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An isogeny between Jacobians of hyperelliptic curves
In the most easy case $q=3$, the curve $X_t$ is bielliptic (the bielliptic involution given by $x\mapsto 1-x$), and the Jacobian of $X_t$ is then $(2,2)$-isogenous to the product
$E_{1,t}\times E_{2, …