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 $F$?



Equation of elliptic curve and field of definition
First, $\tau$ is certainly not unique, it is only well-defined up to linear fractional conjugation by elements of $\text{SL}\_2(\mathbb Z)$. Second, to answer your question, no, $g\_2(\tau)$ and $g\_3(\tau)$ need not be in $F$. What is true is that there exists a complex number $u\in\mathbb C^\*$ such that $u^2g\_2(\tau)\in F$ and $u^3g\_3(\tau)\in F$. 


A good way to think about this is that $g\_2$ and $g\_3$ are really functions on the space of lattices, $g\_2=60G\_4$ and $g\_3=140G\_6$, where 
$$ G\_k(L) = \sum\_{\substack{\omega\in L\\\omega\ne0\\}} \frac{1}{\omega^k}. $$
The uniformization theorem for elliptic curves implies that if $E$ is an elliptic curve given by a Weierstrass equation
$$ y^2 = 4x^3 + ax + b, $$
(with $a,b\in\mathbb C$), then there is a lattice $L$ such that 
$$ g\_2(L)=a\quad\text{and}\quad g\_3(L)=b. $$


