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Equations for the modular curve $ X_1(N)$ and models of elliptic curves with torsion points

Author: Houria Baaziz
Journal: Math. Comp. 79 (2010), 2371-2386
MSC (2010): Primary 11F03; Secondary 11G05, 11G18, 11G30
Published electronically: April 16, 2010
MathSciNet review: 2684370
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Abstract: We describe an algorithm for constructing plane models of the modular curve $ X_1(N)$ and discuss the resulting equations when $ N\leq 51$.

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  • 1. J. W. S. Cassels. Lectures on elliptic curves. London Mathematical Society. Student Texts 24, Cambridge University Press, 1991. MR 1144763 (92k:11058)
  • 2. H. Darmon. Note on a polynomial of Emma Lehmer. Math. Comp. 56, n$ ^{\text{o}}$ 194 (1991), 795-800. MR 1068821 (91i:11149)
  • 3. N. Ishida, N. Ishii. Generators and defining equation of the modular function field of the group $ \Gamma_1(N)$. Acta Arithmetica 101, n$ ^{\text{o}}$ 4 (2002), 303-320. MR 1880045 (2003a:11069)
  • 4. D. S. Kubert, S. Lang. Modular units. Springer-Verlag. New York, Heidelberg, Berlin, 1981. MR 648603 (84h:12009)
  • 5. S. Lang. Elliptic Functions. Addison-Wesley Publishing Company, INC. Advanced Book Program, 1973. MR 0409362 (53:13117)
  • 6. O. Lecacheux. Unités d'une famille de corps cycliques réels de degré $ 6$ liés à la courbe modulaire $ X_1(13)$. J. Number Theory 31 (1989), 54-63. MR 978099 (90i:11062)
  • 7. R. Lercier and F. Morain. Counting points on elliptic curves over $ \mathbb{F}_{p^n}$ using Couveignes's algorithm. Rapport de Recherche LIX/RR/95/09, Ecole Polytechnique, septembre 1995. MR 1367512 (96h:11060)
  • 8. M. A. Reichert. Explicit determination of non-trivial torsion structures of elliptic curves over quadratic number fields. Math. Comput. 46, n$ ^{\text{o}}$ 174 (1986), 637-658. MR 829635 (87f:11039)
  • 9. Y. Yang. Defining equations of modular curves. Advances in Math. 204 (2006), 481-508. MR 2249621 (2007e:11068)
  • 10. L. C. Washington. A family of cyclic quartic fields arising from modular curves. Math. Comp. 57, n$ ^{\text{o}}$ 196 (1991), 763-775. MR 1094964 (92a:11120)

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Additional Information

Houria Baaziz
Affiliation: USTHB Faculté de Mathématiques, BP 32 El Alia Bab Ezzouar Alger, 16111 Algeria

Keywords: Modular curves, elliptic curves, torsion points
Received by editor(s): August 29, 2008
Received by editor(s) in revised form: July 17, 2009
Published electronically: April 16, 2010
Additional Notes: This work was done at the Laboratoire de Mathématiques Nicolas Oresme of the University of Caen. I am deeply indebted to Professor John Boxall for advice and encouragement during my stay at the University of Caen. I am grateful to all members of the Laboratory for their reception and for the provision of facilities. Finally, I would like to thank the referee for his helpful comments on an earlier version of the paper.
Article copyright: © Copyright 2010 American Mathematical Society

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