Entry 78

Level 7. This level is special since we have to consider the curve \(x^3-1 = 7y^2\). Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\) and the McKay-Thompson series of class 7A for the Monster.  $$j_{7A}(\tau) = \left(\left(\frac{d_1}{d_7}\right)^2+7\left(\frac{d_7}{d_1}\right)^2\right)^2$$ Examples. We select \(d=7m\) with class number \(h(-d)=2\) and find \(m=5,13,61\) such that the following are well-behaved integers $$\begin{align}j_{7A}\Big(\tfrac{1+\sqrt{-5/7}}{2}\Big) &= -7\times1^2 = -2^3+1\\ j_{7A}\Big(\tfrac{1+\sqrt{-13/7}}{2}\Big) &= -7\times3^2 = -4^3+1\\ j_{7A}\Big(\tfrac{1+\sqrt{-61/7}}{2}\Big) &= -7\times39^2 = -22^3+1 \end{align}$$ A WolframAlpha search for positive integer solutions to \(x^3-1 = 7y^2\) reveals only these three. However, if we allow \((x,y)\) to be higher algebraic integers with the same odd degree and a solvable Galois group, then it seems there are infinitely many. For example, we select \(d=7m\) with class number \(h(-d)=6\) and find \(m=101\) and others so $$j_{7A}\Big(\tfrac{1+\sqrt{-101/7}}{2}\Big) =-x^3+1=-7y^2$$ where \((x,y)\) are the real roots of cubics $$x^3 - 46x^2 - 380x - 800 = 0\\ y^3 - 145y^2 - 357y - 1235=0$$ For class number \(h(-d)=10\), we find \(m=17\) and others so $$j_{7A}\Big(\tfrac{1+\sqrt{-17/7}}{2}\Big) =-x^3+1=-7y^2$$ where \((x,y)\) are the real roots of solvable quintics $$x^5 - 5x^4 + 4x^3 - 15x^2 - 23x - 11 = 0\\ y^5 - 5y^4 + 5y^3 - 7y^2 - 1=0$$ and so on.