2 \end{align}$$ Note that the last is responsible for the prime-generating polynomial $$F(n)=10n^2+19$$ which has discriminant $d = -5\times 38 = -190$ and is prime for $19$ consecutive values $n=0 - 18$.