Mazur’s theorem on torsion of elliptic curves

Theorem 1 (Mazur).

Let E/Q be an elliptic curveMathworldPlanetmath. Then the torsion subgroup Etorsion(Q) is exactly one of the following groups:


Note: see Nagell-Lutz theorem for an efficient algorithm to compute the torsion subgroup of an elliptic curve defined over .


  • 1 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 2 Barry Mazur, Modular curves and the Eisenstein ideal, IHES Publ. Math. 47 (1977), 33-186.
  • 3 Barry Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
Title Mazur’s theorem on torsionPlanetmathPlanetmath of elliptic curves
Canonical name MazursTheoremOnTorsionOfEllipticCurves
Date of creation 2013-03-22 13:51:59
Last modified on 2013-03-22 13:51:59
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 14H52
Related topic EllipticCurve
Related topic MordellWeilTheorem
Related topic RankOfAnEllipticCurve
Related topic TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve
Related topic ArithmeticOfEllipticCurves
Defines Mazur’s theorem