Introduction: Given any $\tau \in \mathbb H$, there is an elliptic curve $y^2=4x^3-g_2(\tau)x-g_3(\tau)$ which has $\{1,\tau\}$ as its lattice. Wikipedia gives an "explicit" relation between $g$ and $\tau$
\begin{align*} {\displaystyle g_{2}(\tau )=60\sum _{(m,n)\neq (0,0)}\left(m+n\tau \right)^{-4}} &\quad & & {\displaystyle g_{3}(\tau )=140\sum _{(m,n)\neq (0,0)}\left(m+n\tau \right)^{-6}} \end{align*}
Question: Let $\tau=a+b\omega$ where $a,b\in \mathbb Q$ and $\omega=(-1+i\sqrt{3})/2$. Can we show $g_2(\tau), g_3(\tau)$ are algebraic?
Motivation: The triangular lattice of $\mathbb C$ is invariant under translations by $n+m\omega$, where $n,m\in\mathbb Z$. Hence, given an elliptic curve whose periods are $\{1,a+b\omega\}$, we can quotient and get an equilateral triangulation of the elliptic curve (torus).
This is related to Belyi's theorem which says a compact Riemann surface can be equilaterally triangulated iff the Riemann surface can be defined as a zero set of polynomials with algebraic coefficients.
Since the above elliptic curves admit an equilateral triangulation, Belyi says it can be described by polynomials with algebraic coefficients. One can ask: is $g_2(\tau),g_3(\tau)$ algebraic and is there an easy way to see this?