  Let $E$ be an elliptic curve defined over a number field $K$ and let $v$ be a
finite place of $K$. Write $K^{tv}$ the maximal extension of $K$ in which $v$
is totally split and $L$ the field generated over $K^{tv}$ by all torsion
points of $E$. Under some conditions, we will show that the absolute
logarithmic Weil height (resp. N\'eron-Tate height) of any element of $L^*$
(resp. $E(L)$) is either $0$ or bounded from below by a positive constant
depending only on $E, K$ and $v$. This lower bound will be explicit in the
toric case when $K=\mathbb{Q}$.
