[{"abstract":[{"text":"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 \r\nQ whose rational torsion subgroup is isomorphic to Z∕2Z×Z∕8Z. Conditional on GRH and BSD, we compute the rank of 92% of the 202,461 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.","lang":"eng"}],"quality_controlled":"1","_id":"19493","year":"2019","date_updated":"2025-07-10T11:51:49Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1805.10709","open_access":"1"}],"article_type":"original","date_created":"2025-04-05T10:51:07Z","oa":1,"day":"13","oa_version":"Preprint","OA_place":"repository","article_processing_charge":"No","citation":{"apa":"Chan, S., Hanselman, J., &#38; Li, W. (2019). Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. <i>The Open Book Series</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/obs.2019.2.173\">https://doi.org/10.2140/obs.2019.2.173</a>","short":"S. Chan, J. Hanselman, W. Li, The Open Book Series 2 (2019) 173–189.","chicago":"Chan, Stephanie, Jeroen Hanselman, and Wanlin Li. “Ranks, 2-Selmer Groups, and Tamagawa Numbers of Elliptic Curves with ℤ∕2ℤ × ℤ∕8ℤ-Torsion.” <i>The Open Book Series</i>. Mathematical Sciences Publishers, 2019. <a href=\"https://doi.org/10.2140/obs.2019.2.173\">https://doi.org/10.2140/obs.2019.2.173</a>.","ieee":"S. Chan, J. Hanselman, and W. Li, “Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion,” <i>The Open Book Series</i>, vol. 2. Mathematical Sciences Publishers, pp. 173–189, 2019.","mla":"Chan, Stephanie, et al. “Ranks, 2-Selmer Groups, and Tamagawa Numbers of Elliptic Curves with ℤ∕2ℤ × ℤ∕8ℤ-Torsion.” <i>The Open Book Series</i>, vol. 2, Mathematical Sciences Publishers, 2019, pp. 173–89, doi:<a href=\"https://doi.org/10.2140/obs.2019.2.173\">10.2140/obs.2019.2.173</a>.","ista":"Chan S, Hanselman J, Li W. 2019. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. The Open Book Series. 2, 173–189.","ama":"Chan S, Hanselman J, Li W. Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion. <i>The Open Book Series</i>. 2019;2:173-189. doi:<a href=\"https://doi.org/10.2140/obs.2019.2.173\">10.2140/obs.2019.2.173</a>"},"OA_type":"green","publication_identifier":{"issn":["2329-9061"],"eissn":["2329-907X"]},"language":[{"iso":"eng"}],"author":[{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","last_name":"Chan","full_name":"Chan, Yik Tung","first_name":"Yik Tung"},{"last_name":"Hanselman","full_name":"Hanselman, Jeroen","first_name":"Jeroen"},{"last_name":"Li","first_name":"Wanlin","full_name":"Li, Wanlin"}],"publisher":"Mathematical Sciences Publishers","date_published":"2019-02-13T00:00:00Z","title":"Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion","intvolume":"         2","status":"public","publication_status":"published","extern":"1","volume":2,"page":"173-189","publication":"The Open Book Series","doi":"10.2140/obs.2019.2.173","external_id":{"unknown":["1805.10709"]},"type":"journal_article","month":"02","scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}]