Entry 165

The previous entries dealt with even class numbers. For odd class number \(h(-d)=1\), there are the nine Heegner numbers \(d=1,2,3,7,11,19,43,67,163\). Given the Kronecker symbol \(\big(\tfrac{-d}{m}\big)\), we propose 

Conjecture. Let \(d>3\) with class number \(h(-d)=1\). Then \(x\) below is an algebraic number $$\frac1{x_d^2}  = \frac1{K(k_d)}\frac{\sqrt{2\pi}}{\color{blue}4\sqrt{d}} \left(\prod_{m=1}^{d}\Big[\Gamma\big(\tfrac{m}{d}\big)\Big]^{\big(\tfrac{-d}{m}\big)}\right)^{\color{red}{1/2}}$$specifically$$x_d=2^{1/4}G_n=\frac{\eta^2(\tau)}{\eta(\tfrac{\tau}2)\,\eta(\tau)}$$ with Ramanujan's \(G\)-function. This function \(x_d\) has been discussed in Entry 159. For \(d=7\), then \(x_7=\sqrt2\), but for the five \(d\geq11\), then \(x_d\) are the real roots of the following five simple cubics, 

$$\begin{align}& x^3-2x^2+2x-2 = 0\\ & x^3-2x-2 = 0 \\ & x^3-2x^2-2 = 0 \\ & x^3-2x^2-2x-2 = 0 \\ & x^3-6x^2+4x-2=0\end{align}$$ also discussed in Entry 160.