Cartan equivalences for Levi-nondegenerate hypersurfaces M^3 in C^2 belonging to General Class I

We develope in great computational details the classical Cartan equivalence problem for Levi-nondegenerate C^6-smooth real hypersurfaces M^3 in C^2, performing all calculations effectively in terms of a (local) graphing function \varphi. In particular, we present explicitly the unique (complex) essential invariant J of the problem. Its expansion in terms of the 3-variables function \varphi incorporates millions of differential monomials, while, when \varphi is assumed to depend only on 2 variables (rigid case), J writes out in two lines (7 monomials).