TITLE: Special values of the modular J invariant
QUESTION [9 upvotes]: A special value:
$$
J\big(i\sqrt{6}\;\big) = \frac{(14+9\sqrt{2}\;)^3\;(2-\sqrt{2}\;)}{4}
\tag{1}$$
I wrote $J(\tau) = j(\tau)/1728$.  
How up-to-date is the Wikipedia listing of known special values for the modular
j invariant ?
Value (1) is not on it.
Alternatively, is there a compilation of special values elsewhere?   
The value (1) is related to this hypergeometric value, too:
$$
{}_2F_1\left(\frac{1}{6},\frac{1}{3};1;\frac{1}{2}\right) =
\eta(i\sqrt{6}\;)^2 \,2^{1/2} \,3^{3/4} \,(1+\sqrt{2}\;)^{1/6}
\tag{2}$$
Here, $\eta(\tau)$ is the Dedekind eta function 
(I came to this while working on my unanswered question
at math.SE)

REPLY [3 votes]: You are at $z=i\sqrt{6}$. Hence you can use singular modulus $k_r$. It holds in general (when $z=i\sqrt{r}$)
$$
j_r=\frac{256(k_r^2+(k'_r)^4)^3}{(k_rk'_r)^4}\textrm{, }k'_r=\sqrt{1-k_r^2}\textrm{, }\forall r>0.
$$
But $k_6=\lambda^*(6)=(2-\sqrt{3})(\sqrt{3}-\sqrt{2})$ (see here). Hence you get your value.