ring ${{\mathbb F}}_{11}[t]$, with $$\begin{aligned}
R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2}
              { t (t^2+1)^3 }\\
R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3}
              { t^3 (t^2+1)^3}. \end{aligned}$$ We find that $\gcd(R_{2},R_{3}) = (t^2 + 3)(t^2 + 4)$, so that the two curves are geometrically non-isomorphic in characteristic $11$ except possibly when $t^2$ is $-3$ or $-4$.

Finally we note that in characteristic $11$, when $t^2$ is $-3$ or $-4$ the curve $C(t)$ is geometrically isomorphic to the supersingular curve $y^2 = x^6 + x^4 + 4 x^2 + 7$. Thus, $C(t)$ and $C(-t)$ are geometrically isomorphic for these values of $t$.

It was not critical in our construction that the isogeny $\psi:E\to E'$ have degree $2$. Similar constructions can be made with other kinds of isogenies.

Jacobians with real multiplication by $\sqrt{2}$ {#S-RM}
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In this section we review a construction of Bending [@Bending] that produces every genus-$2$ curve over a given field $K$ whose Jacobian has a $K$-rational endomorphism that is fixed by the Rosati involution and whose square is $2$. We will give a variant of Bending’s construction that produces curves over $K$ with a not-necessarily $K$-rational endomorphism that is fixed by Rosati and whose square is $2$. 