Let $E$ and $E'$ be elliptic curves over a field $K$ of characteristic zero. Let $l$ be a large prime and suppose that $K(x(E[l]))=K(x(E'[l]))$ (where these are the fields obtained by adjoining the $x$-coordinates of $l$-torsion points). Then does it follow that $K(E[l])=K(E'[l])$?