Galois lines for a canonical curve of genus 4, I: Non-skew cyclic lines

Let $C \subset \mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic $0$. For a line $l \subset \mathbb{P}^3$, we consider the projection $\pi\_l\colon C \rightarrow \mathbb{P}^1$ with center $l$ and the extension of the function fields $\pi\_l^\ast\colon k(\mathbb{P}^1) \hookrightarrow k(C)$. A line $l$ is assumed to be cyclic for $C$, if the extension $k(C)/\pi\_l^\*(k(\mathbb{P}^1))$ is cyclic. A line $l$ is assumed to be non-skew, if $C \cap l \ne \emptyset$, i.e., $\deg \pi\_l < \deg C = 6$. We investigate the number of non-skew cyclic lines for $C$. As main results, we explicitly give the equation of $C$ in the particular case in which $C$ has two cyclic trigonal morphisms; we prove that the number of cyclic lines with $\deg \pi\_l =4$ is at most\~$1$, and the number of cyclic lines with $\deg \pi\_l =5$ is at most\~$1$.