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%I #19 May 22 2014 09:38:02
%S 7,13,13,17
%N Largest prime p such that some elliptic curve over an extension of the rationals of degree n has a point of finite order p.
%C 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.
%C 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
%H Maarten Derickx, <a href="http://wstein.org/wiki/attachments/seminar(2f)nt(2f)20110318/slides.pdf">Torsion points on elliptic curves over number fields of small degree</a>, UW Number Theory Seminar, 2011
%H Mark van Hoeij, <a href="http://arxiv.org/abs/1202.4355">Low Degree Places on the Modular Curve X1(N)</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Mazur%27s_torsion_theorem">Mazur's torsion theorem</a>
%e Mazur proved that elliptic curves over the rationals can have p-torsion only for p = 2, 3, 5, 7, so a(1) = 7.
%Y Cf. A221362.
%K nonn,more,hard,bref
%O 1,1
%A _Jonathan Sondow_, Oct 12 2013
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