polynomial in $K[x]$ of degree $6$. Let ${{K\llap{$\overline{\phantom{\rmK}}$}}}$ be a separable closure of $K$, and suppose $f$ can be factored as a product $g_1 g_2 g_3$ of three monic quadratic polynomials in ${{K\llap{$\overline{\phantom{\rmK}}$}}}[x]$ that are permuted by ${\operatorname{Gal}}({{K\llap{$\overline{\phantom{\rmK}}$}}}/K)$. For each $i$ write $g_i = x^2 - t_i x + n_i$ and suppose the determinant $$d = \left|
\begin{matrix}
1 & t_1 & n_1 \\
1 & t_2 & n_2 \\
1 & t_3 & n_3
\end{matrix}
\right| ,$$ which is an element of $K$, is nonzero. Define three new polynomials by setting $$\begin{aligned}
h_1 &= g_3 \frac{dg_2}{dx} - g_2 \frac{dg_3}{dx}\\
h_2 &= g_1 \frac{dg_3}{dx} - g_3 \frac{dg_1}{dx}\\
h_3 &= g_2 \frac{dg_1}{dx} - g_1 \frac{dg_2}{dx}.\end{aligned}$$ Then the product $h_1 h_2 h_3$ is a separable polynomial in $K[x]$ of degree $5$ or $6$.

The [*Richelot dual of $C$ associated to the factorization $f = g_1 g_2 g_3$*]{} is the genus-$2$ curve $D$ defined by $d \delta y^2 = h_1 h_2 h_3.$

\[T-simple2\] Let $K$ be a field of characteristic not $2$, let $v$ be an element of $K\setminus\{0,1,4\}$ such that $$(v^2 - v + 4)
(v^2 + v + 2)
(v^2 + 3 v + 4)
(v^3 - 6 v^2 - 7 