Nagell-Lutz theorem

The following theorem, proved independently by E. Lutz and T. Nagell, gives a very efficient method to compute the torsion subgroup of an elliptic curveMathworldPlanetmath defined over .

Theorem 1 (Nagell-Lutz Theorem).

Let E/Q be an elliptic curve with Weierstrass equation:


Then for all non-zero torsionPlanetmathPlanetmath points P we have:

  1. 1.

    The coordinates of P are in , i.e.

  2. 2.

    If P is of order greater than 2, then

  3. 3.

    If P is of order 2 then



  • 1 E. Lutz, Sur l’equation y2=x3-Ax-B dans les corps p-adic, J. Reine Angew. Math. 177 (1937), 431-466.
  • 2 T. Nagell, Solution de quelque problemes dans la theorie arithmetique des cubiques planes du premier genre, Wid. Akad. Skrifter Oslo I, 1935, Nr. 1.
  • 3 James Milne, Elliptic Curves, course notes.
  • 4 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
Title Nagell-Lutz theorem
Canonical name NagellLutzTheorem
Date of creation 2013-03-22 13:52:02
Last modified on 2013-03-22 13:52:02
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
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 Nagell-Lutz theorem