Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion

produced a database of elliptic curves over ordered by height in which they computed the rank, the size of the 2-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over whose rational torsion subgroup is isomorphic to 2= .8= Conditional on GRH and BSD, we compute the rank of 92% of the 202;461 curves with parameter height less than 10 3 . We also compute the size of the 2-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.