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 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 103 . 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
Milestones
Received: 28 February 2018
Revised: 23 September 2018
Accepted: 24 September 2018
Published: 13 February 2019
Authors
Stephanie Chan
Department of Mathematics
University College London
London
United Kingdom
Jeroen Hanselman
Institute of Pure Mathematics
Ulm University Ulm
Germany
Wanlin Li
Department of Mathematics
University of Wisconsin
Madison, WI
United States