Regarding the Weierstrass $\wp$-function of the hexagonal lattice Playing with the Weierstrass $\wp$-function of the hexagonal (or triangular) lattice $\mathbb{T}$,
$$
\wp'(z)^2 = 4 \wp(z)^3 - 1,
$$
I noticed that the zeros of $\wp'(z) + \sqrt{3}$ are
$$
\frac{\varpi}{3} + \omega,
\quad
e^{\pi i/3} \frac{2\varpi}{3} + \omega,
\quad
e^{2\pi i/3} \frac{\varpi}{3} + \omega,
\qquad
\omega \in \mathbb{T}
$$
(clearly they are all simple), where $\varpi$ is the real period of $\wp$ given by
$$
\varpi = \int_{4^{-1/3}}^{\infty} \frac{dx}{\sqrt{x^3 - 1/4}} = \frac{1}{2 \pi}\, \Gamma(1/3)^3
$$
(the zeros of $\wp'(z) - \sqrt{3}$ can be also determined).
Since
$$
\wp'(z)^2 - 3 = 4 \left[\wp(z)^3 - 1\right]
$$
and $\wp(x)$ is real for $0 < x < \varpi$, we also get
$$
\wp\left(\pm\frac{\varpi}{3}\right) = 1
$$
or, equivalently,
$$
\int_1^{\infty} \frac{dx}{\sqrt{4x^3 - 1}} = \int_{\wp\left(\frac{\varpi}{3}\right)}^{\infty} \frac{dx}{\sqrt{4x^3 - 1}}
= \frac{\varpi}{3}
= \frac{1}{6 \pi}\, \Gamma(1/3)^3
$$
(the detailed calculations can be found in arXiv:2105.04307).
Since I am not an expert in elliptic functions, my question is whether the above facts are known.
 A: There is also a more algebraic perspective.
For this, it is easier (at least for me) to ``reverse engineer'' things by starting with your points, and then deducing something about the elliptic function values.
I will working on the elliptic curve $E: y^2 =  4 x^3 - 1$, which is identified with the quotient  $\mathbb C/\mathbb T$ via $z \bmod \mathbb T \mapsto \bigl( \wp(z), \wp'(z) \bigr).$
Let $P$ be the point of $E$ which is the image of $\varpi/3$.  It is a point of order $3$ in $E$.   Now $E$ has ``complex multiplication'' by the ring $\mathcal O$ obtained by adjoining $\zeta = e^{2\pi i/3}$ to $\mathbb Z$  (because the lattice $\mathbb T$ is a submodule of $\mathbb C$ with respect to this  ring).  The element $\zeta$ sends $(x,y)$ to $(\zeta x, y)$.
In the ring $\mathcal O$, the number $3$ is not prime: it factors as
$3 = (1 - \zeta)(1 - \zeta^{-1}) = (1-\zeta)^2 \cdot (-\zeta^{-1}).$
So  among the $3$-torsion points, there are some even more special points, namely those killed by multiplication by $(1-\zeta)$ (and not  just by $(1-\zeta)^2$).  Looking at the formula for multiplication by $\zeta,$
we  see these are the points with $x = 0$, so the points $(0,\pm i).$
There are $9$  points of order dividing $3$ altogether, namely the origin (i.e. the point at  infinity), these $2$  points $(0,\pm i)$, and then the remaining $6$ points $(x,y)$ which are killed by $3$, but not  by $(1-\zeta)$.
If $P$ is one of these latter points, then we see that $[1-\zeta] P = (0,\pm i).$
We can compute the $x$-coordinate of $[1-\zeta] P$ explicitly, in terms of the $x$-coordinate of $P$, just using the usual addition formula.  The
answer is
$$x^3 - 1 = 0.$$
So we find that these $6$ points satisfy $x^3 = 1$, and then (from the equation for $E$) $y^2 = 3$.
In terms of the lattice, these points are the cosets of  $\varpi/3,$
$2 \varpi/3$, $\zeta \varpi/3$, $\zeta 2 \varpi/3$, $\zeta^{-1}\varpi/3,$ and $\zeta^{-1} 2  \varpi/3$.  (The remaining cosets of order $3$ are exactly
$0,$ $(1 + \zeta) \varpi/3$,  and $2(1+\zeta)\varpi/3$,  and when you multiply these by $(1-\zeta)$,  you actually land in $\mathbb T$.)
So there are $6$ lattice cosets, and $6$ sets of $(x,y)$-coordinates, which match when you apply $z \bmod \mathbb T \mapsto \bigl(\wp(z),\wp'(z)\bigr).$
The cosets $\varpi/3$, $\zeta \varpi/3$, and $\zeta^{-1} \varpi/3$ are related by successive multiplications by $\zeta,$ and so have the same $y$-value.  (Either $\sqrt{3}$ or $-\sqrt{3}$.)
Similarly the cosets $2\varpi/3$, $\zeta 2 \varpi/3,$ and $\zeta^{-1} 2 \varpi/3$ have the same $y$-value.    (Again, either $\sqrt{3}$ or $-\sqrt{3}$.)
To see that $\varpi/3$ actually maps to $-\sqrt{3}$, it it  just a matter  of checking  that $\wp'(\varpi/3)$ is negative, which I'd guess one can fairly easily do.

This answer may look elaborate, but I've tried to include details to help non-experts in this  perspective.  I think it's fair to say that typically all such identities are part of the general theory of complex multiplication, and derivable by this kind of analysis.
