Search: id:A146879
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%I A146879
%S A146879 1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,4,2,5,3,4,4
%N A146879 Minimal degree of X_1(n)
%C A146879 a(n) is the least d>0 for which there exists a plane curve f(x,y)=0 of 
               degree d in x or y which is birationally equivalent to the modular 
               curve X_1(n). There exists infinitely many non-isomorphic elliptic 
               curves defined over number fields of degree a(n) which contain a 
               point of order n. a(n)=1 if and only if X_1(n) has genus 0 and these 
               values of n represent the possible finite orders of a point on an 
               elliptic curve over Q.
%C A146879 By Mazur's theorem, these are 1,2,3,4,5,6,7,8,9,10 and 12. a(n)=2 if 
               and only if X_1(n) is elliptic or hyperelliptic, which occurs only 
               for n=11,13,14,15,16 and 18 [Mestre 1981]. The lower bound a(17)>
               3 follows from [Parent 1999] and the upper bound a(17)<=4 appears 
               (for example) in [Reichert 1986]. a(20)=3 since it cannot be 1 or 
               2 and an explicit example of degree 3 is known (see below).
%C A146879 From [Jeon-Kim-Schweizer 2006] it follows that this is the only case 
               when a(n)=3. The results a(21)=4 and a(22)=4 then follow from explicit 
               examples [Sutherland 2008]. a(24) is either 4 or 5 and a(n) is not 
               4 for any n other than 17, 21, 22, or 24 by the results of [Jeon-Kim-Park 
               2006]. a(23) must be 5, 6, or 7. See [Sutherland 2008] for these 
               and other upper bounds for n <= 50.
%D A146879 J.-F. Mestre, "Corps euclidiens, unites exceptionnelles et courbes elliptiques,
               " J. Number Theory, vol. 13, 1981, pp. 123-137
%D A146879 Markus Reichert, "Explicit Determination of Nontrivial Torsion Structures 
               of Elliptic Curves Over Quadratic Number Fields," Math. Comp. 46 
               (1986), pp. 637-658.
%D A146879 Daeyeol Jeon, Chang Heon Kim and Andreas Schweizer, "On the torsion of 
               elliptic curves over cubic number fields," Acta Arithmetica 113 (2004), 
               pp. 291-301.
%H A146879 Andrew V. Sutherland, <a href="http://arxiv.org/abs/0811.0296">Constructing 
               elliptic curves with prescribed torsion over finite fields</a>, preprint, 
               2008.
%e A146879 a(20)<=3 because y^3+(x^2+3)y^2+(x^3+4)y+2=0 is an explicit plane model 
               for X_1(20) and a(20)=3 because it is not 1 or 2 (these are all known).
%Y A146879 A029937
%Y A146879 Sequence in context: A025801 A140426 A060548 this_sequence A058762 A029252 
               A094876
%Y A146879 Adjacent sequences: A146876 A146877 A146878 this_sequence A146880 A146881 
               A146882
%K A146879 hard,more,nonn
%O A146879 1,11
%A A146879 Andrew V. Sutherland (drew(AT)math.mit.edu), Nov 03 2008

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