{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T13:02:12Z","timestamp":1773147732060,"version":"3.50.1"},"reference-count":32,"publisher":"Wiley","issue":"A","license":[{"start":{"date-parts":[[2016,8,26]],"date-time":"2016-08-26T00:00:00Z","timestamp":1472169600000},"content-version":"unspecified","delay-in-days":238,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157016000152_inline2\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157016000152_inline3\"\/><jats:tex-math>$\\mathbb{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, ordered by height. We describe databases of elliptic curves over <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157016000152_inline4\"\/><jats:tex-math>$\\mathbb{Q}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, ordered by height, in which we compute ranks and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157016000152_inline5\"\/><jats:tex-math>$2$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.<\/jats:p>","DOI":"10.1112\/s1461157016000152","type":"journal-article","created":{"date-parts":[[2016,8,26]],"date-time":"2016-08-26T11:30:53Z","timestamp":1472211053000},"page":"351-370","source":"Crossref","is-referenced-by-count":13,"title":["Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks"],"prefix":"10.1112","volume":"19","author":[{"given":"Jennifer S.","family":"Balakrishnan","sequence":"first","affiliation":[]},{"given":"Wei","family":"Ho","sequence":"additional","affiliation":[]},{"given":"Nathan","family":"Kaplan","sequence":"additional","affiliation":[]},{"given":"Simon","family":"Spicer","sequence":"additional","affiliation":[]},{"given":"William","family":"Stein","sequence":"additional","affiliation":[]},{"given":"James","family":"Weigandt","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2016,8,26]]},"reference":[{"key":"S1461157016000152_r1","unstructured":"1. J.\u00a0S. Balakrishnan , W. Ho , N. Kaplan , S. Spicer , W. Stein and J. Weigandt , http:\/\/wstein.org\/papers\/2016-height\/."},{"key":"S1461157016000152_r18","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2014.02.033"},{"key":"S1461157016000152_r24","first-page":"209","article-title":"Formules explicites et minorations de conducteurs de vari\u00e9t\u00e9s alg\u00e9briques","volume":"58","author":"Mestre","year":"1986","journal-title":"Compositio Math."},{"key":"S1461157016000152_r17","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.2001.10504442"},{"key":"S1461157016000152_r12","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.1996.0125"},{"key":"S1461157016000152_r13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01232033"},{"key":"S1461157016000152_r6","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2015.181.1.3"},{"key":"S1461157016000152_r14","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1990-15937-3"},{"key":"S1461157016000152_r4","unstructured":"4. M. Bhargava and W. Ho , \u2018On the average sizes of Selmer groups in families of elliptic curves\u2019, Preprint."},{"key":"S1461157016000152_r20","article-title":"Counting elliptic curves with prescribed torsion","author":"Harron","journal-title":"J. reine angew. Math."},{"key":"S1461157016000152_r30","unstructured":"30. The SageMath Developers. Sage Mathematics Software (Version 6.9), 2015. http:\/\/www.sagemath.org."},{"key":"S1461157016000152_r28","unstructured":"28. S.\u00a0V. Spicer , The zeros of elliptic curve $L$ -functions: analytic algorithms with explicit time complexity, PhD Thesis, University of Washington, 2015."},{"key":"S1461157016000152_r25","unstructured":"25. J. Park , B. Poonen , J. Voight and M. Wood , \u2018Heuristics for boundedness of ranks of elliptic curves over $\\mathbb{Q}$ \u2019, Preprint, 2016, arXiv:1602.01431."},{"key":"S1461157016000152_r27","unstructured":"27. SageMath, Inc. SageMathCloud, 2016. https:\/\/cloud.sagemath.com."},{"key":"S1461157016000152_r11","first-page":"135","volume-title":"ANTS X\u2014Proceedings of the Tenth Algorithmic Number Theory Symposium","author":"Bober","year":"2013"},{"key":"S1461157016000152_r31","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.2008.10129019"},{"key":"S1461157016000152_r32","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-05-00503-5"},{"key":"S1461157016000152_r3","unstructured":"3. M. Bennett and A. Rechnitzer , \u2018Computing elliptic curves over $\\mathbb{Q}$ : bad reduction at one prime\u2019, Preprint."},{"key":"S1461157016000152_r21","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-04-12235-3"},{"key":"S1461157016000152_r26","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-2011-00710-8"},{"key":"S1461157016000152_r22","doi-asserted-by":"publisher","DOI":"10.1090\/coll\/053"},{"key":"S1461157016000152_r10","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0097580"},{"key":"S1461157016000152_r19","first-page":"108","volume-title":"Number theory, Carbondale","author":"Goldfeld","year":"1979"},{"key":"S1461157016000152_r7","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2015.181.2.4"},{"key":"S1461157016000152_r9","unstructured":"9. M. Bhargava and A. Shankar , \u2018The average size of the $5$ -Selmer group of elliptic curves is $6$ , and the average rank is less than $1$ \u2019, Preprint, 2013, arXiv:1312.7859."},{"key":"S1461157016000152_r2","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-07-01138-X"},{"key":"S1461157016000152_r29","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-45455-1_22"},{"key":"S1461157016000152_r16","unstructured":"16. J. Cremona , ecdata: 2016-02-07, http:\/\/dx.doi.org\/10.5281\/zenodo.45657, February 2016."},{"key":"S1461157016000152_r23","volume-title":"Random matrices, Frobenius eigenvalues, and monodromy","author":"Katz","year":"1999"},{"key":"S1461157016000152_r5","doi-asserted-by":"publisher","DOI":"10.4310\/CJM.2015.v3.n3.a1"},{"key":"S1461157016000152_r8","unstructured":"8. M. Bhargava and A. Shankar , \u2018The average number of elements in the $4$ -Selmer groups of elliptic curves is $7$ \u2019, Preprint, 2013, arXiv:1312.7333."},{"key":"S1461157016000152_r15","volume-title":"Algorithms for modular elliptic curves","author":"Cremona","year":"1997"}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157016000152","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T20:53:54Z","timestamp":1555707234000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157016000152\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":32,"journal-issue":{"issue":"A","published-print":{"date-parts":[[2016]]}},"alternative-id":["S1461157016000152"],"URL":"https:\/\/doi.org\/10.1112\/s1461157016000152","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]}}}