Thirteenth Algorithmic Number Theory Symposium ANTS-XIII University of Wisconsin, Madison July 16 – 20, 2018 Thirteenth Algorithmic Number Theory Symposium (ANTS-XIII) July 16 – 20, 2018

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

Stephanie Chan, Jeroen Hanselman and Wanlin Li

Abstract: In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-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 x Z/8Z. 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.

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