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# Mazur’s theorem on torsion of elliptic curves

###### Theorem 1 (Mazur).

Let $E/\mathbb{Q}$ be an elliptic curve. Then the torsion subgroup $E_{{\operatorname{torsion}}}(\mathbb{Q})$ is exactly one of the following groups:

$\mathbb{Z}/N\mathbb{Z}\quad 1\leq N\leq 10\quad or\quad N=12$ |

$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2N\mathbb{Z}\quad 1\leq N\leq 4$ |

Note: see Nagell-Lutz theorem for an efficient algorithm to compute the torsion subgroup of an elliptic curve defined over $\mathbb{Q}$.

# References

- 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.

Defines:

Mazur's theorem

Keywords:

torsion, elliptic curve

Related:

EllipticCurve, MordellWeilTheorem, RankOfAnEllipticCurve, TorsionSubgroupOfAnEllipticCurveInjectsInTheReductionOfTheCurve, ArithmeticOfEllipticCurves

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

14H52*no label found*

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