Why is $E[2]$ of $E: y^2 = x^3 - p$ over $\mathbb{Q}_p$ ramified?

Question

I would greatly appreciate if someone could explain me the following. Given an elliptic curve $E: y^2 = x^3 - p$ over $\mathbb{Q}_p$, I am struggling to show why $E[2]$ is ramified. I would greatly appreciate any comments or suggestions on how to start this. Thank you.

Answer 1

The nonzero points in $E[2]$ are the $(\alpha,0)$ where $\alpha$ runs through the cube roots of $p$. Each $\alpha$ has $1/3$ of the valuation of $p$, so it apparent that $\Bbb Q_p(\alpha)$ is totally ramified of degree three over $\Bbb Q_p$.