cp's OEIS Frontend

A146879 Minimal degree of X_1(n).

%I A146879 #28 Nov 14 2022 23:04:47
%S A146879 1,1,1,1,1,1,1,1,1,1,2,1,2,2,2,2,4,2,5,3,4,4,7,4,5,6,6,6,11,6,12,8,10,
%T A146879 10,12,8,18,12,14,12
%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 exist 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.
%C A146879 For n = 23 to 40, a(n) has been computed by M. Derickx and M. van Hoeij. For n = 41 to 100, upper bounds for a(n) have been computed by M. van Hoeij (see link). - _Mark van Hoeij_, Apr 17 2012
%H A146879 Daeyeol Jeon, Chang Heon Kim and Andreas Schweizer, <a href="http://dx.doi.org/10.4064/aa113-3-6">On the torsion of elliptic curves over cubic number fields</a>, Acta Arithmetica 113 (2004), pp. 291-301.
%H A146879 Mark van Hoeij, <a href="http://www.math.fsu.edu/~hoeij/files/X1N/gonality">Upper bounds</a>
%H A146879 J.-F. Mestre, <a href="https://doi.org/10.1016/0022-314X(81)90001-9">Corps euclidiens, unités exceptionnelles et courbes elliptiques</a>, J. Number Theory, vol. 13, 1981, pp. 123-137
%H A146879 Markus Reichert, <a href="https://doi.org/10.1090/S0025-5718-1986-0829635-X">Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields</a>, Math. Comp. 46 (1986), pp. 637-658.
%H A146879 Andrew V. Sutherland, <a href="http://arxiv.org/abs/0811.0296">Constructing elliptic curves with prescribed torsion over finite fields</a>, preprint, arXiv:0811.0296 [math.NT], 2008-2012.
%H A146879 A. V. Sutherland, <a href="http://math.mit.edu/~drew/MazursTheoremSubsequentResults.pdf">Notes on torsion subgroups of elliptic curves over number fields</a>, 2012. - From _N. J. A. Sloane_, Feb 02 2013
%H A146879 A. V. Sutherland, <a href="http://www-math.mit.edu/~drew/MazursTheoremSubsequentResults.pdf">Torsion subgroups of elliptic curves over number fields</a>, 2012. - From _N. J. A. Sloane_, Feb 03 2013
%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 Cf. A029937.
%K A146879 hard,more,nonn
%O A146879 1,11
%A A146879 _Andrew V. Sutherland_, Nov 03 2008