Entry 167

For odd class number \(h(-d)\), the kind that is \(d \equiv 7\,\text{mod}\,8\) seems more well-behaved than \(d \equiv 3\,\text{mod}\,8\). For class number \(3\), there are only two of the first kind: \(d = 23,31\). Hence, $$\begin{align}K(k_{23}) &=\frac{\sqrt{2\pi}}{2\sqrt{23}}\,x^{4/3} \left(\prod_{m=1}^{23}\Big[\Gamma\big(\tfrac{m}{23}\big)\Big]^{\big(\tfrac{-23}{m}\big)}\right)^{\color{red}{1/6}}\\ K(k_{31}) &=\frac{\sqrt{2\pi}}{2\sqrt{31}}\,y^{4/3} \left(\prod_{m=1}^{31}\Big[\Gamma\big(\tfrac{m}{31}\big)\Big]^{\big(\tfrac{-31}{m}\big)}\right)^{\color{red}{1/6}}\end{align}$$ where \((x,y)\) are the real roots of the cubics $$x^3-x-1=0\\ y^3-y^2-1=0$$ or the plastic ratio and supergolden ratio, respectively. But for \(d=59\) which is of the second kind, then the radical involved will be an algebraic number of degree \(3\times3 =9\).