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The purpose of this paper is to list the refined Humbert invariants for agiven automorphism group of a curve $C/K$ of genus 2 over an algebraicallyclosed field $K$ with characteristic $0$. This invariant is an algebraicgeneralization of the (usual) \textit{Humbert invariant}. It is a positivedefinite quadratic form associated to the curve $C$, and it encodes manygeometric properties of the curve. The paper has a special interest in thecases where $Aut(C)\simeq D_4$ or $D_6$. In these cases, several applicationsof the main results are discussed, including the curves with elliptic subcoversof a given degree.

The Refined Humbert Invariant for an Automorphism Group of a Genus 2 Curve