  Let $\mathbb F_q$ be a finite field with $q$ elements, where $q$ is a power
of an odd prime $p$. In this paper we associate circulant matrices and
quadratic forms with the Artin-Schreier curve $y^q - y= x \cdot F(x) -
\lambda,$ where $F(x)$ is a $\mathbb F_q$-linearized polynomial and $\lambda
\in \mathbb F_q$. Our results provide a characterization of the number of
affine rational points of this curve in the extension $\mathbb F_{q^r}$ of
$\mathbb F_q$, for $\gcd(q,r)=1$. In the case $F(x) = x^{q^i}-x$ we give a
complete description of the number of affine rational points in terms of
Legendre symbols and quadratic characters.
