Vol. 2, 2019

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

Stephanie Chan, Jeroen Hanselman and Wanlin Li

Vol. 2 (2019), No. 1, 173–189
Abstract

In 2016, Balakrishnan, Ho, Kaplan, Spicer, Stein and Weigandt produced a database of elliptic curves over $ℚ$ 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 $ℚ$ whose rational torsion subgroup is isomorphic to $ℤ∕2ℤ×ℤ∕8ℤ$. Conditional on GRH and BSD, we compute the rank of $92%$ of the $202,\phantom{\rule{0.3em}{0ex}}461$ curves with parameter height less than $1{0}^{3}$. 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.

Keywords
elliptic curve, rank, Selmer group, Tamagawa number
Mathematical Subject Classification 2010
Primary: 11G05, 11Y40