# Explicit Lower Height Bounds in Q(E_tor) and Generalizations

Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\Q(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this talk we will prove an explicit lower bound for the height of non-zero elements in $\Q(E_{\text{tor}})$ 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 upper bound for a supersingular prime for an elliptic curve. We will also mention generalizations of the Theorem and give an outline for further work.