We study effectively the Cartan geometry of Levi-nondegenerate C^6-smooth hypersurfaces M^3$ in C^2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M^3 of C^2 to the Heisenberg sphere H^3, such M being necessarily real analytic. Finally using the Groebner basis techniques, we present an an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a field K of characteristic zero.