Finding Elliptic Curves With Many Integral Points

TL;DR

This paper develops parameterizations of elliptic curves to identify those with many integral points, performing computational searches to find curves with specific integral properties and classifying self-descriptive numbers.

Contribution

It introduces new parameterizations for elliptic curves with many integral points and provides a complete classification of self-descriptive numbers.

Findings

Found over 200 curves with more than 100 integral points

Identified curves with points of small height and many integral multiples

Classified all self-descriptive numbers based on zero counts

Abstract

In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small height, curves with many integral multiples of a point, curves with high multiples of a point integral, and over two hundred curves with more than one hundred integral points. In addition, a novel and complete classification of "self-descriptive numbers" is constructed by bounding the number of zeros such a number must contain.