Entry 125

Assume \(\tau=n\sqrt{-\color{blue}{1}}\) for some positive integer \(n\). Given \(_2F_1(a,b;c;z)\) where \(a+b=c=\frac12\) for the case \(a=\tfrac1{12}\). Let \(j(\tau)\) be the j-function and \(z_1 = (1-2w)^2\) where \(w\) is a root of $$\frac{12^3}{4w(1-w)} =j(\tau)$$ Or more simply \(z_1=1-\frac{1728}{j(\tau)}\), then \((z_1, z_2)\) are algebraic numbers in

$$\,_2F_1\left(\tfrac1{12},\tfrac5{12};\tfrac12;z_1\right) = z_2$$

Examples: 

If \(n=2\) so \(\tau=2\sqrt{-1}\) and \(j(\tau)=66^3\), then,

$$_2F_1\left(\frac1{12},\frac5{12};\frac12;\frac{1323}{1331}\right)=\frac34\cdot11^{1/4}$$ 

If \(n=3\) so \(\tau=3\sqrt{-1}\), then,

$$\qquad _2F_1\left(\frac1{12},\frac5{12};\frac12;\frac{(14\sqrt6)^2\,(72+43\sqrt3)}{(21+20\sqrt3)^3}\right)=\frac23(21+20\sqrt3)^{1/4}$$