Entry 128

Assume $\tau=n\sqrt{-\color{blue}{4}}$ for some positive integer $n$. Given $_2F_1(a,b;c;z)$ where $a+b=c=\frac12$ for the case $a=\tfrac1{4}$. Let $z_1 = (1-2w)^2$ where $w$ is $$w=\frac{16}{16+\Big(\tfrac{\eta(\tau/4)}{\eta(\tau)}\Big)^8}$$ Then $(z_1, z_2)$ are algebraic numbers in

$$_2F_1\left(\tfrac14,\tfrac14;\tfrac12;z_1\right) = z_2$$

Examples: 

If $n=2$ so $\tau=2\sqrt{-4}$, then,

$$_2F_1\left(\frac14,\frac14;\frac12;\,\frac{9\,(3+\sqrt2)^2}{(1+\sqrt2)^6}\right)=\frac38\big(2+\sqrt2\big)$$  

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

$$\quad _2F_1\left(\frac14,\frac14;\frac12;\,\frac{8\sqrt3}{(2+\sqrt3)^2}\right)=\frac23\big(3+2\sqrt3\big)^{1/2}$$