  Genus 5 curves can be hyperelliptic, trigonal, or non-hyperelliptic
non-trigonal, whose model is a complete intersection of three quadrics in
$\mathbb{P}^4$. We present and explain algorithms we used to determine, up to
isomorphism over $\mathbb{F}_2$, all genus 5 curves defined over
$\mathbb{F}_2$, and we do that separately for each of the three mentioned
types. We consider these curves in terms of isogeny classes over $\mathbb{F}_2$
of their Jacobians or their Newton polygons, and for each of the three types,
we compute the number of curves over $\mathbb{F}_2$ weighted by the size of
their $\mathbb{F}_2$-automorphism groups.
