

A146879


Minimal degree of X_1(n).


1



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, 10, 12, 8, 18, 12, 14, 12
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OFFSET

1,11


COMMENTS

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 nonisomorphic 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.
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).
From [JeonKimSchweizer 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 [JeonKimPark 2006]. a(23) must be 5, 6, or 7. See [Sutherland 2008] for these and other upper bounds for n <= 50.
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


LINKS

Table of n, a(n) for n=1..40.
Daeyeol Jeon, Chang Heon Kim and Andreas Schweizer, On the torsion of elliptic curves over cubic number fields, Acta Arithmetica 113 (2004), pp. 291301.
Mark van Hoeij, Upper bounds
J.F. Mestre, Corps euclidiens, unités exceptionnelles et courbes elliptiques, J. Number Theory, vol. 13, 1981, pp. 123137
Markus Reichert, Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields, Math. Comp. 46 (1986), pp. 637658.
Andrew V. Sutherland, Constructing elliptic curves with prescribed torsion over finite fields, preprint, arXiv:0811.0296 [math.NT], 20082012.
A. V. Sutherland, Notes on torsion subgroups of elliptic curves over number fields, 2012.  From N. J. A. Sloane, Feb 02 2013
A. V. Sutherland, Torsion subgroups of elliptic curves over number fields, 2012.  From N. J. A. Sloane, Feb 03 2013


EXAMPLE

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).


CROSSREFS

Cf. A029937.
Sequence in context: A025801 A060548 A140426 * A231577 A277210 A304777
Adjacent sequences: A146876 A146877 A146878 * A146880 A146881 A146882


KEYWORD

hard,more,nonn


AUTHOR

Andrew V. Sutherland, Nov 03 2008


STATUS

approved



