A census of genus 6 curves over 𝔽₂

<p> We compile a complete list of isomorphism class representatives of curves of genus 6 over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We use explicit descriptions of canonical curves in each stratum of the Brill–Noether stratification of the moduli space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M 6"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {M}_6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , due to Mukai in the generic case. Our computed value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="number-sign script upper M 6 left-parenthesis double-struck upper F 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> # </mml:mi> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\#\mathcal {M}_6(\mathbb {F}_2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> agrees with the Lefschetz trace formula as recently computed by Bergstrom–Canning–Petersen–Schmitt. We also report progress on compiling a corresponding list in genus 7. </p>