

A221362


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


2



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
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OFFSET

1,4


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.


REFERENCES

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


LINKS

Table of n, a(n) for n=1..37.
B. Mazur, Rational isogenies of prime degree, Inventiones Math. 44, 2 (June 1978), 129162.
Wikipedia, Elliptic curve
Wikipedia, Mazur's torsion theorem


FORMULA

a(n) = 0 for n > 16.
a(A059765(n)) > 0.  Jonathan Sondow, May 10 2014


EXAMPLE

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.


CROSSREFS

Cf. A059765 (possible sizes of the torsion subgroup of an elliptic curve over Q), A146879.
Sequence in context: A327785 A105242 A336709 * A114116 A054532 A260415
Adjacent sequences: A221359 A221360 A221361 * A221363 A221364 A221365


KEYWORD

nonn,fini,full,easy


AUTHOR

Jonathan Sondow, Jan 12 2013


STATUS

approved



