{"id":"https://openalex.org/W2231413792","doi":"https://doi.org/10.1090/mcom/3163","title":"Higher descents on an elliptic curve with a rational 2-torsion point","display_name":"Higher descents on an elliptic curve with a rational 2-torsion point","publication_year":2016,"publication_date":"2016-04-14","ids":{"openalex":"https://openalex.org/W2231413792","doi":"https://doi.org/10.1090/mcom/3163","mag":"2231413792"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/3163","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3163","pdf_url":"https://www.ams.org/mcom/2017-86-307/S0025-5718-2016-03163-4/S0025-5718-2016-03163-4.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":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2017-86-307/S0025-5718-2016-03163-4/S0025-5718-2016-03163-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020020594","display_name":"Tom Fisher","orcid":"https://orcid.org/0000-0003-2565-8805"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Tom Fisher","raw_affiliation_strings":["University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom","University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB,United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom","institution_ids":["https://openalex.org/I241749"]},{"raw_affiliation_string":"University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB,United Kingdom","institution_ids":["https://openalex.org/I241749"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5020020594"],"corresponding_institution_ids":["https://openalex.org/I241749"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.03864455,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"86","issue":"307","first_page":"2493","last_page":"2518"},"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/T11693","display_name":"Cryptography and Residue Arithmetic","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9889000058174133,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/elliptic-curve","display_name":"Elliptic curve","score":0.7205641269683838},{"id":"https://openalex.org/keywords/discriminant","display_name":"Discriminant","score":0.6540247201919556},{"id":"https://openalex.org/keywords/torsion","display_name":"Torsion (gastropod)","score":0.6389719247817993},{"id":"https://openalex.org/keywords/descent","display_name":"Descent (aeronautics)","score":0.5291514992713928},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.49064508080482483},{"id":"https://openalex.org/keywords/torsion-subgroup","display_name":"Torsion subgroup","score":0.47291335463523865},{"id":"https://openalex.org/keywords/supersingular-elliptic-curve","display_name":"Supersingular elliptic curve","score":0.4561026692390442},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4231991469860077},{"id":"https://openalex.org/keywords/schoofs-algorithm","display_name":"Schoof's algorithm","score":0.4192991554737091},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.37277787923812866},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3273956775665283},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.29445070028305054},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.185476154088974},{"id":"https://openalex.org/keywords/quarter-period","display_name":"Quarter period","score":0.16788065433502197},{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.11791530251502991},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.08148336410522461}],"concepts":[{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.7205641269683838},{"id":"https://openalex.org/C78397625","wikidata":"https://www.wikidata.org/wiki/Q192487","display_name":"Discriminant","level":2,"score":0.6540247201919556},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.6389719247817993},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.5291514992713928},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49064508080482483},{"id":"https://openalex.org/C14669601","wikidata":"https://www.wikidata.org/wiki/Q2293759","display_name":"Torsion subgroup","level":4,"score":0.47291335463523865},{"id":"https://openalex.org/C157567686","wikidata":"https://www.wikidata.org/wiki/Q7644139","display_name":"Supersingular elliptic curve","level":3,"score":0.4561026692390442},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4231991469860077},{"id":"https://openalex.org/C121444067","wikidata":"https://www.wikidata.org/wiki/Q2835817","display_name":"Schoof's algorithm","level":4,"score":0.4192991554737091},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.37277787923812866},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3273956775665283},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.29445070028305054},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.185476154088974},{"id":"https://openalex.org/C197875053","wikidata":"https://www.wikidata.org/wiki/Q7269266","display_name":"Quarter period","level":3,"score":0.16788065433502197},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.11791530251502991},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.08148336410522461},{"id":"https://openalex.org/C81008192","wikidata":"https://www.wikidata.org/wiki/Q1017231","display_name":"Elementary abelian group","level":3,"score":0.0},{"id":"https://openalex.org/C153294291","wikidata":"https://www.wikidata.org/wiki/Q25261","display_name":"Meteorology","level":1,"score":0.0},{"id":"https://openalex.org/C141071460","wikidata":"https://www.wikidata.org/wiki/Q40821","display_name":"Surgery","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1090/mcom/3163","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3163","pdf_url":"https://www.ams.org/mcom/2017-86-307/S0025-5718-2016-03163-4/S0025-5718-2016-03163-4.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:oai:arXiv.org:1509.03234","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.03234","pdf_url":"https://arxiv.org/pdf/1509.03234","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2231413792","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1509.03234.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1509.03234","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1509.03234","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.1090/mcom/3163","is_oa":true,"landing_page_url":"https://doi.org/10.1090/mcom/3163","pdf_url":"https://www.ams.org/mcom/2017-86-307/S0025-5718-2016-03163-4/S0025-5718-2016-03163-4.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":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.6700000166893005}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2231413792.pdf","grobid_xml":"https://content.openalex.org/works/W2231413792.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1511533198","https://openalex.org/W1531996733","https://openalex.org/W1976677460","https://openalex.org/W2000568564","https://openalex.org/W2050932155","https://openalex.org/W2118942434"],"related_works":["https://openalex.org/W2963788355","https://openalex.org/W2500411863","https://openalex.org/W2069924527","https://openalex.org/W2054037211","https://openalex.org/W2050501101","https://openalex.org/W2009528366","https://openalex.org/W2951364177","https://openalex.org/W2964174633","https://openalex.org/W2027809387","https://openalex.org/W2951420852","https://openalex.org/W1511292221","https://openalex.org/W1617714430","https://openalex.org/W1976120821","https://openalex.org/W2042773224","https://openalex.org/W22277317","https://openalex.org/W2002739167","https://openalex.org/W2005713079","https://openalex.org/W1549508854","https://openalex.org/W2110244809","https://openalex.org/W2029148888"],"abstract_inverted_index":{"Let":[0],"<inline-formula":[1,22,39,85,114,167,183,200,216,234,253],"content-type=\"math/mathml\">":[2,23,40,86,115,168,184,201,217,235,254],"<mml:math":[3,24,41,87,116,169,185,202,218,236,255],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[4,25,42,88,117,170,186,203,219,237,256],"alttext=\"upper":[5,26,43,118,171,204,257],"E\">":[6,44,172,205],"<mml:semantics>":[7,28,45,90,124,173,188,206,221,239,263],"<mml:mi>E</mml:mi>":[8,46,174,207,265],"<mml:annotation":[9,30,47,92,134,175,190,208,223,241,276],"encoding=\"application/x-tex\">E</mml:annotation>":[10,48,176,209],"</mml:semantics>":[11,32,49,94,137,177,192,210,225,243,279],"</mml:math>":[12,33,50,95,138,178,193,211,226,244,280],"</inline-formula>":[13,34,51,96,139,179,194,212,227,245,281],"be":[14,53,164],"an":[15],"elliptic":[16,145,251],"curve":[17],"over":[18],"a":[19,181,231],"number":[20],"field":[21],"K\">":[27],"<mml:mi>K</mml:mi>":[29,126],"encoding=\"application/x-tex\">K</mml:annotation>":[31],".":[35],"Descent":[36],"calculations":[37],"on":[38],"can":[52,163],"used":[54],"to":[55,67],"find":[56],"upper":[57,122,261],"bounds":[58],"for":[59,76,144,233,250],"the":[60,63,74,100,160],"rank":[61],"of":[62,78,84,103,154],"Mordell-Weil":[64],"group,":[65],"and":[66,106,141,156],"compute":[68],"covering":[69],"curves":[70,146,252],"that":[71,162,247],"assist":[72],"in":[73,99,111],"search":[75],"generators":[77],"this":[79],"group.":[80],"The":[81],"general":[82],"method":[83,232],"alttext=\"4\">":[89],"<mml:mn>4</mml:mn>":[91],"encoding=\"application/x-tex\">4</mml:annotation>":[93],"-descent,":[97],"developed":[98],"PhD":[101],"theses":[102],"Siksek,":[104],"Womack":[105],"Stamminger,":[107],"has":[108,180,213],"been":[109],"implemented":[110],"Magma":[112],"(when":[113],"K":[119],"equals":[120],"double-struck":[121,260],"Q\">":[123,262],"<mml:mrow>":[125,264],"<mml:mo>=</mml:mo>":[127],"<mml:mrow":[128,266,270],"class=\"MJX-TeXAtom-ORD\">":[129,267,271],"<mml:mi":[130,272],"mathvariant=\"double-struck\">Q</mml:mi>":[131,273],"</mml:mrow>":[132,133,269,274,275],"encoding=\"application/x-tex\">K=\\mathbb":[135],"{Q}</mml:annotation>":[136,278],")":[140],"works":[142],"well":[143],"with":[147,282],"sufficiently":[148],"small":[149],"discriminant.":[150,284],"By":[151],"extending":[152],"work":[153],"Bremner":[155],"Cassels,":[157],"we":[158,229],"describe":[159,230],"improvements":[161],"made":[165],"when":[166,199],"rational":[182,215],"alttext=\"2\">":[187,220],"<mml:mn>2</mml:mn>":[189,222],"encoding=\"application/x-tex\">2</mml:annotation>":[191,224],"-torsion":[195],"point.":[196],"In":[197],"particular,":[198],"full":[214],"-torsion,":[228],"alttext=\"8\">":[238],"<mml:mn>8</mml:mn>":[240],"encoding=\"application/x-tex\">8</mml:annotation>":[242],"-descent":[246],"is":[248],"practical":[249],"E":[258],"slash":[259],"<mml:mo>/</mml:mo>":[268],"encoding=\"application/x-tex\">E/\\mathbb":[277],"large":[283]},"counts_by_year":[{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}