Given the Elliptic integral singular value $K(k_m)$, Dedekind eta $\eta(\tau)$, j-function $j(\tau)$, and hypergeometric $_2F_1\left(a,b;c;z\right)$ with $\color{brown}{a+b=c=\tfrac12}$.
Conjecture: "The proposed equalities below hold for real $N>1$, though for any integer $N>1$, then the hypergeometric function $_2F_1(z)$ and its argument $z$ are algebraic numbers."
I. For $a=\tfrac14$ and $\tau=N\sqrt{-\color{blue}4}$
$$\begin{aligned}\,_2F_1\left(\tfrac14,\tfrac14;\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{(1+\sqrt2)}{4\sqrt2}\frac{_2F_1\left(\tfrac12,\tfrac12;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}4)}\\[2mm] \end{aligned}\tag1$$ $$w=\frac{16}{16+\Big(\tfrac{\eta(\tau/4)}{\eta(\tau)}\Big)^8}$$
Example: If $N=2$ so $\tau=2\sqrt{-4}$, then, $$_2F_1\left(\tfrac14,\tfrac14;\tfrac12;\,\tfrac{9\,(1-2\sqrt2)^2}{(1+\sqrt2)^4}\right)=\tfrac{3}{4\sqrt2}(1+\sqrt2)$$
II. For $a=\tfrac16$ and $\tau=N\sqrt{-\color{blue}3}$
$$\begin{aligned}\,_2F_1\left(\tfrac16,\tfrac13;\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{1}{27^{1/4}}\frac{_2F_1\left(\tfrac13,\tfrac23;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}3)}\\[2mm] \end{aligned}\tag2$$ $$w=\frac{27}{27+\Big(\tfrac{\eta(\tau/3)}{\eta(\tau)}\Big)^{12}}$$
Example: If $N=2$ so $\tau=2\sqrt{-3}$, then, $$_2F_1\left(\tfrac16,\tfrac13;\tfrac12;\,\tfrac{25}{27}\right)=\tfrac{3\sqrt3}{4}$$
III. For $a=\tfrac18$ and $\tau=N\sqrt{-\color{blue}2}$
$$\begin{aligned}\,_2F_1\left(\tfrac18,\tfrac38;\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{\sqrt{1+\sqrt2}}{128^{1/4}}\frac{_2F_1\left(\tfrac14,\tfrac34;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}2)}\\[2mm] \end{aligned}\tag3$$ $$w=\frac{64}{64+\Big(\tfrac{\eta(\tau/2)}{\eta(\tau)}\Big)^{24}}$$
Example: If $N=3$ so $\tau=3\sqrt{-2}$, then, $$_2F_1\left(\tfrac18,\tfrac38;\tfrac12;\tfrac{2400}{2401}\right)=\tfrac{2\sqrt7}{3}$$
IV. For $a=\tfrac1{12}$ and $\tau=N\sqrt{-\color{blue}1}$
$$\begin{aligned}\,_2F_1\left(\tfrac1{12},\tfrac5{12};\tfrac12;\,(1-2w)^2\right) &=\frac{N+1}{2}\frac{1}{12^{1/4}}\frac{_2F_1\left(\tfrac16,\tfrac56;1;\,w\right)}{\pi^{-1}\,K(k_\color{blue}1)}\\[2mm] \end{aligned}\tag4$$ $$\frac{12^3}{4w(1-w)} =j(\tau)$$
Example: If $N=2$ so $\tau=2\sqrt{-1}$, then, $$_2F_1\left(\tfrac1{12},\tfrac5{12};\tfrac12;\tfrac{1323}{1331}\right)=\tfrac{3\,\sqrt[4]{11}}{4}$$
This is a highly compactified version of the results by Zucker and Joyce in "Special values of the hypergeometric series II, III" but I used a common form for the argument $z = (1-2w)^2$ as well as similar eta quotients to better illustrate their affinity. However, I only derived this empirically.
Q: How do we rigorously prove the four conjectures?
