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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. 0
7, 13, 13, 17 (list; graph; refs; listen; history; text; internal format)
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)

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 * A145009 A090229 A259222

Adjacent sequences:  A229828 A229829 A229830 * A229832 A229833 A229834

KEYWORD

nonn,more,hard,bref

AUTHOR

Jonathan Sondow, Oct 12 2013

STATUS

approved

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Last modified October 23 10:17 EDT 2021. Contains 348211 sequences. (Running on oeis4.)