by Ciro Santilli (@cirosantilli, 34)
The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.
For this reason, the rank of an elliptic curve over the rational numbers is always defined.
TODO example.
Mordell's theorem guarantees that the rank (number of elements in the generating set of the group) is always well defined for an elliptic curve over the rational numbers. But as of 2023 there is no known algorithm which calculates the rank of any curve!
It is not even known if there are elliptic curves of every rank or not: Largest known ranks of an elliptic curve over the rational numbers, and it has proven extremely hard to find new ones over time.
TODO list of known values and algorithms? The Birch and Swinnerton-Dyer conjecture would immediately provide a stupid algorithm for it.
web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:
(1)
TODO why this non standard formulation?

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  1. Number of elements of an elliptic curve over the rational numbers
  2. Elliptic curve over the rational numbers
  3. Domain of an elliptic curve
  4. Elliptic curve
  5. Algebraic geometry
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