Elliptic curves of large rank and small conductor

TL;DR

This paper reports the discovery of elliptic curves over rationals with large ranks (6 to 11) and minimal conductors, surpassing previous records and detailing their search methods and specific examples.

Contribution

It presents new elliptic curves with high rank and minimal conductor, improving known records and providing detailed data and search methodology.

Findings

Found elliptic curves with ranks 6 to 11 and minimal conductors.

Improved previous records for elliptic curves of given ranks.

Tabulated lowest conductor and discriminant curves for ranks 5 to 11.

Abstract

For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.