Orbits of binary cubic forms under GL(R ⊕ a) are canonically in bijection with 3-torsion ideal classes of quadratic rings over any Dedekind domain R with char(R) ≠ 3, with the Steinitz class [a] as an explicit parameter.
The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
We prove that, on average, the monogenicity or $n$-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.
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A parametrization of $3$-class groups of quadratic rings over Dedekind domains
Orbits of binary cubic forms under GL(R ⊕ a) are canonically in bijection with 3-torsion ideal classes of quadratic rings over any Dedekind domain R with char(R) ≠ 3, with the Steinitz class [a] as an explicit parameter.