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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
Maarten Derickx, Torsion points on elliptic curves over number fields of small degree, UW Number Theory Seminar, 2011
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 * A352444 A145009 A352352
KEYWORD
nonn,more,hard,bref
AUTHOR
Jonathan Sondow, Oct 12 2013
STATUS
approved

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Last modified March 28 13:17 EDT 2024. Contains 371254 sequences. (Running on oeis4.)