On monic abelian cubics

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 $\max \{|a_2|,|a_1|,|a_0|\}\le X$ 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