# Constructing families of 3-Selmer companions

## TL;DR

This paper constructs families of elliptic curves that are non-isogenous 3-Selmer companions under certain conditions.

## Contribution

The novelty is constructing multiple families of 3-Selmer companion elliptic curves parameterized by integers.

## Key findings

- Pairs of families of elliptic curves are constructed as 3-Selmer companions.
- Each family is parameterized by integers with possible congruence conditions.
- The curves in each pair are non-isogenous.

## Abstract

Mazur and Rubin introduced the notion of n-Selmer companion elliptic curves and gave several examples of pairs of non-isogenous Selmer companions. We construct several pairs of families of elliptic curves, each parameterised by \documentclass[12pt]{minimal}
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\begin{document}$$t\in \mathbb {Z}$$\end{document}t∈Z, such that the two curves in a pair corresponding to a given t are non-isogenous 3-Selmer companions, possibly provided that t satisfies a simple congruence condition.