For every compact symplectic manifold $M$ with a Hamiltonian circle action and isolated fixed points, a simple algebraic identity involving the first Chern class is derived. When the manifold satisfies an extra “positivity condition”, this enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. If $\dim(M)$ is less than or equal to 6 and the number of fixed points is minimal, this algorithm quickly determines all the possible families of isotropy weights, simplifying the proofs due to Ahara and Tolman. In this case all the possible equivariant cohomology rings and Chern classes are determined.