Rank: Difference between revisions
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[[Mordell’s theorem|Mordell proved]] by a method of [[height]]s and descent that all elliptic curves with rational coefficients have finite rank. | [[Mordell’s theorem|Mordell proved]] by a method of [[height]]s and descent that all elliptic curves with rational coefficients have finite rank. | ||
There is said to be an isomorphism of abelian groups | |||
:<math>E(K)\cong \mathbb{Z}^r\oplus E(K)_{\text{tors}}</math> | |||
== Computing the rank == | |||
The rank of “randomly chosen” elliptic curves is claimed to be very small, 0 or 1, on average about 1/2, although curves of arbitrarily large rank are conjectured to exist. | |||
An elliptic curve has a rank of 0 if only a finite number of rational points lie on it <ref>LMFDB: Elliptic curve search results https://www.lmfdb.org/EllipticCurve/Q/?rank=0&torsion=16&search_type=List (And that’s a [[Weierstraß normal form]] with an extra term of ''xy'' thrown in just for good measure.)</ref><ref>Quora: What are the elliptic curves with rank 0 which have most rational points, and what are the elliptic curves with rank 0 which have most integral points? https://www.quora.com/What-are-the-elliptic-curves-with-rank-0-which-have-most-rational-points-and-what-are-the-elliptic-curves-with-rank-0-which-have-most-integral-points</ref>. | |||
See ''[[Birch and Swinnerton-Dyer conjecture]]''. | |||
Latest revision as of 11:26, 3 January 2025
The rank of an elliptic curve is the largest number of rational points on it which are linearly independent with respect to its point group operation.
This is also the smallest number of rational points on the elliptic curve which generate all the others.
Mordell proved by a method of heights and descent that all elliptic curves with rational coefficients have finite rank.
There is said to be an isomorphism of abelian groups
Computing the rank
The rank of “randomly chosen” elliptic curves is claimed to be very small, 0 or 1, on average about 1/2, although curves of arbitrarily large rank are conjectured to exist.
An elliptic curve has a rank of 0 if only a finite number of rational points lie on it [1][2].
See Birch and Swinnerton-Dyer conjecture.
- ↑ LMFDB: Elliptic curve search results https://www.lmfdb.org/EllipticCurve/Q/?rank=0&torsion=16&search_type=List (And that’s a Weierstraß normal form with an extra term of xy thrown in just for good measure.)
- ↑ Quora: What are the elliptic curves with rank 0 which have most rational points, and what are the elliptic curves with rank 0 which have most integral points? https://www.quora.com/What-are-the-elliptic-curves-with-rank-0-which-have-most-rational-points-and-what-are-the-elliptic-curves-with-rank-0-which-have-most-integral-points
