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Elliptic Curves with Good Reduction Outside of the First Six Primes

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Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))

Abstract

We present a database of rational elliptic curves, up to -isomorphism, with good reduction outside {2, 3, 5, 7, 11, 13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.

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Notes

  1. 1.

    However, work in progress by the second author gives the same set of curves using a different method.

  2. 2.

    Allan MacLeod has communicated to us the Mordell–Weil bases of 10 more of these curves, using his own implementations of similar techniques to those outlined above. Nevertheless a fast general method to find bases for all of our curves remains elusive.

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Acknowledgements

It is our pleasure to thank Edgar Costa for various useful comments and for computing the analytic ranks of all curves in our database, as well as the leading coefficients and root numbers of the associated L-series. They are available from the same GitHub repository. We would also like to thank the anonymous referees for their helpful comments on earlier versions of this article.

The first author was supported by Simons Foundation grant #550023 and a Hariri Institute Graduate Student Fellowship.

The second author was partially supported by Excellence Initiative of Université de Bordeaux (IdEx project DiGeMANT), and by Simons Foundation grant #550023.

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Correspondence to Benjamin Matschke .

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Best, A.J., Matschke, B. (2021). Elliptic Curves with Good Reduction Outside of the First Six Primes. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_5

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