Entry 126

Assume $\tau=n\sqrt{-\color{blue}{2}}$ for some positive integer $n$. Given $_2F_1(a,b;c;z)$ where $a+b=c=\frac12$ for the case $a=\tfrac1{8}$. Let $z_1 = (1-2w)^2$ where $w$ is $$w=\frac{64}{64+\Big(\tfrac{\eta(\tau/2)}{\eta(\tau)}\Big)^{24}}$$ Then $(z_1, z_2)$ are algebraic numbers in

$$_2F_1\left(\tfrac18,\tfrac38;\tfrac12;z_1\right) = z_2$$

Examples: 

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

$$_2F_1\left(\frac18,\frac38;\frac12;\frac{(70\sqrt{2})^2\,(1+\sqrt2)^2}{(2+3\sqrt2)^6}\right)=\frac34\left(\frac{2+3\sqrt2}2\right)^{1/2}$$

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

$$_2F_1\left(\frac18,\frac38;\frac12;\frac{2400}{2401}\right)=\tfrac23\cdot7^{1/2}$$