  One of the many interesting algebraic objects associated to a given rational
elliptic curve, $E$, is its full-torsion representation
$\rho_E:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\mathrm{GL}_2(\hat{\mathbf{Z}})$.
Generalizing this idea, one can create another full-torsion Galois
representation,
$\rho_{(E_1,E_2)}:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\left(\mathrm{GL}_2(\hat{\mathbf{Z}})\right)^2$
associated to a pair $(E_1,E_2)$ of rational elliptic curves. The goal of this
paper is to provide an infinite number of concrete examples of pairs of
elliptic curves whose associated full-torsion Galois representation
$\rho_{(E_1,E_2)}$ has maximal image. The size of the image is inversely
related to the size of the intersection of various division fields defined by
$E_1$ and $E_2$. The representation $\rho_{(E_1,E_2)}$ has maximal image when
these division fields are maximally disjoint, and most of the paper is devoted
to studying these intersections.
