Entry 164

To summarize, given fundamental discriminant \(d\), class number \(h(-d)=n\), complete elliptic integral of the first kind \(K(k_p)\), and Kronecker symbol \(\big(\tfrac{-d}{m}\big)\). 

Conjecture 1. Let even \(d=4p\) for prime \(p\equiv 1\,\text{mod}\,4\) with even class number \(h(-d)=n\) and $$x = \frac1{K(k_{\color{blue}{d/4}})}\frac{\sqrt{2\pi}}{2\sqrt{d}} \left(\prod_{m=1}^{d}\Big[\Gamma\big(\tfrac{m}{d}\big)\Big]^{\big(\tfrac{-d}{m}\big)}\right)^{\color{red}{1/(2n)}}$$ Conjecture 2. Let odd \(\,d=p\,\) for prime \(p\,\equiv\, 3\,\text{mod}\,4\,\) with odd class number \(h(-d)=n\) and $$y = \frac1{K(k_{d})}\frac{\sqrt{2\pi}}{2\sqrt{d}} \left(\prod_{m=1}^{d}\Big[\Gamma\big(\tfrac{m}{d}\big)\Big]^{\big(\tfrac{-d}{m}\big)}\right)^{\color{red}{1/(2n)}}$$ then \((x,y)\) are algebraic numbers as seen in entries \(160-163\) and \(165-168\). They seem to have a closed-form in term of eta quotients but I haven't found it yet.