  This note presents explicit equations (up to birational equivalence over
$\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over
$\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985
Harvard lecture notes on curves over finite fields, J-P.~Serre already showed
the existence of such a curve: he used class field theory to describe the
function field $\mathbb{F}_2(X)$ as a certain abelian extension of the function
field $\mathbb{F}_2(E)$ of some elliptic curve $E/\mathbb{F}_2$. Although
various more recent texts recall Serre's construction, explicit equations as
well as a description of intermediate curves $X\to Y\to E$ over $\mathbb{F}_2$
seem to be new. We also describe explicit equations for a curve over
$\mathbb{F}_2$ of genus $8$ with $11$ rational points, and for a curve over
$\mathbb{F}_2$ of genus $22$ with $21$ rational points.
