In p.192 The Arithmetic of Elliptic Curves, the proposition3.1 is described when $m$ is relatively prime to $p=char(k)$. How is this fact when $p|m$?
For example,Example3.3.2.
Let $E/\mathbb{Q}$ be the elliptic curve
\begin{equation} E:y^2= x^3+3. \end{equation} And the discriminant $\mathbb{\Delta}=-2^4\cdot3^5 \neq0$ mod $p\geq5$.
Then $|\tilde{E}(\mathbb{F}_5)|=6$ , $|\tilde{E}(\mathbb{F}_7)|=13$.
From these calculations, this book concludes that $E(\mathbb{Q})$ has no nontrivial torsion.
However , according to the proposition 3.1 ,I think that this proves only when m is relatively prime 5 or 7.