{"id":"https://openalex.org/W7225004121","doi":"https://doi.org/10.5281/zenodo.23830820","title":"Mazur Torsion Groups and Crystallographic Classification Correspondence \u2014 E8 Intelligence Research","display_name":"Mazur Torsion Groups and Crystallographic Classification Correspondence \u2014 E8 Intelligence Research","publication_year":2026,"publication_date":"2026-10-31","ids":{"openalex":"https://openalex.org/W7225004121","doi":"https://doi.org/10.5281/zenodo.23830820"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23830820","doi":"https://doi.org/10.5281/zenodo.23830820","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23830820","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23830820","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158599967","display_name":"Andrew Stewart Caldin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Caldin, Andrew Stewart","raw_affiliation_strings":["Independent Researcher, United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher, United Kingdom","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5158599967"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.8949999809265137,"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":0.8949999809265137,"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":0.09120000153779984,"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"}},{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.002099999925121665,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/rational-elliptic-curves","display_name":"rational elliptic curves","score":0.949999988079071},{"id":"https://openalex.org/keywords/crystallographic-point-groups","display_name":"crystallographic point groups","score":0.9160000085830688},{"id":"https://openalex.org/keywords/bravais-lattices","display_name":"Bravais lattices","score":0.9940000176429749},{"id":"https://openalex.org/keywords/modular-curves","display_name":"modular curves","score":0.9039999842643738},{"id":"https://openalex.org/keywords/supersingular-primes","display_name":"supersingular primes","score":0.9340000152587891},{"id":"https://openalex.org/keywords/monster-group","display_name":"Monster group","score":0.9729999899864197},{"id":"https://openalex.org/keywords/e8","display_name":"E8","score":0.765999972820282}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7875999808311462},{"id":"https://openalex.org/C77461463","wikidata":"https://www.wikidata.org/wiki/Q7827201","display_name":"Torsion (gastropod)","level":2,"score":0.7059999704360962},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6226000189781189},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4657999873161316},{"id":"https://openalex.org/C57796230","wikidata":"https://www.wikidata.org/wiki/Q899720","display_name":"Point group","level":2,"score":0.4180000126361847},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.4056999981403351},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38119998574256897},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.3725000023841858},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.34360000491142273},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3273000121116638},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.32269999384880066},{"id":"https://openalex.org/C59766168","wikidata":"https://www.wikidata.org/wiki/Q3001220","display_name":"Modular curve","level":3,"score":0.31119999289512634},{"id":"https://openalex.org/C14669601","wikidata":"https://www.wikidata.org/wiki/Q2293759","display_name":"Torsion subgroup","level":4,"score":0.3028999865055084},{"id":"https://openalex.org/C81651864","wikidata":"https://www.wikidata.org/wiki/Q874429","display_name":"Group theory","level":2,"score":0.290800005197525},{"id":"https://openalex.org/C156330621","wikidata":"https://www.wikidata.org/wiki/Q3393476","display_name":"Rational point","level":3,"score":0.26660001277923584},{"id":"https://openalex.org/C20725272","wikidata":"https://www.wikidata.org/wiki/Q245462","display_name":"Cyclic group","level":3,"score":0.2535000145435333}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23830820","doi":"https://doi.org/10.5281/zenodo.23830820","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23830820","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23830821","doi":"https://doi.org/10.5281/zenodo.23830821","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23830821","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23830820","doi":"https://doi.org/10.5281/zenodo.23830820","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23830820","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7224996331","https://openalex.org/W2094731488","https://openalex.org/W3087500227","https://openalex.org/W4416955039","https://openalex.org/W7223558003","https://openalex.org/W3144872786","https://openalex.org/W2002774326","https://openalex.org/W7115003719","https://openalex.org/W4400025410","https://openalex.org/W3120485641"],"abstract_inverted_index":{"FINDING:":[0],"Mazur's":[1,37],"torsion":[2,8,58,145],"theorem":[3],"classifies":[4],"the":[5,14,18,137,162],"15":[6,16,57,126,130,144,148,152],"possible":[7],"subgroups":[9],"of":[10,161],"rational":[11,90,106],"elliptic":[12],"curves;":[13],"count":[15],"and":[17],"group":[19,26],"structures":[20],"exhibit":[21],"correspondence":[22],"with":[23],"crystallographic":[24,29],"point":[25,30],"classification":[27],"(32":[28],"groups,":[31],"14":[32],"Bravais":[33],"lattices).":[34],"MATH:":[35],"-":[36,56,66,78,84,108,128,143],"theorem:":[38],"For":[39],"E/Q,":[40],"E(Q)_tors":[41],"\u2245":[42],"Z/nZ":[43],"for":[44,52],"n":[45,53],"\u2208":[46,54,123],"{1,2,3,4,5,6,7,8,9,10,12}":[47,69],"or":[48,96],"Z/2Z":[49],"\u00d7":[50],"Z/2nZ":[51],"{1,2,3,4}":[55],"groups":[59,146],"total:":[60],"11":[61],"cyclic":[62],"+":[63],"4":[64],"non-cyclic":[65],"Cyclic":[67],"orders:":[68],"\u2014":[70,125],"notably":[71],"excludes":[72],"11,":[73],"13,":[74],"14,":[75],"15,":[76],"16...":[77],"Non-cyclic:":[79],"Z/2\u00d7Z/2,":[80],"Z/2\u00d7Z/4,":[81],"Z/2\u00d7Z/6,":[82],"Z/2\u00d7Z/8":[83],"Modular":[85],"curve":[86],"connection:":[87],"X\u2081(N)":[88],"has":[89,118],"points":[91,107],"iff":[92],"N":[93,97],"\u2264":[94],"10":[95],"=":[98],"12":[99],"(Mazur);":[100],"X\u2081(11),":[101],"X\u2081(13)...":[102],"have":[103],"no":[104],"non-cuspidal":[105],"Ogg's":[109,133],"conjecture":[110],"(now":[111],"theorem):":[112],"primes":[113,127,131,141,150],"p":[114,122],"such":[115],"that":[116],"X\u2080\u207a(p)":[117],"genus":[119],"0":[120],"are":[121],"{2,3,5,7,11,13,17,19,23,29,31,41,47,59,71}":[124],"The":[129],"in":[132],"list":[134],"connect":[135],"to":[136],"Monster":[138],"group's":[139],"supersingular":[140,149],"CONNECTION:":[142],"\u2194":[147,151],"Author:":[153],"Andrew":[154],"Stewart":[155],"Caldin,":[156],"Independent":[157],"Researcher,":[158],"UK.":[159],"Part":[160],"E8":[163],"Intelligence":[164],"Research":[165],"series.":[166],"Platform:":[167],"e8intelligence.com":[168]},"counts_by_year":[],"updated_date":"2026-11-02T09:45:03.842371","created_date":"2026-11-01T00:00:00"}