Summary: How does one compute the Mordell-Weil group of an elliptic curve $E / \mathbb{Q}$, in the case where the torsion points are only defined over larger fields?

More detail: I've been reading the description in chapter X of Silverman's *Arithmetic of Elliptic Curves* of how to compute specific Mordell-Weil groups. Proposition X.1.4 describes a way to compute the Mordell-Weil group assuming all $2$-torsion points are rational. Proposition X.4.9 extends this to the case where only one $2$-torsion point is rational.

I can sort of imagine that this generalizes to rational $m$-torsion for general $m$, albeit probably somewhat painfully (and there seem to be some papers on the subject of doing $5$-descents, $7$-descents, etc). However, this still requires *some* torsion point to be rational, which need not be the case. Plus, if I understand how programs like mwrank work, they use only $2$-descent.

Thus, it seems like the thing to do is to carry out a $2$-descent over a larger field which does have $2$-torsion points, and then use the result to reconstruct what the Mordell-Weil group is. My first question: is this actually how you do it? My second question: once you accomplish a $2$-descent over a field $K / \mathbb{Q}$, is it always straightforward to infer what $E(\mathbb{Q})$ is?