cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A059765 Possible sizes of the torsion group of an elliptic curve over the rationals Q. This is a finite sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16
Offset: 1

Views

Author

Noam Katz (noamkj(AT)hotmail.com), Feb 21 2001

Keywords

Examples

			a(1) corresponds to the trivial group.
a(2) corresponds to the cyclic group C_2.
a(3) corresponds to the cyclic group C_3.
a(4) corresponds to the cyclic group C_4 and the product C_2 x C_2.
a(5) corresponds to the cyclic group C_5.
a(6) corresponds to the cyclic group C_6.
a(7) corresponds to the cyclic group C_7.
a(8) corresponds to the cyclic group C_8 and the product C_2 x C_4.
a(9) corresponds to the cyclic group C_9.
a(10) corresponds to the cyclic group C_10.
a(12) corresponds to the cyclic group C_12 and the product C_2 x C_6.
a(16) corresponds to the product C_2 x C_8.

References

  • Joseph H. Silverman, The Arithmetic of Elliptic Curves, Graduates texts in mathematics 106 Springer-Verlag.

Links

Crossrefs

Cf. A221362.

Formula

Numbers n such that A221362(n) > 0. - Jonathan Sondow, May 10 2014

A229831 Largest prime p such that some elliptic curve over an extension of the rationals of degree n has a point of finite order p.

Original entry on oeis.org

7, 13, 13, 17
Offset: 1

Views

Author

Jonathan Sondow, Oct 12 2013

Keywords

Comments

a(1) = 7 is due to Mazur; a(2) = 13 to Kamienny, Kenku, and Momose; a(3) = 13 to Parent; and a(4) = 17 to Kamienny, Stein, and Stoll. See Derickx 2011.
For each n = 1..32, an explicit elliptic curve with a point of order p(n) has been found over a number field of degree n where p(n) = 7, 13, 13, 17, 19, 37, 23, 23, 31, 37, 31, 43, 37, 43, 43, 37, 43, 43, 43, 61, 47, 67, 47, 73, 53, 79, 61, 53, 53, 73, 61, 97. So p(n) is a lower bound for a(n). I suspect most of them are sharp but that would be difficult to prove. - Mark van Hoeij, May 21 2014

Examples

			Mazur proved that elliptic curves over the rationals can have p-torsion only for p = 2, 3, 5, 7, so a(1) = 7.

Links

Crossrefs

Cf. A221362.
Showing 1-2 of 2 results.

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