Nagell–Lutz theorem

In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.

Definition of the terms

Suppose that the equation

defines a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side:

Statement of the theorem

If P = (x,y) is a rational point of finite order on C, for the elliptic curve group law, then:


The Nagell–Lutz theorem generalizes to arbitrary number fields and more general cubic equations.[1] For curves over the rationals, the generalization says that, for a nonsingular cubic curve whose Weierstrass form

has integer coefficients, any rational point P=(x,y) of finite order must have integer coordinates, or else have order 2 and coordinates of the form x=m/4, y=n/8, for m and n integers.


The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937).

See also


  1. See, for example, Theorem VIII.7.1 of Joseph H. Silverman (1986), "The arithmetic of elliptic curves", Springer, ISBN 0-387-96203-4.
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