A221362 Number of distinct groups of order n that are the torsion subgroup of an elliptic curve over the rationals Q.

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Jonathan Sondow, Jan 12 2013

Keywords

• nonn
• fini
• full
• easy

Comments

Barry Mazur proved that the torsion subgroup of an elliptic curve over Q is one of the 15 following groups: Z/NZ for N = 1, 2, …, 10, or 12, or Z/2Z × Z/2NZ with N = 1, 2, 3, 4.

Examples

			a(4) = 2 because a subgroup of order 4 in an elliptic curve over Q is isomorphic to one of the 2 groups Z/4Z or Z/2Z × Z/2Z.
		

References

• J. H. Silverman, The Arithmetic of Elliptic Curves, Graduates Texts in Mathematics 106, Springer-Verlag, 1986 (see Theorem 7.5).

Links

• B. Mazur, Rational isogenies of prime degree, Inventiones Math. 44, 2 (June 1978), 129-162.
• Wikipedia, Elliptic curve
• Wikipedia, Mazur's torsion theorem

Crossrefs

Cf. A059765 (possible sizes of the torsion subgroup of an elliptic curve over Q), A146879.

Formula

a(n) = 0 for n > 16.

a(A059765(n)) > 0. - Jonathan Sondow, May 10 2014