The ideal generating this is from what I call the "super three-vortex problem", the equations for the central configurations of a 1/r^2 potential:
I am mainly interested in the nonzero solutions of this system. To get at those, we can saturate the ideal - or in practical terms we can introduce a new variable, w, and add the equation
w*s12*s13*s23 - 1 = 0to the ideal. The 3D Groebner fan of the resulting system can be seen here.