Elliptic curves with good reduction outside of the first six primes

2020/07/21 by Alex J. Best, Best, Alex J., Benjamin Matschke +1
Computer Science · Mathematics · Social Sciences · #11G05 (Primary) 11Y50 (Secondary) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.2007.10535

openalex publication_date 2020/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a database of rational elliptic curves, up to Q-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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