  Let $k$ be a field of characteristic $0$, and let $\alpha_{1}$, $\alpha_{2}$,
and $\alpha_{3}$ be algebraically independent and transcendental over $k$. Let
$K$ be the transcendental extension of $k$ obtained by adjoining the elementary
symmetric functions of the $\alpha_{i}$'s. Let $E$ be the elliptic curve
defined over $K$ which is given by the equation $y^{2} = (x - \alpha_{1})(x -
\alpha_{2})(x - \alpha_{3})$. We define a tower of field extensions $K = K_{0}'
\subset K_{1}' \subset K_{2}' \subset ...$ by giving recursive formulas for the
generators of each $K_{n}'$ over $K_{n - 1}'$. We show that $K_{\infty}'$ is a
certain central subextension of the field $K(E[2^{\infty}]) := \bigcup_{n =
0}^{\infty} K(E[2^{n}])$, and a generator of $K(E[2^{\infty}])$ over
$K_{\infty}'(\mu_{2})$ is given. Moreover, if we assume that $k$ contains all
$2$-power roots of unity, for each $n$, we show that $K(E[2^{n}])$ contains
$K_{n}'$ and is contained in a certain quadratic extension of $K_{n + 1}'$.
