A146879 Minimal degree of X_1(n).

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

Offset: 1

Author

Andrew V. Sutherland, Nov 03 2008

Keywords

• hard
• more
• nonn

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 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.

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 [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.

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

Examples

			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).
		

Links

• 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.
• Mark van Hoeij, Upper bounds
• J.-F. Mestre, Corps euclidiens, unités exceptionnelles et courbes elliptiques, J. Number Theory, vol. 13, 1981, pp. 123-137
• Markus Reichert, Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields, Math. Comp. 46 (1986), pp. 637-658.
• Andrew V. Sutherland, Constructing elliptic curves with prescribed torsion over finite fields, preprint, arXiv:0811.0296 [math.NT], 2008-2012.
• 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

Crossrefs

Cf. A029937.