Given an abelian variety $A$ defined over $\mathbb{Q}$, for a positive integer (we can suppose prime) $\ell$, let $A[\ell]$ denote the group
of points of $A$ that are annihilated by $\ell$, the division field $\mathbb{Q}(A[\ell])$ is obtained by adjoining to $\mathbb{Q}$ the **coordinates** of the points of $A[k]$.

Why the $\ell$-th cyclotomic field is contained in $\mathbb{Q}(A[\ell])$?

I saw somewhere that this is because of the existenece of the Weil pairing (for simplicity, we can suppose that $A$ has a principal polarization)

$$ e_\ell: A[\ell]\times A[\ell] \rightarrow \mu_\ell $$

From here
I know that there exist points $P,Q\in A[\ell]$ such that $e_\ell(P,Q)$ is a primitive $\ell$-th root of unity, but I dont understand how this is related to the **coordinates** of the points in $A[\ell]$.
Thanks in advance.