and $\alpha$. We also show that all these quantities can be bounded effectively in terms of standard invariants of the elliptic curve and of the height of $\alpha$, see Remark \[rmk:MainTheoremIsEffective\].

\[thm:UniformIntroduction\] There is a universal constant $C$ with the following property. Let $E/{\mathbb{Q}}$ be an elliptic curve, and let $\alpha \in E({\mathbb{Q}})$ be a point such that the class of $\alpha$ in the free abelian group $E({\mathbb{Q}})/E({\mathbb{Q}})_{\operatorname{tors}}$ is not divisible by any $n>1$. Then $$\frac{N^2}{\left[ {\mathbb{Q}}_{M,N} : {\mathbb{Q}}_M \right]} \quad \text{ divides } \quad C$$ for all pairs of positive integers $(M,N)$ with $N$ dividing $M$.

Structure of the paper
----------------------

We start with some necessary general preliminaries in Section \[sec:Preliminaries\], leading up to a factorisation of the constant $C$ of Theorem \[thm:Main\] as a product of certain contributions which we dub the *$\ell$-adic* and *adelic* failures (corresponding to $E$, $\alpha$, and a fixed prime $\ell$). In the same section we also introduce some of the main actors of this paper, in the form of several Galois representations associated with the torsion and Kummer extensions. In Section \[sec:PropertiesTorsionRepresentation\] we then recall some important properties of the torsion representations that will be needed in the rest of the paper. In 