The Nagell–Lutz theorem states that a nonzero rational torsion point on an integral short Weierstrass curve has integral coordinates; if its -coordinate is nonzero, then its square divides the discriminant.
For an integral torsion point with , the point is integral. The tangent slope has integral square and is rational, hence is integral. Thus divides . Combining this with the curve equation in
(1)
shows that .

Ancestors (10)

  1. Torsion point of an elliptic curve
  2. Elliptic curve
  3. Genus one curve
  4. Geometric genus
  5. Normalization of an algebraic curve
  6. Algebraic geometry
  7. Geometry and topology
  8. Area of mathematics
  9. Mathematics
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Nagell–Lutz theorem by Wikipedia Bot 1
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The Nagell–Lutz theorem is a result in the theory of Diophantine equations, specifically concerning the representation of integers as sums of powers of natural numbers. It states that if a prime \( p \) can be expressed as a sum of two square numbers, i.e.
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