by

Ciro Santilli (@cirosantilli, 32)

Number of elements of an elliptic curve over the rational numbers

Can be finite or infinite! TODO examples. But it is always a finitely generated group.

Table of contents
- Mordell's theorem
  - Rank of an elliptic curve over the rational numbers Mordell's theorem
    - Largest known ranks of an elliptic curve over the rational numbers Rank of an elliptic curve over the rational numbers

Mordell's theorem (1922)

The elliptic curve group of all elliptic curve over the rational numbers is always a finitely generated group.

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.

Rank of an elliptic curve over the rational numbers ( $r$ )

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!

TODO list of known values and algorithms? The Birch and Swinnerton-Dyer conjecture would immediately provide a stupid algorithm for it.

Largest known ranks of an elliptic curve over the rational numbers

web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:

y^{2} + x y + y = x^{3} - x^{2} - 20067762415575526585033208209338542750930230312178956502 x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429

TODO why this non standard formluation?

 Ancestors

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