be defined over an arbitrary field $K$ whose characteristic is not $2$. If $t$ is an element of $K$ with $t(t+1)(t^2+1)\neq 0$ then the equation $$(t + 1) y^2 = (2 x^2 - t) (4 t^2 x^4 + 4(t^2 + t + 1)x^2 + 1)$$ defines a curve of genus $2$ that we will denote $C(t)$. Clearly the quotient of $C(t)$ by the involution $(x,y)\mapsto (-x,y)$ is an elliptic curve, so the Jacobian of $C(t)$ splits over $K$.

\[T-nonsimple2\] Let $K$ be a field of characteristic not $2$ and suppose $t$ is an element of $K$ such that $t(t^2-1)(t^2+1)$ is nonzero. Then $C(t)$ and $C(-t)$ are curves of genus $2$ over $K$ whose Jacobians are isomorphic over $K$. Furthermore, $C(t)$ and $C(-t)$ are geometrically non-isomorphic unless $K$ has characteristic $11$ and $t^2 \in \{-3, -4\}$.

Our next family takes a little more effort to describe. In order to do so we must define the [*Richelot duals*]{} of a genus-$2$ curve over a field $K$ of characteristic not $2$ (see [@Cassels-Flynn Ch. 9], [@Bending §3]). Suppose $C$ is a genus-$2$ curve over $K$ defined by an equation $\delta y^2 = f$, where $\delta\in K^*$ and where $f$ is a monic separable 