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A146879
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Minimal degree of X_1(n)
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0
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 5, 3, 4, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,11
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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 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.
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REFERENCES
| J.-F. Mestre, "Corps euclidiens, unites 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.
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.
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LINKS
| Andrew V. Sutherland, Constructing elliptic curves with prescribed torsion over finite fields, preprint, 2008.
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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).
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CROSSREFS
| A029937
Sequence in context: A025801 A060548 A140426 * A058762 A029252 A094876
Adjacent sequences: A146876 A146877 A146878 * A146880 A146881 A146882
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KEYWORD
| hard,more,nonn
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AUTHOR
| Andrew V. Sutherland (drew(AT)math.mit.edu), Nov 03 2008
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