  We study the regulator of the Mordell-Weil group of elliptic curves over
number fields, functions fields of characteristic zero or function fields of
characteristic $p>0$. We prove a new Northcott property for the regulator of
elliptic curves of rank at least $4$ defined over a number field. In the
function field case, we obtain that non-trivial families of elliptic curves
with bounded regulators and bounded ranks are limited families. We conclude by
asking new questions.
