Gauss type complex quadrature formulae, power moment problem and elliptic curves

A complex-valued Borel measure $\omega$ on $\mathbb C$ is called $n$-reducible if there is a quadrature formula with $n$ complex nodes which is exact for all polynomials of degree $\le 2n-1$. A criterion of $n$-reducibility is given on the base of a solvability criterion for a complex power moment problem. The latter is an analytic version of a Sylvester theorem from the theory of binary form invariants. The $2$-reducibility of measures $\omega$ with $|{\mathrm{supp}\,\omega}|=3$ is closely related to the modular invariants of elliptic curves.