On the Clifford algebra of a binary form

The Clifford algebra $C_f$ of a binary form $f$ of degree $d$is the $k$-algebra $k\{x, y\}/I$, where $I$ is the ideal generated by $\{(\alpha x + \beta y)^d - f(\alpha, \beta) \mid \alpha, \beta \in k\}$. $C_f$ has a natural homomorphic image $A_f$ that is a rank $d^2$ Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit $\Theta$-divisor in $\ensuremath{{Pic}_{C/k}^{d + g - 1}}$, where $C$ is the curve $(w^d - f(u, v))$ and $g$is the genus of $C$.