This is an addition to Noam's answer. In higher dimensions the same type of reasoning---using explicit conditions on the coefficients of reduced quadratic forms---is not feasible. Nevertheless, it is possible to prove the following result:
Theorem. Let $E \subset \mathbb{R^n}$ be an $n$-dimensional ellipsoid centered at the origin and containing no other integer point. There exists a transformation $T \in GL(n,\mathbb{Z})$ such that $T(E)$ is contained in the ball of radius $$ \left(\frac{3}{2}\right)^{(n-1)(n-2)/2} \frac{2^n}{\epsilon_n}\sqrt{n^{n-1}} $$ centered at the origin.
Here $\epsilon_n$ is the volume of the unit ball of dimension $n$.