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In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt produced a database ofelliptic curves over $\mathbb{Q}$ ordered by height in which they computed therank, the size of the $2$-Selmer group, and other arithmetic invariants. Theyobserved that after a certain point, the average rank seemed to decrease as theheight increased. Here we consider the family of elliptic curves over$\mathbb{Q}$ whose rational torsion subgroup is isomorphic to$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Conditional on GRH andBSD, we compute the rank of $92\%$ of the $202461$ curves with parameter heightless than $10^3$. We also compute the size of the $2$-Selmer group and theTamagawa product, and prove that their averages tend to infinity for thisfamily.

Ranks, $2$-Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$-torsion