Minimal field of definition (reviewed)

Let $V/k$ be an algebraic variety defined over a field $k$ and let $S$ be the set of subfields $k_0\subseteq k$ for which there exists an algebraic variety $V_0/k_0$ whose base change to $k$ is isomorphic to $V$.

Any field $k_0\in S$ that contains no other elements of $S$ is a minimal field of definition for $V$.

In general, an algebraic variety may have more than one minimal field of definition; this does not occur for elliptic curves but it does occur for curves of genus 2.