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2021 | OriginalPaper | Chapter

Elliptic Curves with Good Reduction Outside of the First Six Primes

Authors : Alex J. Best, Benjamin Matschke

Published in: Arithmetic Geometry, Number Theory, and Computation

Publisher: Springer International Publishing

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Abstract

We present a database of rational elliptic curves, up to https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-80914-0_5/510803_1_En_5_IEq1_HTML.gif -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.
Footnotes
1
However, work in progress by the second author gives the same set of curves using a different method.
 
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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Metadata
Title
Elliptic Curves with Good Reduction Outside of the First Six Primes
Authors
Alex J. Best
Benjamin Matschke
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-80914-0_5

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