login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 27 20:49 EST 2021. Contains 349395 sequences. (Running on oeis4.)