Entry 116

For fundamental discriminants \(d=4m\) with class number \(h(-d)=8\), there are exactly 29 \(m\) that are even. The three \(m = 210, 330, 462\) were discussed in Entry 114 since they have special properties. However, the four largest are \(m = 598, 658, 742, 862\). Given the nome \(q = e^{\pi i\tau}\), we can use the Ramanujan G and g functions $$\begin{align}2^{1/4}G_n &= q^{-\frac{1}{24}}\prod_{k>0}(1+q^{2k-1}) = \frac{\eta^2(\tau)}{\eta\big(\tfrac{\tau}{2}\big)\eta(2\tau)}\\ 2^{1/4}g_n &= q^{-\frac{1}{24}}\prod_{k>0}(1-q^{2k-1}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)}\end{align}$$ with \(\tau=\sqrt{-n}\) where Ramanujan uses \(G_n\) and \(g_n\) for odd \(n\) and even \(n\), respectively. For these four \(m = n\), then $$\begin{align}(g_{598})^2 &=  \left(\frac{\;\eta\big(\tfrac{\tau}{2}\big)}{2^{1/4}\eta(\tau)}\right)^2 = \Big(6+\sqrt{26}+\sqrt{(6+\sqrt{26})^2-1}\Big)(1+\sqrt2)^2\Big(\tfrac{3+\sqrt{13}}2\Big)\\ (g_{658})^2 &=  \left(\frac{\;\eta\big(\tfrac{\tau}{2}\big)}{2^{1/4}\eta(\tau)}\right)^2 = \, ?? \\ (g_{762})^2 &= \left(\frac{\;\eta\big(\tfrac{\tau}{2}\big)}{2^{1/4}\eta(\tau)}\right)^2 = \Big(\tfrac{11+\sqrt{106}}2+\sqrt{\big(\tfrac{11+\sqrt{106}}2\big)^2-1}\Big)\,(1+\sqrt2)^2\,\Big(\tfrac{7+\sqrt{53}}2\Big)\\ (g_{862})^2 &=  \left(\frac{\;\eta\big(\tfrac{\tau}{2}\big)}{2^{1/4}\eta(\tau)}\right)^2  = \, ??\end{align}$$ I know the octics for the other two, though I don't know how to factor them into quartic units.