Rank - Revision history

Rational Point: /* Computing the rank */ extra term

2025-01-03T11:26:57Z

Computing the rank: extra term

← Older revision		Revision as of 11:26, 3 January 2025
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	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.		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</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>.		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]]''.		See ''[[Birch and Swinnerton-Dyer conjecture]]''.

Rational Point at 11:09, 3 January 2025

2025-01-03T11:09:44Z

← Older revision		Revision as of 11:09, 3 January 2025
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	== Computing the rank ==		== 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.		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</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]]''.		See ''[[Birch and Swinnerton-Dyer conjecture]]''.

Rational Point: more info

2025-01-03T06:11:12Z

more info

← Older revision		Revision as of 06:11, 3 January 2025
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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.

			See ''[[Birch and Swinnerton-Dyer conjecture]]''.

Rational Point: def

2024-12-29T22:19:29Z

def

New page

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’s theorem|Mordell proved]] by a method of [[height]]s and descent that all elliptic curves with rational coefficients have finite rank.