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.

7, 13, 13, 17

Offset

1,1

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

Links

Table of n, a(n) for n=1..4.

Maarten Derickx, Torsion points on elliptic curves over number fields of small degree, UW Number Theory Seminar, 2011

Mark van Hoeij, Low Degree Places on the Modular Curve X1(N), arXiv:1202.4355 [math.NT], 2012-2014.

Wikipedia, Mazur's torsion theorem

Example

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

Crossrefs

Cf. A221362.

Sequence in context: A242584 A135555 A243044 * A372206 A372083 A352444

Adjacent sequences: A229828 A229829 A229830 * A229832 A229833 A229834

Keyword

nonn,more,hard,bref

Author

Jonathan Sondow, Oct 12 2013

Status

approved