Checking whether two rational points of infinite order are generating the torsion free part of an elliptic curve Let an elliptic curve be given.
As the title says I want to know if we can show that two independent points $P$ and $Q$ are generators of the torsion free part of $E$.
For instance let $E:y^2=x^3-17x$. Then I determined with the method from "Rational Points on Elliptic Curves" from Silverman and Tate, that the $E$ has the independent integer points 
$$P=(-1,4)\text{ and } Q=(9,24)$$
and rank$(E(\mathbb{Q}))=2$.
How can I show that $\langle (0,0),P,Q\rangle =E(\mathbb Q)$?
 A: Your question asks "How can I show ...". As Jesper Petersen notes, if by that you mean "How can I show by using a software package", then one can use Sage (or undoubtedly Magma or mwrank, etc.) But if you mean "How do I understand how one shows...", that's different. The answer is that the standard descent argument shows that if $P_1,...,P_r$ generate $E(\mathbb Q)/2 E(\mathbb Q)$, for example, then there are explicitly computable constants $C_1$ and $C_2(E)$ so that the effectively computable set 
$$ \{P\in E(\mathbb Q) : h(P)\le C_1\max h(P_i) + C_2(E)\}$$ 
contains a set of generators for $E(\mathbb Q)$. So one computes this set, gets a finite set of points, and whittles it down to a generating set. A nice observation (of Don Zagier, I believe) is that if one uses canonical heights, then
$$ \{P\in E(\mathbb Q) : \hat h(P)\le 2\max \hat h(P_i)\} $$
(or something like that) contains a set of generators for $E(\mathbb Q)$. This is an exercise on lattices and positive definite quadratic forms. Then one can use estimates relating $\hat h$ and $h$ to search for a set containing generators. This is the basic method. Various ideas can be used to speed the process, but ultimately there is going to be a search over a finite set of potential generators.
