{"id":"https://openalex.org/W2073214093","doi":"https://doi.org/10.1090/s0025-5718-01-01320-5","title":"Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves","display_name":"Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves","publication_year":2001,"publication_date":"2001-05-11","ids":{"openalex":"https://openalex.org/W2073214093","doi":"https://doi.org/10.1090/s0025-5718-01-01320-5","mag":"2073214093"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-01-01320-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-01-01320-5","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-01-01320-5/S0025-5718-01-01320-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-01-01320-5/S0025-5718-01-01320-5.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041729023","display_name":"E. V. Flynn","orcid":"https://orcid.org/0000-0002-3340-8868"},"institutions":[{"id":"https://openalex.org/I146655781","display_name":"University of Liverpool","ror":"https://ror.org/04xs57h96","country_code":"GB","type":"education","lineage":["https://openalex.org/I146655781"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"E. Flynn","raw_affiliation_strings":["Department of Mathematical Sciences, University of Liverpool, P.O.Box 147, Liverpool L69 3BX, England","University of Liverpool England#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Liverpool, P.O.Box 147, Liverpool L69 3BX, England","institution_ids":["https://openalex.org/I146655781"]},{"raw_affiliation_string":"University of Liverpool England#TAB#","institution_ids":["https://openalex.org/I146655781"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042757399","display_name":"Franck Lepr\u00e9vost","orcid":"https://orcid.org/0000-0001-8808-2730"},"institutions":[{"id":"https://openalex.org/I16269868","display_name":"Santa Clara University","ror":"https://ror.org/03ypqe447","country_code":"US","type":"education","lineage":["https://openalex.org/I16269868"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Franck Lepr\u00e9vost","raw_affiliation_strings":["Santa Clara University, USA"],"affiliations":[{"raw_affiliation_string":"Santa Clara University, USA","institution_ids":["https://openalex.org/I16269868"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071293587","display_name":"Edward F. Schaefer","orcid":"https://orcid.org/0000-0002-7228-0932"},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]},{"id":"https://openalex.org/I16269868","display_name":"Santa Clara University","ror":"https://ror.org/03ypqe447","country_code":"US","type":"education","lineage":["https://openalex.org/I16269868"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Edward Schaefer","raw_affiliation_strings":["Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053","University of Berkeley - USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053","institution_ids":["https://openalex.org/I16269868"]},{"raw_affiliation_string":"University of Berkeley - USA","institution_ids":["https://openalex.org/I95457486"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071010301","display_name":"William Stein","orcid":null},"institutions":[{"id":"https://openalex.org/I44260953","display_name":"Heinrich Heine University D\u00fcsseldorf","ror":"https://ror.org/024z2rq82","country_code":"DE","type":"education","lineage":["https://openalex.org/I44260953"]},{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["DE","US"],"is_corresponding":false,"raw_author_name":"William Stein","raw_affiliation_strings":["Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138","Universit\u00e4t D\u00fcsseldorf, Germany"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Universit\u00e4t D\u00fcsseldorf, Germany","institution_ids":["https://openalex.org/I44260953"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069201482","display_name":"Michael Stoll","orcid":"https://orcid.org/0000-0001-5868-2962"},"institutions":[{"id":"https://openalex.org/I44260953","display_name":"Heinrich Heine University D\u00fcsseldorf","ror":"https://ror.org/024z2rq82","country_code":"DE","type":"education","lineage":["https://openalex.org/I44260953"]},{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]}],"countries":["DE","US"],"is_corresponding":false,"raw_author_name":"Michael Stoll","raw_affiliation_strings":["Mathematisches Institut der Heinrich-Heine-Universit\u00c3\u00a4t, Universit\u00c3\u00a4tsstr. 1, 40225 D\u00c3\u00bcsseldorf, Germany","University of Southern , California , USA"],"affiliations":[{"raw_affiliation_string":"Mathematisches Institut der Heinrich-Heine-Universit\u00c3\u00a4t, Universit\u00c3\u00a4tsstr. 1, 40225 D\u00c3\u00bcsseldorf, Germany","institution_ids":["https://openalex.org/I44260953"]},{"raw_affiliation_string":"University of Southern , California , USA","institution_ids":["https://openalex.org/I1174212"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5042822250","display_name":"Joseph L. Wetherell","orcid":null},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Joseph Wetherell","raw_affiliation_strings":["Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, California 90089-1113"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Southern California, 1042 W. 36th Place, Los Angeles, California 90089-1113","institution_ids":["https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5041729023"],"corresponding_institution_ids":["https://openalex.org/I146655781"],"apc_list":null,"apc_paid":null,"fwci":4.8204,"has_fulltext":true,"cited_by_count":59,"citation_normalized_percentile":{"value":0.95271961,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"70","issue":"236","first_page":"1675","last_page":"1697"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.989300012588501,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.942799985408783,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8602609634399414},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.6812589168548584},{"id":"https://openalex.org/keywords/genus","display_name":"Genus","score":0.648131251335144},{"id":"https://openalex.org/keywords/modular-form","display_name":"Modular form","score":0.54266756772995},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.5057394504547119},{"id":"https://openalex.org/keywords/modular-curve","display_name":"Modular curve","score":0.4833793044090271},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4822464883327484},{"id":"https://openalex.org/keywords/torsion","display_name":"Torsion (gastropod)","score":0.44530245661735535},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.44204768538475037},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.39882686734199524},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3812231719493866},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.2240559458732605},{"id":"https://openalex.org/keywords/zoology","display_name":"Zoology","score":0.09417706727981567},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.054904431104660034}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8602609634399414},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.6812589168548584},{"id":"https://openalex.org/C157369684","wikidata":"https://www.wikidata.org/wiki/Q34740","display_name":"Genus","level":2,"score":0.648131251335144},{"id":"https://openalex.org/C75764964","wikidata":"https://www.wikidata.org/wiki/Q870797","display_name":"Modular form","level":2,"score":0.54266756772995},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.5057394504547119},{"id":"https://openalex.org/C59766168","wikidata":"https://www.wikidata.org/wiki/Q3001220","display_name":"Modular curve","level":3,"score":0.4833793044090271},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4822464883327484},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.44530245661735535},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.44204768538475037},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39882686734199524},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3812231719493866},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2240559458732605},{"id":"https://openalex.org/C90856448","wikidata":"https://www.wikidata.org/wiki/Q431","display_name":"Zoology","level":1,"score":0.09417706727981567},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.054904431104660034},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0}],"mesh":[],"locations_count":8,"locations":[{"id":"doi:10.1090/s0025-5718-01-01320-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-01-01320-5","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-01-01320-5/S0025-5718-01-01320-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:uuid:e23f4731-cd7f-4e6b-807d-65e7fc6a982b","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"http://eprints.maths.ox.ac.uk/259","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.105.9320","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.105.9320","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/2001-70-236/S0025-5718-01-01320-5/S0025-5718-01-01320-5.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.24.2676","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.2676","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://math.usc.edu/~wetherel/papers/Bsd1Final.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.506.2489","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.506.2489","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://modular.math.washington.edu/papers/evidence/evidence3.pdf","raw_type":"text"},{"id":"pmh:oai:ora.ox.ac.uk:uuid:0c6d7e1f-5a0c-432f-85e3-ac20f39a6c62","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:0c6d7e1f-5a0c-432f-85e3-ac20f39a6c62","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symplectic Elements at Oxford","raw_type":"Journal article"},{"id":"pmh:oai:ora.ox.ac.uk:uuid:e23f4731-cd7f-4e6b-807d-65e7fc6a982b","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:e23f4731-cd7f-4e6b-807d-65e7fc6a982b","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://eprints.maths.ox.ac.uk/259","raw_type":"Journal article"},{"id":"pmh:oai:orbilu.uni.lu:10993/12604","is_oa":false,"landing_page_url":"https://orbilu.uni.lu/handle/10993/12604","pdf_url":null,"source":{"id":"https://openalex.org/S4306401815","display_name":"Open Repository and Bibliography (University of Luxembourg)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I186903577","host_organization_name":"University of Luxembourg","host_organization_lineage":["https://openalex.org/I186903577"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics of Computation, 70(236), 1675-1697. American Mathematical Society (2001).","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-01-01320-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-01-01320-5","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-01-01320-5/S0025-5718-01-01320-5.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Life in Land","score":0.41999998688697815,"id":"https://metadata.un.org/sdg/15"}],"awards":[{"id":"https://openalex.org/G5422210062","display_name":"Mathematical Sciences Postdoctoral Research Fellowships","funder_award_id":"9705959","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320311089","display_name":"National Security Agency","ror":"https://ror.org/0047bvr32"},{"id":"https://openalex.org/F4320319997","display_name":"Nuffield Foundation","ror":"https://ror.org/0281jqk77"},{"id":"https://openalex.org/F4320324094","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2073214093.pdf","grobid_xml":"https://content.openalex.org/works/W2073214093.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W192016536","https://openalex.org/W1484733457","https://openalex.org/W1528669012","https://openalex.org/W1530016667","https://openalex.org/W1533469128","https://openalex.org/W1537838966","https://openalex.org/W1559532354","https://openalex.org/W1559870539","https://openalex.org/W1584131488","https://openalex.org/W1586485728","https://openalex.org/W1868013030","https://openalex.org/W1969031248","https://openalex.org/W1976994099","https://openalex.org/W1993772258","https://openalex.org/W2002926215","https://openalex.org/W2003929417","https://openalex.org/W2004401181","https://openalex.org/W2006068756","https://openalex.org/W2006077852","https://openalex.org/W2011844852","https://openalex.org/W2025553790","https://openalex.org/W2027134735","https://openalex.org/W2053499741","https://openalex.org/W2061962229","https://openalex.org/W2086576039","https://openalex.org/W2088875371","https://openalex.org/W2099085851","https://openalex.org/W2118942434","https://openalex.org/W2168648858","https://openalex.org/W2277535877","https://openalex.org/W2604330053","https://openalex.org/W2745753443","https://openalex.org/W2774686533","https://openalex.org/W3104093056","https://openalex.org/W3135300355","https://openalex.org/W3144374870","https://openalex.org/W4229896862","https://openalex.org/W4232950085","https://openalex.org/W4234505739","https://openalex.org/W4248269519","https://openalex.org/W4255866901","https://openalex.org/W4285719527","https://openalex.org/W6633402805"],"related_works":["https://openalex.org/W1537136047","https://openalex.org/W1996244739","https://openalex.org/W123909518","https://openalex.org/W51342543","https://openalex.org/W4286912068","https://openalex.org/W1984378493","https://openalex.org/W3169715737","https://openalex.org/W375329393","https://openalex.org/W2025736736","https://openalex.org/W1867247190"],"abstract_inverted_index":{"This":[0],"paper":[1],"provides":[2],"empirical":[3],"evidence":[4],"for":[5,11,58],"the":[6,31,40,43,52,59,66,88,91,94],"Birch":[7],"and":[8,56],"Swinnerton-Dyer":[9],"conjectures":[10,22],"modular":[12],"Jacobians":[13],"of":[14,20,35,42,78,84,90,93],"genus":[15],"2":[16,85],"curves.":[17],"The":[18],"second":[19],"these":[21,36],"relates":[23],"six":[24,37],"quantities":[25,38,55],"associated":[26],"to":[27,47,71],"a":[28,76],"Jacobian":[29],"over":[30],"rational":[32],"numbers.":[33],"One":[34],"is":[39,68,75],"size":[41,89],"Shafarevich-Tate":[44,95],"group.":[45],"Unable":[46],"compute":[48],"that,":[49],"we":[50,98],"computed":[51],"five":[53],"other":[54],"solved":[57],"last":[60],"one.":[61],"In":[62,80],"all":[63],"32":[64],"cases,":[65],"result":[67],"very":[69],"close":[70],"an":[72],"integer":[73],"that":[74],"power":[77,83],"2.":[79],"addition,":[81],"this":[82],"agrees":[86],"with":[87],"2-torsion":[92],"group,":[96],"which":[97],"could":[99],"compute.":[100]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}