Elliptic curves with good reduction outside of the first six primes

TL;DR

This paper constructs and analyzes a comprehensive database of rational elliptic curves with good reduction outside the first six primes, providing evidence for its completeness and exploring related number theoretic properties.

Contribution

It introduces a new database of elliptic curves with specific reduction properties and heuristically argues for its completeness based on conjectures.

Findings

Database likely contains all such elliptic curves with the given reduction constraints.

Distribution data of elliptic curve invariants are provided.

Connections to S-unit equations and maximal conductor curves are discussed.

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.