This chapter deals with functionality that is specific to elliptic curves defined over the rationals or over number fields. As a general goal, these functions are developed in parallel. Naturally, machinery for curves over Q has been developed much further than for general number fields, although the latter has been a major focus of recent releases.
The material on computing the Mordell-Weil group (of rational points) and descent methods is presented in the final part of the chapter, and deals with all base fields at the same time. The other topics are presented separately for Q and for number fields, although many of the functions are the same for both.
Note: functions declared for number fields can also be used over Q. To do so, it may be necessary to construct Q as RationalsAsNumberField() rather than Rationals().