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 ?
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.

$\begingroup$ The Sage code given in the previous post (now deleted) was not working. Can you suggest some other code for computing the $p$torsion over $\mathbb{Q}_p$ ? $\endgroup$ – Suman May 26 '16 at 10:07

$\begingroup$ Yes, it was working. And for small $p$ it is very good to use that code. Otherwise the above translates easily to code. But this site is not meant to answer that sort of questions. $\endgroup$ – Chris Wuthrich May 26 '16 at 10:13

$\begingroup$ I tried for $p=11$. May be that's why it was not working. $\endgroup$ – Suman May 27 '16 at 8:05

2$\begingroup$ sage: E = EllipticCurve("57a1") sage: Ep = E.base_extend(Qp(11,100)) sage: f = Ep.torsion_polynomial(11) sage: r = f.roots() sage: len(r) 0 shows that there are no11torsion on that curve over $\mathbb{Q}_{11}$. $\endgroup$ – Chris Wuthrich May 27 '16 at 9:17