Connection between lattice basis vector and algebraic coefficients of a complex elliptic curve

Question

Background

1. An elliptic curve over $\mathbb C$ is the zero locus of $y^2 = 4x^3-g_2x-g_3$ in $\mathbb C^2$ or $\mathbb CP^2$.

2. A complex number $\tau \in \mathbb H$, uniquely determines an elliptic curve $y^2 = 4x^3-g_2(\tau)x-g_3(\tau)$, where $g_2,g_3$ are given by the following formulas. $$g_2(\tau) = 60\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-4} \quad \quad \text{and }\quad \quad g_3(\tau) = 140\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-6}.$$ Conversely, given an elliptic curve $y^2 = 4x^3-g_2x-g_3$, there is an essentially unique $\tau$ such that $g_2,g_3 = g_2(\tau),g_3(\tau)$.

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Question

Let ${\bar{\mathbb{Q}}}$ be the set of all algebraic numbers, i.e. $x_0\in {\bar{\mathbb{Q}}}$ if and only if $x_0$ is a root of a polynomial in $\mathbb{Q}[x]$.

Next consider all those elliptic curves "defined over ${\bar{\mathbb{Q}}}$", i.e. those elliptic curves for which $g_2,g_3 \in {\bar{\mathbb{Q}}}$. The question is: Is there a nice description of the the corresponding $\tau$?

More specifically, we define $S$ as follows. $$S:=\{\tau \in \mathbb H\mid g_2(\tau),g_3(\tau)\in \bar{\mathbb{Q}} \}$$

Question. What can we say about $S$?

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Remarks

1. What I am looking for is something like $S= \bar{\mathbb{Q}} \cap \mathbb{H}$. (It is perhaps false that $S= \bar{\mathbb{Q}} \cap \mathbb{H}$, because Wikipedia mentions $g_2(2i) = \frac{11\,\Gamma \left(\frac14\right)^8}{2^{8} \pi^2},\ g_3(2i) = \frac{7\,\Gamma \left(\frac14\right)^{12}}{2^{12} \pi^3}$ and these are transcendental numbers.)

2. This question is of interest because Belyi's theorem states that an elliptic curve $T$ admits an holomorphic map $f: T \rightarrow \mathbb{C}P^1$ that is branched over three points if and only if $T$ can be defined as the zero loci of a set of polynomials with coefficients in $\bar{\mathbb{Q}}$. In particular, if $g_,g_3 \in \bar{\mathbb{Q}}$, then $y^2=x^3-4g_2x-g_3$ admits a branched cover over $\mathbb{C}P^1$ with only three branched points.

Answer 1

It's probably better to ask whether $j(\tau)\in\overline{\mathbb Q}$, since that happens if and only if there is some linear fractional transformation $\gamma\in\text{SL}_2(\mathbb Z)$ so that $g_2(\gamma\tau)$ and $g_3(\gamma\tau)$ are in $\overline{\mathbb Q}$. Then the answer is known for $\tau\in\mathbb H\cap\overline{\mathbb Q}$.

Theorem. Let $\tau\in\mathbb H\cap\overline{\mathbb Q}$. Then $j(\tau)\in\overline{\mathbb Q}$ if and only if $\bigl[\mathbb Q(\tau):\mathbb Q\bigr]=2$, which is if and only if the associated elliptic curve has complex multiplication.