$4^i$, can be computed in Magma using the function [ScaledIgusaInvariants]{}.) Let $$\begin{aligned}
R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2}
              { t (t^2+1)^3 } \\
R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3}
              { t^3 (t^2+1)^3} \\
R_{5} &= \frac{J_{10}(t) J_2(-t)^5 - J_{10}(-t) J_2(t)^5}
              {t^3 (t^2+1)^7},\end{aligned}$$ all of which we view as elements of ${{\mathbb Z}}[t]$. If $C(t)$ and $C(-t)$ are isomorphic for a given value of $t$ in a given field $K$, then the polynomials $R_{2}, R_{3},$ and $R_{5}$ must all evaluate to $0$ at this value. But we compute that $$\gcd({\operatorname{resultant}}(R_{2},R_{3}), {\operatorname{resultant}}(R_{2},R_{5}))
  = 2^{980} 3^{48} 11^8,$$ so if the characteristic of $K$ is neither $3$ nor $11$ then the two curves $C(t)$ and $C(-t)$ are geometrically non-isomorphic for every value of $t$ in $K$ with $t(t^2+1)(t^2-1)\neq 0$.

We repeat the above calculation in the ring ${{\mathbb F}}_3[t]$, only now we define $$\begin{aligned}
R_{2} &= \frac{J_4(t) J_2(-t)^2 - J_4(-t) J_2(t)^2}
              {t (t^2-1)^2 (t^2+1)^7} \\
R_{3} &= \frac{J_6(t) J_2(-t)^3 - J_6(-t) J_2(t)^3}
              {t^3 (t^2+1)^9}. \end{aligned}$$ We find that $\gcd(R_{2},R_{3}) = 1$, so the two curves $C(t)$ and $C(-t)$ are geometrically non-isomorphic for every value of $t$ in characteristic $3$, as long as $t(t^2+1)(t^2-1)\neq 0$.

Next we repeat the above calculation in the 