Entry 127

Assume $\tau=n\sqrt{-\color{blue}{3}}$ for some positive integer $n$. Given $_2F_1(a,b;c;z)$ where $a+b=c=\frac12$ for the case $a=\tfrac1{6}$. Let $z_1 = (1-2w)^2$ where $w$ is $$w=\frac{27}{27+\Big(\tfrac{\eta(\tau/3)}{\eta(\tau)}\Big)^{12}}$$ Then $(z_1, z_2)$ are algebraic numbers in

$$_2F_1\left(\tfrac16,\tfrac13;\tfrac12;z_1\right) = z_2$$

Example: 

If $n=2$ so $\tau=2\sqrt{-3}$, then,

$$_2F_1\left(\frac16,\frac13;\frac12;\,\frac{25}{27}\right)=\frac{3\sqrt3}{4}$$  

If $n=4$ so $\tau=4\sqrt{-3}$, then,

$$\qquad _2F_1\left(\frac16,\frac13;\frac12;\,\frac{45^2\,(\sqrt2+\sqrt3)^2}{(1+\sqrt6)^8}\right)=\frac58\big(1+\sqrt6\big)$$