field is $K={\mathbb{Q}}$. Our first result is that, under the assumption $\operatorname{End}_K(E)={\mathbb{Z}}$, the integer $C$ can be bounded (explicitly) in terms of the $\ell$-adic Galois representations attached to $E$ and of divisibility properties of the point $\alpha$, and that this statement becomes false if we remove the hypothesis $\operatorname{End}_K(E)={\mathbb{Z}}$.

On the other hand, the assumption $\operatorname{End}_{\mathbb{Q}}(E)={\mathbb{Z}}$ is always satisfied when $K={\mathbb{Q}}$, and we show that in this case $C$ can be taken to be independent of $E$ and $\alpha$, provided that $\alpha$ and all its translates by torsion points are not divisible by any $n>1$ in the group $E({\mathbb{Q}})$. This is a rather surprising statement, especially given that such a strong uniformity result is not known for the closely connected problem of studying the degrees of the torsion fields $K_M$ over $K$.

Main results
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Our main results are the following.

\[thm:Main\] Assume that $\operatorname{End}_K(E)={\mathbb{Z}}$. There is an explicit constant $C$, depending only on $\alpha$ and on the $\ell$-adic torsion representations associated to $E$ for all primes $\ell$, such that $$\begin{aligned}
\frac{N^2}{\left[K_{M,N}:K_M\right]}\quad\text{divides}\quad C\end{aligned}$$ for all pairs of positive integers $(M,N)$ with $N$ dividing $M$.

The proof gives an explicit expression for $C$ that depends on computable parameters associated with $E$ 