Computing Mordell-Weil Groups without Rational Torsion

Question

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?

Answer 1

When there is a $2$-torsion point present then one should indeed use these isogenies to do a descent. One can do this over a number field as long as it is not too difficult to calculate the class group and the units. To focus on points that are actually defined over $\mathbb{Q}$ imposes a norm equation for some étale algebra. A series of 3 papers entitled "Explicit $n$-descent on elliptic curves" explains the general method (The third is https://arxiv.org/abs/1107.3516) which also works well for elliptic curves over $\mathbb{Q}$ and $n=3$.

Otherwise there is also the very efficient method using the invariants $I$ and $J$ and their syzygy. This avoids working with larger fields alltogether. This was already used by Birch and Swinnerton-Dyer. It is well explained in Chapter 3 of Cremona's book http://homepages.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html . This explains his implementation of "mwrank". It is also described in Smart's "The algorithmic resolution of Diophantine equations".