Let $E$ be an elliptic curve defined over a fixed number field $F$. Note that there exists a unique complex number $\tau$ in the upper half plane so that

$E(\mathbb{C})\simeq\mathbb{C}/L_\tau$

where $L_\tau=\mathbb{Z}+\mathbb{Z}\tau$. From this, we can find an equation for $E$ of the form

$y^2=4x^3-g_2(\tau)x-g_3(\tau)$.

My question is, are $g_2(\tau)$ and $g_3(\tau)$ contained in $F$? If not, does $g_2(\tau)$ and $g_3(\tau)$ have something to do with $F$?