Real versus complex plane curves

We prove that a smooth, complex plane curve $C$ of odd degree can be defined by a polynomial with real coefficients if and only if $C$ is isomorphic to its complex conjugate. Counterexamples are known for curves of even degree. More generally, we prove that a plane curve $C$ over an algebraically closed field $K$ of characteristic $0$ with field of moduli $k_{C}\subset K$ is defined by a polynomial with coefficients in $k'$, where $k'/k_{C}$ is an extension with $[k':k_{C}]\le 3$ and $[k':k_{C}]\mid \operatorname{deg} C$.