Explicit Small Heights in Infinite Non-Abelian Extensions

Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower bound for the height of non-zero elements in $\mathbb{Q}(E_{\text{tor}})$ that are not a root of unity, only depending on the conductor of the elliptic curve. As a side result we will give an explicit bound for a small supersingular prime for an elliptic curve.