# abstract algebra – Plane quartics and morphisms

Suppose that $$C$$ is a non-hyperelliptic genus 3 curve defined over $$mathbb{Q}$$.
It is well known that non-hyperelliptic genus 3 curves are plane quartics, and these are also trigonal. So we have a morphism $$C mapsto mathbb{P}^{1}$$ of degree $$3$$, and the way in which one usually constructs this map is choosing a point $$P$$ in $$C$$ and projecting from $$P$$ to $$mathbb{P}^{1} subseteq mathbb{P}^{2}.$$

I am now wondering whether one can deduce that the degree 3 map $$C mapsto mathbb{P}^{1}$$ can always be defined over $$mathbb{Q}$$. If we have a rational point $$P$$ on $$C$$, then I guess you can always find a rational map $$C mapsto mathbb{P}^{1}$$ defined over $$mathbb{Q}$$ by the above construction. Moreover, if $$C$$ admits a model of the form $$ax^4+ay^4+ text{lower order terms} =0$$ for $$a$$ some fourth power, then I guess this is doable too.

I am now wondering whether one can always ensure that there’s a degree $$3$$ map which is defined over $$mathbb{Q}$$ if $$C$$ is defined over $$mathbb{Q}$$.