  The distribution of degree $d$ points on curves is well understood,
especially for low degrees. We refine this study to include information on the
Galois group in the simplest interesting case: $d = 3$. For curves of genus at
least 5, we show cubic points with Galois group $C_3$ arise from
well-structured morphisms, along with providing computable tests for the
existence of such morphisms. We prove the same for curves of lower genus under
some geometric or arithmetic assumptions.
