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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 (list; graph; refs; listen; history; text; internal format)
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, 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
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
KEYWORD
nonn,fini,full,easy
AUTHOR
Jonathan Sondow, Jan 12 2013
STATUS
approved

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Last modified April 17 07:13 EDT 2024. Contains 371756 sequences. (Running on oeis4.)