Entry 68

II. Level 2. The McKay-Thompson series of class 2A for the Monster

$$\begin{align}j_{2A}(\tau) &=\left(\left(\frac{d_1}{d_2}\right)^{12}+2^6\left(\frac{d_2}{d_1}\right)^{12}\right)^2\\ &= \left(\left(\frac{d_1\,d_2}{d_{1/2}\;d_4}\right)^4-4\left(\frac{d_{1/2}\;d_4}{d_1\,d_2}\right)^4\right)^4\end{align}$$ The second form shows they may be \(4\)th powers. Examples: 

$$j_{2A}\big(\tfrac12\sqrt{-10}\big)=12^4\\ \; j_{2A}\big(\tfrac12\sqrt{-58}\big)=396^4\\  j_{2A}\Big(\tfrac{1+\sqrt{-5}}2\Big) = -\big(4\sqrt2\big)^4\\ j_{2A}\Big(\tfrac{1+\sqrt{-13}}2\Big) = -\big(12\sqrt2\big)^4\\ j_{2A}\Big(\tfrac{1+\sqrt{-37}}2\Big) = -\big(84\sqrt2\big)^4$$ which have class number \(h(-d)=2\). For class number \(h(-d)=4\)

$$\begin{align}j_{2A}\Big(\tfrac{1+\sqrt{-17}}2\Big) &= -2^{11}\big(4+\sqrt{17}\big)^2 \big({-1}+\sqrt{17}\big)\\ j_{2A}\Big(\tfrac{1+\sqrt{-73}}2\Big) &= -2^{9}\cdot3^4\big(111+13\sqrt{73}\big)^3\\ j_{2A}\Big(\tfrac{1+\sqrt{-97}}2\Big) &= -2^{11}\cdot3^4\big(59+6\sqrt{97}\big)^4\big({-9}+\sqrt{97}\big)\\ j_{2A}\Big(\tfrac{1+\sqrt{-193}}2\Big) &= -2^{11}\cdot3^4\big(208+15\sqrt{193}\big)^4\big(903+65\sqrt{193}\big)\end{align}$$ All the quadratic irrationals with odd powers are odd fundamental solutions to Pell equations \(x^2-dy^2=-16\). For example, the initial solution to \(x^2-193y^2=-16\) is \((x,y)=(903,\,65)\) which appears above.