Frobenius in $W(E_8)$ for $y^2=x^3+x+D(t)$ over $\mathbb F_q(t)$: ranks, Tate--Shafarevich groups and trace statistics

Let $q$ be a power of a prime $p\ge5$ and let $\mathcal F_5(q)$ be the set of curves $E_D\colon y^2=x^3+x+D(t)$ over $\mathbb F_q(t)$ with $D$ monic of degree $5$ and $4+27D^2$ squarefree. Each $E_D$ is the generic fibre of a rational elliptic surface with Mordell--Weil lattice $E_8$, Frobenius acts