Regulators of rank one quadratic twists

TL;DR

This paper studies the regulators of rank one quadratic twists of elliptic curves, proposing conjectures on their average size, and introduces an efficient algorithm for computing related invariants, supported by numerical data.

Contribution

It formulates conjectures on the average regulators of rank one quadratic twists and provides an efficient computational method for key invariants.

Findings

Numerical data supports conjectures on regulator sizes

Algorithm efficiently computes invariants like Tate-Shafarevich group

Analysis enhances understanding of quadratic twist families

Abstract

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.