${\operatorname{char}}K = 19$ and $v + 1 = 0$[;]{}

-   ${\operatorname{char}}K = 89$ and $v + 36 = 0$[;]{}

-   ${\operatorname{char}}K = 1033$ and $v + 508 = 0$.

Furthermore, if $K$ has characteristic $0$ and if $v$ is not an algebraic number, then the Jacobians of $C$ and $C'$ are absolutely simple.

In fact, when $K$ has characteristic $0$ it is very easy to find [*algebraic*]{} numbers $v$ in $K$ for which the Jacobians in Theorem \[T-simple2\] are absolutely simple. For example, suppose $R$ is a subring of $K$ for which there is a homomorphism $\varphi$ to an extension of ${{\mathbb F}}_{13}$. We show in the proof of Theorem \[T-simple2\] that in this case the Jacobians of $D$ and $D'$ are geometrically irreducible whenever $v$ lies in $\varphi^{-1}(2)$ or $\varphi^{-1}(6)$.

Theorem \[T-simple2\] gives a $1$-parameter family of pairs of non-isomorphic curves with isomorphic Jacobians. In fact, we shall see that there is a family of such pairs of curves parametrized by an elliptic surface; over ${{\mathbb Q}}$, this surface has positive rank.

Our third family of pairs of curves with isomorphic Jacobians is again easy to write down. Suppose $K$ is a field of characteristic not $2$ and suppose 