  Let $E$ be an elliptic curve defined over the rationals without complex
multiplication. The field $F$ generated by all torsion points of $E$ is an
infinite, non-abelian Galois extension of the rationals which has unbounded,
wild ramification above all primes. We prove that the absolute logarithmic Weil
height of an element of $F$ is either zero or bounded from below by a positive
constant depending only on $E$. We also show that the N\'eron-Tate height has a
similar gap on $E(F)$ and use this to determine the structure of the group
$E(F)$.
