<s>[INST] Does the elliptic function $\operatorname{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\big|\frac{1}{2}\right)$ have a closed form? Given the complete elliptic integral of the first kind $K(k)$ for the modulus $k$,
can the elliptic function $$\text{cn}\left(\frac{2}{3}K\left(\frac{1}{2}\right)\bigg|\frac{1}{2}\right)$$ be expressed in closed form?
 [/INST] Assuming $1/2$ is $k$, as in Maple ... Maple
JacobiCN(2/3*EllipticK(1/2),1/2) evaluates to
$0.473058826656122429170671314726$.
From ISC we find that
this is a solution of
$$
Z^4-2Z^3-6Z+3=0
$$
which may be written
$$
{\frac { \left( 1+2\,\sqrt [3]{6} \right) ^{3/4}+\sqrt [4]{1+2\,
\sqrt [3]{6}}-\sqrt {-2\,\sqrt [3]{6}\sqrt {1+2\,\sqrt [3]{6}}+2\,
\sqrt {1+2\,\sqrt [3]{6}}+14}}{2\sqrt [4]{1+2\,\sqrt [3]{6}}}}
$$
On the other hand, if $1/2$ is $m=k^2$, then in Maple we want
JacobiCN(2/3*EllipticK(1/sqrt(2)),1/sqrt(2)) which evaluates to 
$0.43542054468233904782250442376$.
This is a solution of $-1+2Z+2Z^3-Z^4=0$.
</s>