The greatest common valuation of $φ_{n}$ and $ψ_{n}^{2}$ at points on elliptic curves

Abstract: Given a minimal model of an elliptic curve, $E/K$, over a finite extension, $K$, of ${\mathbb Q}{p}$ for any rational prime, $p$, and any point $P \in E(K)$ of infinite order, we determine precisely $\min \left( v \left( \phi{n}(P) \right), v \left( \psi_{n}^{2}(P) \right) \right)$, where $v$ is a normalised valuation on $K$ and $\phi_{n}(P)$ and $\psi_{n}(P)$ are polynomials arising from multiplication by $n$ for this model of the curve.