$t$ is an element of $K$ with $t(t+1)(t^2+1)(t^2+t+1)\neq 0$. Let $H(t)$ be the genus-$3$ hyperelliptic curve defined by the homogeneous equations $$\begin{aligned}
W^2 Z^2 &= - \frac{ (t^2+1) }{t(t+1)(t^2+t+1)}X^4
           - \frac{4(t^2+1) }{ (t+1)(t^2+t+1)}Y^4
           + \frac{        1}{t              }Z^4 \label{E-H1}  \\
     0  &= - X^2 + 2 t Y^2 + (t+1) Z^2            \label{E-H2}\end{aligned}$$ and let $Q(t)$ be the plane quartic $$\begin{gathered}
\label{E-Q}
X^4 + 4t^2 Y^4 + (t+1)^2 Z^4  + (8t^2 + 4t + 8)X^2Y^2 \\ 
       - (4t^2 + 2t + 2)X^2Z^2 + (4t^2 + 4t +8)Y^2Z^2 = 0.\end{gathered}$$

\[T-nonsimple3\] Let $K$ be a field of characteristic not $2$ and let $t$ be an element of $K$ such that $t(t+1)(t^2+1)(t^2+t+1)\neq 0$. Then the Jacobians of the two genus-$3$ curves $H(t)$ and $Q(t)$ are isomorphic to one another over $K$.

In Section \[S-curves\] we mention some simple facts about abelian surfaces with two non-isomorphic principal polarizations and we show how Richelot isogenies can in principle be used to produce such surfaces from an abelian surface that has nontrivial automorphisms. In Section \[S-nonsimple\] we prove Theorem \[T-nonsimple2\]. In Section \[S-RM\] we review a result of Bending that shows how to obtain genus-$2$ curves over a given field $K$ whose Jacobians have real multiplication by 