An old paper (Über hypergeometrische Funktionen, deren letztes Element speziell ist) by W. Heymann (from a talk by Wadim Zudilin) discusses the value of an ${}_2F_1$ at special $t$’s. He does $t=-1/3,1/4,1/5$ and related values and mentions Gauss for $t=1,-1,1/2$ and Kummer for $t=-1/8,1/9,8/9$. For example we have this identity $$ F_h = F \left( -\frac{h}{2}, -\frac{h}{2} + \frac{1}{2}, h + \frac{3}{2}, -\frac{1}{3} \right) = \frac{2}{3} \cdot \left( \frac{8}{9} \right)^h \cdot \frac{\Gamma \left( \frac{1}{3} \right) \cdot \Gamma \left( h + \frac{3}{2} \right)}{\sqrt{\pi} \cdot \Gamma \left( h + \frac{4}{3} \right)} $$ If we take $h=1/2$ we get an HGM defined over $\mathbb Q$, we get the gamma vector $[-4,1,1,2]$ and corresponding Weierstrass model $$ y^2+xy=x^3+\frac t{64}x.