Abstract
Tabulating elliptic curves has been carried out since the earliest days of machine computation in number theory. After some historical remarks, we report on significant recent progress in enlarging the database of elliptic curves defined over ℚ to include all those of conductor N≤130000. We also give various statistics, summarize the data, describe how it may be obtained and used, and mention some recent work regarding the verification of Manin’s “c=1” conjecture.
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References
Agashe, A., Ribet, K., Stein, W.A. (with an appendix by J. E. Cremona): The Manin Constant, Congruence Primes and the Modular Degree (preprint, 2006)
Birch, B.J., Kuyk, W. (eds.): Modular Functions of One Variable IV. Lecture Notes in Mathematics, vol. 476. Springer, Heidelberg (1975), http://modular.math.washington.edu/scans/antwerp/
Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on Elliptic Curves I. J. Reine Angew. Math. 212, 7–25 (1963)
Brumer, A., McGuinness, O.: The behaviour of the Mordell-Weil group of elliptic curves. Bull. Amer. Math. Soc (N.S.) 23(2), 375–382 (1990)
Cremona, J.E.: Modular symbols for Γ1(N) and elliptic curves with everywhere good reduction. Math. Proc. Cambridge Philos. Soc. 111(2), 199–218 (1992)
Cremona, J.E.: Algorithms for modular elliptic curves. Cambridge University Press, Cambridge (1992)
Cremona, J.E.: Algorithms for modular elliptic curves, 2nd edn. Cambridge University Press, Cambridge (1997), http://www.maths.nott.ac.uk/personal/jec/book/fulltext/
Cremona, J.E.: Computing the degree of the modular parametrization of a modular elliptic curve. Math. Comp. 64(211), 1235–1250 (1995)
Cremona, J.E.: Tables of Elliptic Curves, http://www.maths.nott.ac.uk/personal/jec/ftp/data/
Cremona, J.E.: mwrank, a program for 2-descent on elliptic curves over ℚ, http://www.maths.nott.ac.uk/personal/jec/mwrank/
Cremona, J.E., Siksek, S.: Computing a Lower Bound for the Canonical Height on Elliptic Curves over ℚ. In: ANTS VII Proceedings 2006. Springer, Heidelberg (2006)
Duke, W.: Elliptic curves with no exceptional primes. C. R. Acad. Sci. Paris Sér. I Math. 325(8), 813–818 (1997)
Edixhoven, B.: On the Manin constants of modular elliptic curves. In: Arithmetic algebraic geometry (Texel, 1989), Progr. Math., vol. 89, pp. 25–39. Birkhäuser, Boston (1991)
LiDIA: A C++ Library For Computational Number Theory, http://www.informatik.tu-darmstadt.de/TI/LiDIA/
NTL: A Library for doing Number Theory, http://www.shoup.net/ntl/
The MAGMA Computational Algebra System, version 2.12-16, http://magma.maths.usyd.edu.au/magma/
pari/gp, version 2.2.13, Bordeaux (2006), http://pari.math.u-bordeaux.fr/
Stein, W.A., Joyner, D.: SAGE: System for Algebra and Geometry Experimentation. Comm. Computer Algebra 39, 61–64 (2005)
Stein, W.A.: SAGE, Software for Algebra and Geometry Experimentation, http://sage.scipy.org/sage
Stein, W.A., Grigorov, G., Jorza, A., Patrikis, S., Patrascu, C.: Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves (preprint, 2006)
Stein, W.A., Watkins, M.: A database of elliptic curves—first report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002)
Stevens, G.: Stickelberger elements and modular parametrizations of elliptic curves. Invent. Math. 98(1), 75–106 (1989)
Tingley, D.J.: Elliptic curves uniformized by modular functions, University of Oxford D. Phil. thesis (1975)
Vélu, J.: Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris Sér. A-B 273, A238–A241 (1971)
Watkins, M.: Computing the modular degree of an elliptic curve. Experiment. Math. 11(4), 487–502 (2002)
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Cremona, J. (2006). The Elliptic Curve Database for Conductors to 130000. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_2
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DOI: https://doi.org/10.1007/11792086_2
Publisher Name: Springer, Berlin, Heidelberg
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