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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 Table of n, a(n) for n=1..37. B. Mazur, Rational isogenies of prime degree, Inventiones Math. 44, 2 (June 1978), 129-162. 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

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