elliptic curves over $\mathbb{Q}_p$

Question

Let $E: Y^2 = X^3 + AX + B$ be an elliptic curve over $\mathbb{Q}_p$, i.e. $A,B \in \mathbb{Q}_p$ and $4A^3 + 27B^2 \neq 0$. Then, according to page 47 of Cassels' Lectures on Elliptic Curves, if $(x,y)$ is a point on the curve such that $x,y \not\in \mathbb{Z}_p$, then $|y|^2 = |x|^3$ where $|\cdot|$ denotes the $p$-adic absolute value.

Why is this the case?

Thanks.

Answer 1

Clearing out denominators, assume without loss of generality that $A, B\in \mathbb{Z}_p$. Since $|\cdot|_p$ is non-archimedean and $x, y\not\in \mathbb{Z}_p$, we have $|y|^2 = |x^3 + Ax + b| = |x|^3$.