Hypergeometric formulas for the j-function Given the j-function $j(\tau)$ and elliptic lambda function $\lambda(\tau)$. Define,
$$g = -1+2\frac{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\,\lambda(2\tau)\big) }{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\,\lambda(\tau)\big) }$$
then,
$$j(\tau) = \frac{4^4(g^4-g^2+1)^3 }{(g^4-g^2)^2}\tag{1}$$
Question: Any other formula that uses $\,_2F_1\big(a,b;c;z\big)$ for other $a,b,c$?
$\color{blue}{Edit}$: (In response to ccorn's answer.)
For any non-zero constant $N(N-1728)\neq0$, one can solve the following two equations,
$$N = \frac{(x^2+10x+5)^3}{x}\tag{2}$$
$$N = \frac{-(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^3}{r^5(r^{10} + 11r^5 - 1)^5}\tag{3}$$
for unknowns $x,r$ as,
$$x = \frac{-125r^5}{r^{10}+11r^5-1}$$
where,
$$r = \frac{N\,^{-11/60}\,_2F_1\left(\tfrac{31}{60},\tfrac{11}{60},\tfrac{6}{5},\tfrac{1728}{N}\right)}{N\,^{1/60}\,_2F_1\left(\tfrac{19}{60},\tfrac{-1}{60},\tfrac{4}{5},\tfrac{1728}{N}\right)} = \frac{(N-1728)^{-11/60}\,_2F_1\left(\tfrac{41}{60},\tfrac{11}{60},\tfrac{6}{5},\tfrac{1728}{1728-N}\right)}{(N-1728)^{1/60}\,_2F_1\left(\tfrac{29}{60},\tfrac{-1}{60},\tfrac{4}{5},\tfrac{1728}{1728-N}\right)}$$
The eqns (2) and (3) are the Jacobi sextic and icosahedral equation, respectively, both of which do not have a solvable Galois group. (Eq. (1) has a solvable group.)
The utility of the question is then it finds an equation, with one free parameter $N$ (the special case $N = j(\tau)$ being only a subset), that can be solved in terms of $_2F_1(a,b;c;z)$ where $z$ is a function of $N$. (This also  implies the general quintic, via the Jacobi sextic, is solvable in terms of $r$.)
So it would be nice to find more polynomial examples like (1), but has a higher degree.
 A: The following is an too-long-for-a-comment remark of mine to the original question.
It seems outdated now.
Sorry, I do not get the point of this question.
In your example, $\lambda$ is a modular function for some subgroup of the full modular group, and therefore $j$ and $\lambda$ fulfill some bivariate polynomial equation.
In fact, $j$ can be expressed as a rational function of $\lambda$.
Plumbing hypergeometrics in such a simple relation looks like an obfuscated math contest to me.
In particular, the given example obfuscates that
$$g=k'=\sqrt{1-\lambda}$$
So probably my two cents here will not be able to assist your quest.
Of course, I could provide another obfuscated math example:
$$j=\gamma_3^2\left(\frac{{}_2F_1\left(\frac{1}{12},\frac{5}{12};1;\frac{12^3}{\gamma_2^3}\right)}{{}_2F_1\left(\frac{1}{12},\frac{7}{12};1;-\frac{12^3}{\gamma_3^2}\right)}\right)^{12}$$
where $\gamma_2$, $\gamma_3$ are
Weber functions.
If $\tau+m$ is in the full modular group's fundamental domain
for some $m\in\mathbb{Z}$,
the numerator in the fraction is $\sqrt[4]{E_4}$,
and the denominator is $\sqrt[6]{E_6}$,
where $E_4=\gamma_2\eta^8$ and $E_6=\gamma_3\eta^{12}$
are Eisenstein series
and $\eta$ is the
Dedekind eta function.
This thing just obfuscates
$$j=\gamma_2^3=\gamma_3^2+12^3$$
Therefore I do not see the point of such jugglings.
The theorem that relates modular forms with functions like ${}_2F_1$ is given,
for example, in


*

*Don Zagier: Elliptic modular forms and their applications.
In: Kristian Ranestad (ed.): The 1-2-3 of modular forms. Springer 2008,
DOI: 10.1007/978-3-540-74119-0.


It implies that a weight-$1$ modular form (in the wider sense, that is,
with respect to subgroups of the full modular group,
or with a multiplier system) fulfills an ordinary second-order
linear differential equation if the independent variable of that equation
is chosen to be a suitable modular function.
Often, the differential equation turns out to be the hypergeometric one,
and with a proper handling of initial conditions, the modular form can be
expressed locally in terms of ${}_2F_1$, whose main argument
is then necessarily some modular function.
The key point here is that the main argument of ${}_2F_1$, let us name it $t$,
must itself be a modular function and thus have weight zero.
But then $j$ and $t$ are algebraically related over $\mathbb{C}$.
Therefore, I expect every expression of $j$ in terms of ratios of ${}_2F_1$
to be just obfuscations of a simpler algebraic relation.
In some cases, it may serve as an illustration of hypergeometric transformations
though.
A: Note the following relation 

$$ _2F_1\left( 1/2,1/2,1, x  \right)= \frac{2}{\pi} K(\sqrt{x}).  $$

where $K(x)$ is the Complete elliptic integral of the first kind and $_2F_1$ is the hypergeometric function.
