rank of an elliptic curve

Let K be a number fieldMathworldPlanetmath and let E be an elliptic curveMathworldPlanetmath over K. By E(K) we denote the set of points in E with coordinates in K.

Theorem 1 (Mordell-Weil).

The proof of this theorem is fairly involved. The main two ingredients are the so called “weak Mordell-Weil theorem” (see below), the concept of height function for abelian groups and the “descent” theorem.
See [2], Chapter VIII, page 189. ∎

Theorem 2 (Weak Mordell-Weil).

E(K)/mE(K) is finite for all m2.

The Mordell-Weil theoremMathworldPlanetmath implies that for any elliptic curve E/K the group of points has the following structure:


where Etorsion(K) denotes the set of points of finite order (or torsion groupPlanetmathPlanetmath), and R is a non-negative integer which is called the rank of the elliptic curve. It is not known how big this number R can get for elliptic curves over . The largest rank known for an elliptic curve over is 28 http://www.math.hr/ duje/tors/tors.htmlElkies (2006).

Note: see Mazur’s theorem for an account of the possible torsion subgroups over .


  1. 1.

    The elliptic curve E1/:y2=x3+6 has rank 0 and E1()0.

  2. 2.

    Let E2/:y2=x3+1, then E2()/6. The torsion group is generated by the point (2,3).

  3. 3.

    Let E3/:y2=x3+109858299531561, then E3()/35. See http://math.bu.edu/people/alozano/Torsion.htmlgeneratorsPlanetmathPlanetmathPlanetmath here.

  4. 4.

    Let E4/:y2+1951/164xy-3222367/40344y=x3+3537/164x2-40302641/121032x, then E4()10. See http://math.bu.edu/people/alozano/Examples.htmlgenerators here.


  • 1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
  • 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Title rank of an elliptic curve
Canonical name RankOfAnEllipticCurve
Date of creation 2013-03-22 13:49:12
Last modified on 2013-03-22 13:49:12
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 14
Author alozano (2414)
Entry type Definition
Classification msc 14H52
Synonym rank
Related topic EllipticCurve
Related topic HeightFunction
Related topic MordellWeilTheorem
Related topic SelmerGroup
Related topic MazursTheoremOnTorsionOfEllipticCurves
Related topic NagellLutzTheorem
Related topic ArithmeticOfEllipticCurves
Defines weak Mordell-Weil theorem
Defines rank of an elliptic curve