(Too long for a comment) Replacing $\xi$ with a variable $x$, Mathematica gives the following:


\begin{align*} 
\Delta(2)&=-1+x\\
\Delta(3)&=(-1+x)^3 x (1+x)\\
\Delta(4)&=(-1+x)^6 x^4 (1+x)^2 \left(1+x+x^2\right)\\
\Delta(5)&=(-1+x)^{10} x^{10} (1+x)^4 \left(1+x^2\right) \left(1+x+x^2\right)^2\\
\Delta(6)&=(-1+x)^{15} x^{20} (1+x)^6 \left(1+x^2\right)^2 \left(1+x+x^2\right)^3 \left(1+x+x^2+x^3+x^4\right)
\end{align*}
 etc...

There is a nice pattern: for $d\ge3$, $\Delta(d)$ contains as factors exactly the cyclotomic polynomials of degree 1 to $d-1$, and $x$, raised to some powers.