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,

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