If we apply [@HLP Prop. 4] we find that $(B,\mu)$ is isomorphic over $K$ to the polarized Jacobian of the curve $y^2 = h_t$, where $$h_t =  2^{38} t^6 (t+1)^3 (t^2+1)^{12}
         (2 x^2 - t)
         (4 t^2 x^4 + 4(t^2 + t + 1)x^2 + 1).$$ Furthermore, $(B,\mu')$ is isomorphic to the polarized Jacobians of the curve $y^2 = h_{-t}$. Scaling $h_t$ and $h_{-t}$ by squares in $K$, we find that $y^2 = h_t$ is isomorphic to the curve $C(t)$ of Theorem \[T-nonsimple2\] and that $y^2 = h_{-t}$ is isomorphic to the curve $C(-t)$.

To complete the proof of Theorem \[T-nonsimple2\] we must show that $C(t)$ and $C(-t)$ are geometrically non-isomorphic, except for the special cases listed in the theorem. The simplest way to do this is to use Igusa invariants (see [@Igusa], [@Mestre]). Facilities for computing Igusa invariants are included in the computer algebra package Magma [@Magma].

Let us begin by working over the ring ${{\mathbb Z}}[t]$, where $t$ is an indeterminate. Let $J_2(t), J_4(t), J_6(t), J_8(t),$ and $J_{10}(t)$ be the Igusa invariants of the twist $y^2 =  (2x^2 - t)(4x^4 + 4(t^2 + t + 1)x^2 + 1)$ of $C(t)$. (The invariants $J_{2i}(t)$ of this curve, scaled by 