# On monic abelian cubics

## TL;DR

This paper establishes bounds on the number of monic irreducible cubic polynomials with integer coefficients that are Galois over Q, counts abelian monic binary cubic forms, and characterizes splitting fields of 2-torsion points of semi-stable elliptic curves.

## Contribution

It provides new upper bounds on the count of certain Galois cubic polynomials and classifies abelian binary cubic forms, along with a characterization of elliptic curve torsion fields.

## Key findings

- Bound of O(X (log X)^2) for Galois monic cubics with coefficients bounded by X
- Counting of abelian monic binary cubic forms up to Bhargava-Shankar height
- Characterization of splitting fields of 2-torsion points of semi-stable elliptic curves

## Abstract

In this paper we prove the assertion that the number of monic cubic polynomials $F(x) = x^3 + a_2 x^2 + a_1 x + a_0$ with integer coefficients and irreducible, Galois over $\mathbb{Q}$ satisfying $\max\{|a_2|, |a_1|, |a_0|\} \leq X$ is bounded from above by $O(X (\log X)^2)$. We also count the number of abelian monic binary cubic forms with integer coefficients up to a natural equivalence relation ordered by the so-called Bhargava-Shankar height. Finally, we prove an assertion characterizing the splitting field of 2-torsion points of semi-stable abelian elliptic curves