Reduced minimal models and torsion

TL;DR

This paper classifies the reduced minimal models of elliptic curves over rationals with torsion points, linking their form to congruences on invariants and reduction properties at small primes.

Contribution

It provides an explicit classification of reduced minimal models for elliptic curves with torsion points, based on invariants and reduction behavior at 2 and 3.

Findings

Reduced minimal model is uniquely determined by a congruence on $c_6$ modulo 24.

Classification applies to families of elliptic curves with torsion points.

Reduction at 2 and 3 influences the form of the reduced minimal model.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. The reduced minimal model of $E$ is a global minimal model $y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}$ which satisfies the additional conditions that $a_{1},a_{3}\in \{0,1\}$ and $a_{2}\in\{0,\pm1\}$. The reduced minimal model of $E$ is unique, and in this article, we explicitly classify the reduced minimal model of an elliptic curve $E/\mathbb{Q}$ with a non-trivial torsion point. We obtain this classification by first showing that the reduced minimal model of $E$ is uniquely determined by a congruence on $c_6$ modulo $24$. We then apply this result to parameterized families of elliptic curves to deduce our main result. We also show that the reduction at $2$ and $3$ of $E$ affects the reduced minimal model of $E$.