Entry 135

Mathworld has a list of the modular lambda function \(\lambda(\tau)\) with the particular case \(\tau=\sqrt{-14}\) as the rather complicated $$ \sqrt{\lambda(\sqrt{-14})}=-11-8\sqrt2-2(2+\sqrt2)\sqrt{5+4\sqrt2}\qquad \\\ \qquad+\sqrt{11+8\sqrt2}\,\Big(2+2\sqrt2+\sqrt2\sqrt{5+4\sqrt2}\Big)\qquad $$ which is approximately \(0.011208.\) It can be calculated in Mathematica or WolframAlpha as ModularLambda[tau]. However, we can simplify and factor that into two quartic units as $$\begin{align}\frac1{\lambda(\sqrt{-14})} & =\frac1{2^8}(8+3\sqrt7)\,(\sqrt7+\sqrt8)\,\Big(2^{1/4}+\sqrt{4+\sqrt2}\Big)^8\\ &=7960.423255\dots\end{align}$$ It is then just a matter of getting the reciprocal and square roots. One can do so similarly for discriminants \(d=4m\) with class number \(h(-d)=4\) and even \(m=14,62,142\) as described in Entry 134.