# 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 curve defined over $\mathbb{Q}$.

###### Theorem 1 (Nagell-Lutz Theorem).

Let $E/\mathbb{Q}$ be an elliptic curve with Weierstrass equation:

 $y^{2}=x^{3}+Ax+B,\quad A,B\in\mathbb{Z}$

Then for all non-zero torsion points $P$ we have:

1. 1.

The coordinates of $P$ are in $\mathbb{Z}$, i.e.

 $x(P),y(P)\in\mathbb{Z}$
2. 2.

If $P$ is of order greater than $2$, then

 $y(P)^{2}\quad divides\quad 4A^{3}+27B^{2}$
3. 3.

If $P$ is of order $2$ then

 $y(P)=0\quad and\quad x(P)^{3}+Ax(P)+B=0$

## References

• 1 E. Lutz, Sur l’equation $y^{2}=x^{3}-Ax-B$ 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, http://www.jmilne.org/math/CourseNotes/math679.htmlonline course notes.
• 4 Joseph H. Silverman, . 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