Let $E$ be an elliptic curve over $\mathbb{Q}$. 
Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}\_p$ using SAGE or any other means ? 
At least can we say whether $E(\mathbb{Q}\_p)[p]=0$ or not ?



Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$
Suppose $p>2$ and that $E$ has good reduction. If the reduction $\tilde E(\mathbb{F}\_p)$ has no $p$-torsion then there is no $p$-torsion in $E(\mathbb{Q}\_p)$. Otherwise look at the exact sequence
$$ 0\to E(\mathbb{Q}\_p)[p]\to \tilde E(\mathbb{F}\_p)[p]\to \hat E(p\mathbb{Z}\_p)/p\hat E(p\mathbb{Z}\_p).$$
Here $\hat E$ is the formal group. So to check if a $p$-torsion point $\tilde P$ in the reduction lifts to a $p$-torsion point in $\mathbb{Q}\_p$ do the following. Take any lift $P\in E(\mathbb{Q}\_p)$. Then $Q=pP\in \hat E(p\mathbb{Z}\_p)$ is the image of $\tilde P$ under the boundary map. Now check if $Q$ belongs to $p\hat E(p\mathbb{Z}\_p) = \hat E(p^2\mathbb{Z}\_p)$ by looking at the valuation at a parameter $t(Q)=-x(Q)/y(Q)$ of the formal group.


Refined versions of this will work for any type of reduction over any local field.


