Entry 170

There are only seven fundamental \(d=4p\) with class number \(6\), namely \(p = 29, 53, 61, 109, 157, 277, 397\). Hence their Ramanujan \(G\)-functions are

$$\begin{align}G_{29} &= \left(\frac{y_1^2+\sqrt{y_1^4+4}}2\right)^{1/4} \\ G_{53} &= \left(\frac{y_2^2+\sqrt{y_2^4+4}}2\right)^{1/4} \\ G_{61} &=  \left(\frac{y_3^2+\sqrt{y_3^4+36}}6\right)^{1/4} \end{align}$$ where$$y^3 - y^2 - 4y - 4 = 0\\ y^3 - 7y^2 + 13y - 11 = 0\\ y^3 - 6y^2 - 27y - 54 = 0$$ and similarly for the other \(p\).