# Application of Riemann-Roch in genus 3

Referring to the notation of Application of Riemann-Roch how is it possible to show that in genus $g=3$ then $X$ is a double covering space of the Riemann sphere ramified in 8 points or it is isomorphic to a curve in $\mathbb{P}^2(\mathbb{C})$ of deg 4?

The 8 points is clear (it follows from Rieman-Hurwitz). Appliyng Riemann-Roch with $D=K$ i find $l(K)=5$ and with $D=0$ i find $l(K)=3$, so how can i use these facts (provided i didn't make errors )?

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