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Properties

Label 32.0.72742794030...0625.3
Degree $32$
Signature $[0, 16]$
Discriminant $3^{16}\cdot 5^{24}\cdot 17^{28}$
Root discriminant $69.09$
Ramified primes $3, 5, 17$
Class number $340$ (GRH)
Class group $[2, 170]$ (GRH)
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112550881, -51358169, -39942885, -19509848, 45419760, -32655165, 1253609, 1917717, 16511370, -6671092, 4223756, -5443694, 1201420, -657436, 1310904, -641829, -155159, 175569, 161874, -22352, -10544, -31498, -208, 736, 6444, -417, -811, 81, 78, 8, -9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 9*x^30 + 8*x^29 + 78*x^28 + 81*x^27 - 811*x^26 - 417*x^25 + 6444*x^24 + 736*x^23 - 208*x^22 - 31498*x^21 - 10544*x^20 - 22352*x^19 + 161874*x^18 + 175569*x^17 - 155159*x^16 - 641829*x^15 + 1310904*x^14 - 657436*x^13 + 1201420*x^12 - 5443694*x^11 + 4223756*x^10 - 6671092*x^9 + 16511370*x^8 + 1917717*x^7 + 1253609*x^6 - 32655165*x^5 + 45419760*x^4 - 19509848*x^3 - 39942885*x^2 - 51358169*x + 112550881)
 
gp: K = bnfinit(x^32 - x^31 - 9*x^30 + 8*x^29 + 78*x^28 + 81*x^27 - 811*x^26 - 417*x^25 + 6444*x^24 + 736*x^23 - 208*x^22 - 31498*x^21 - 10544*x^20 - 22352*x^19 + 161874*x^18 + 175569*x^17 - 155159*x^16 - 641829*x^15 + 1310904*x^14 - 657436*x^13 + 1201420*x^12 - 5443694*x^11 + 4223756*x^10 - 6671092*x^9 + 16511370*x^8 + 1917717*x^7 + 1253609*x^6 - 32655165*x^5 + 45419760*x^4 - 19509848*x^3 - 39942885*x^2 - 51358169*x + 112550881, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 9 x^{30} + 8 x^{29} + 78 x^{28} + 81 x^{27} - 811 x^{26} - 417 x^{25} + 6444 x^{24} + 736 x^{23} - 208 x^{22} - 31498 x^{21} - 10544 x^{20} - 22352 x^{19} + 161874 x^{18} + 175569 x^{17} - 155159 x^{16} - 641829 x^{15} + 1310904 x^{14} - 657436 x^{13} + 1201420 x^{12} - 5443694 x^{11} + 4223756 x^{10} - 6671092 x^{9} + 16511370 x^{8} + 1917717 x^{7} + 1253609 x^{6} - 32655165 x^{5} + 45419760 x^{4} - 19509848 x^{3} - 39942885 x^{2} - 51358169 x + 112550881 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(72742794030721358896825924818930661340866148471832275390625=3^{16}\cdot 5^{24}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.09$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(128,·)$, $\chi_{255}(1,·)$, $\chi_{255}(2,·)$, $\chi_{255}(4,·)$, $\chi_{255}(134,·)$, $\chi_{255}(8,·)$, $\chi_{255}(13,·)$, $\chi_{255}(16,·)$, $\chi_{255}(154,·)$, $\chi_{255}(26,·)$, $\chi_{255}(32,·)$, $\chi_{255}(161,·)$, $\chi_{255}(166,·)$, $\chi_{255}(169,·)$, $\chi_{255}(157,·)$, $\chi_{255}(179,·)$, $\chi_{255}(52,·)$, $\chi_{255}(53,·)$, $\chi_{255}(59,·)$, $\chi_{255}(64,·)$, $\chi_{255}(67,·)$, $\chi_{255}(77,·)$, $\chi_{255}(206,·)$, $\chi_{255}(208,·)$, $\chi_{255}(83,·)$, $\chi_{255}(212,·)$, $\chi_{255}(217,·)$, $\chi_{255}(103,·)$, $\chi_{255}(104,·)$, $\chi_{255}(106,·)$, $\chi_{255}(236,·)$, $\chi_{255}(118,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{20} - \frac{1}{12} a^{10} - \frac{1}{2} a^{5} - \frac{5}{12}$, $\frac{1}{12} a^{21} - \frac{1}{12} a^{11} - \frac{1}{2} a^{6} - \frac{5}{12} a$, $\frac{1}{12} a^{22} - \frac{1}{12} a^{12} - \frac{1}{2} a^{7} - \frac{5}{12} a^{2}$, $\frac{1}{12} a^{23} - \frac{1}{12} a^{13} - \frac{1}{2} a^{8} - \frac{5}{12} a^{3}$, $\frac{1}{12} a^{24} - \frac{1}{12} a^{14} - \frac{1}{2} a^{9} - \frac{5}{12} a^{4}$, $\frac{1}{12} a^{25} - \frac{1}{12} a^{15} + \frac{1}{12} a^{5} - \frac{1}{2}$, $\frac{1}{24} a^{26} - \frac{1}{24} a^{23} - \frac{1}{24} a^{22} - \frac{1}{24} a^{21} - \frac{1}{24} a^{20} - \frac{1}{4} a^{19} - \frac{1}{24} a^{16} - \frac{1}{4} a^{15} + \frac{1}{24} a^{13} - \frac{5}{24} a^{12} + \frac{1}{24} a^{11} + \frac{1}{24} a^{10} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{7}{24} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{7}{24} a^{3} + \frac{11}{24} a^{2} - \frac{1}{24} a + \frac{11}{24}$, $\frac{1}{24} a^{27} - \frac{1}{24} a^{24} - \frac{1}{24} a^{23} - \frac{1}{24} a^{22} - \frac{1}{24} a^{21} - \frac{1}{24} a^{17} - \frac{1}{4} a^{16} + \frac{1}{24} a^{14} - \frac{5}{24} a^{13} + \frac{1}{24} a^{12} + \frac{1}{24} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{7}{24} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{7}{24} a^{4} + \frac{11}{24} a^{3} - \frac{1}{24} a^{2} + \frac{11}{24} a - \frac{1}{4}$, $\frac{1}{19174872} a^{28} + \frac{82883}{4793718} a^{27} - \frac{55257}{3195812} a^{26} + \frac{212023}{19174872} a^{25} + \frac{85161}{6391624} a^{24} + \frac{55691}{19174872} a^{23} - \frac{522349}{19174872} a^{22} - \frac{118171}{9587436} a^{21} - \frac{140779}{9587436} a^{20} + \frac{77289}{798953} a^{19} + \frac{2962895}{19174872} a^{18} + \frac{121433}{9587436} a^{17} + \frac{14725}{3195812} a^{16} - \frac{3648355}{19174872} a^{15} - \frac{16311}{71816} a^{14} - \frac{1704539}{19174872} a^{13} - \frac{640727}{19174872} a^{12} - \frac{521057}{2396859} a^{11} - \frac{364802}{2396859} a^{10} + \frac{16714}{798953} a^{9} - \frac{5441753}{19174872} a^{8} + \frac{423043}{9587436} a^{7} + \frac{54249}{1597906} a^{6} - \frac{914615}{19174872} a^{5} - \frac{380951}{6391624} a^{4} + \frac{7150451}{19174872} a^{3} + \frac{54575}{19174872} a^{2} - \frac{274889}{4793718} a - \frac{1579669}{9587436}$, $\frac{1}{13997984410701346072836197052664007649384} a^{29} + \frac{93752035851139668179521868055361}{4665994803567115357612065684221335883128} a^{28} + \frac{198830968592955704431824027977637267151}{13997984410701346072836197052664007649384} a^{27} - \frac{8284740881417968275911225568069955333}{583249350445889419701508210527666985391} a^{26} + \frac{271759201595424392750306025873164185559}{6998992205350673036418098526332003824692} a^{25} + \frac{518554855435583876976071038846887647045}{13997984410701346072836197052664007649384} a^{24} + \frac{25880632324528632361108052458135326019}{1166498700891778839403016421055333970782} a^{23} + \frac{457329430483070027969313479345282888539}{13997984410701346072836197052664007649384} a^{22} - \frac{246125850814168450355492393344739398693}{6998992205350673036418098526332003824692} a^{21} + \frac{416991353141292384909072375293997525785}{13997984410701346072836197052664007649384} a^{20} - \frac{161180413062113697506951149303177686679}{13997984410701346072836197052664007649384} a^{19} + \frac{118685925905680158733490410418381540241}{4665994803567115357612065684221335883128} a^{18} + \frac{2775380751180856347447141085102873201799}{13997984410701346072836197052664007649384} a^{17} - \frac{111251222383724857961465639795651004559}{2332997401783557678806032842110667941564} a^{16} - \frac{982566955300763426404016579644673496053}{6998992205350673036418098526332003824692} a^{15} - \frac{399955854856959006787432895357792487779}{13997984410701346072836197052664007649384} a^{14} + \frac{64576123481064524203920348985491336707}{2332997401783557678806032842110667941564} a^{13} - \frac{161348620697536636972016841950031373435}{13997984410701346072836197052664007649384} a^{12} - \frac{785031820645626415047817560110509142093}{6998992205350673036418098526332003824692} a^{11} - \frac{1274391550971725403809515674421761259901}{13997984410701346072836197052664007649384} a^{10} + \frac{1207086501022846357978035058846535576545}{13997984410701346072836197052664007649384} a^{9} - \frac{492010346454049006492445733768198817257}{4665994803567115357612065684221335883128} a^{8} + \frac{942764928896878162059895533023190586693}{13997984410701346072836197052664007649384} a^{7} + \frac{130279860637446521663939946362662224792}{583249350445889419701508210527666985391} a^{6} - \frac{982316264321107798472448793523753745455}{3499496102675336518209049263166001912346} a^{5} + \frac{4490363102551725293537084563344953676221}{13997984410701346072836197052664007649384} a^{4} + \frac{512874402185656852095594775569768310535}{2332997401783557678806032842110667941564} a^{3} - \frac{6453036746363019801211098734572614350435}{13997984410701346072836197052664007649384} a^{2} - \frac{1213080955234659108917001843605899009841}{3499496102675336518209049263166001912346} a + \frac{57471116379772277014615599335316146189}{135902761268945107503264049054990365528}$, $\frac{1}{2883584788604477291004256592848785575773104} a^{30} + \frac{17}{480597464767412881834042765474797595962184} a^{29} + \frac{11793366042111789443143513014803873}{720896197151119322751064148212196393943276} a^{28} - \frac{9355551024188015509524817373937874716497}{480597464767412881834042765474797595962184} a^{27} - \frac{985462171360050410921041795310760838087}{720896197151119322751064148212196393943276} a^{26} + \frac{25924105291803397304288984073750813460239}{961194929534825763668085530949595191924368} a^{25} + \frac{12040247481155245132907193300845889017633}{720896197151119322751064148212196393943276} a^{24} + \frac{53659403398247302752372403618268125349053}{1441792394302238645502128296424392787886552} a^{23} + \frac{1604513659103594953308716339404059825577}{180224049287779830687766037053049098485819} a^{22} + \frac{10036220302171457299959552081519845392663}{720896197151119322751064148212196393943276} a^{21} - \frac{52546696507714920840832975998347973306773}{1441792394302238645502128296424392787886552} a^{20} + \frac{58985101188858222796559178147701052899525}{480597464767412881834042765474797595962184} a^{19} + \frac{77351338286190706603842201339174238794245}{360448098575559661375532074106098196971638} a^{18} + \frac{19703413015697343627298916805309162150989}{480597464767412881834042765474797595962184} a^{17} - \frac{608846085067582787758654155380923944643}{3834554240165528312505660362830831882677} a^{16} - \frac{77854832430592306843161536838876560163789}{961194929534825763668085530949595191924368} a^{15} - \frac{25092749752053222087597996609920558630279}{720896197151119322751064148212196393943276} a^{14} - \frac{218094546219252960426273373402210048866943}{1441792394302238645502128296424392787886552} a^{13} + \frac{1556955018911620386263117816828220747167}{180224049287779830687766037053049098485819} a^{12} - \frac{38548554642610994263380312328408442448904}{180224049287779830687766037053049098485819} a^{11} + \frac{273661058050790181779048349848014663629509}{1441792394302238645502128296424392787886552} a^{10} - \frac{11938528935180285780964788504204677822703}{480597464767412881834042765474797595962184} a^{9} + \frac{40880512717082394990290161431749782186859}{720896197151119322751064148212196393943276} a^{8} - \frac{57358705938551534113434810399082349255569}{480597464767412881834042765474797595962184} a^{7} + \frac{54580073220901243552512542395520290244882}{180224049287779830687766037053049098485819} a^{6} - \frac{296639854022798673881395484529840765060637}{961194929534825763668085530949595191924368} a^{5} - \frac{140118329765008622153474235544129676435521}{720896197151119322751064148212196393943276} a^{4} + \frac{242948963402779201199443477636662006336307}{1441792394302238645502128296424392787886552} a^{3} - \frac{7613953830485507159462079248124827821093}{15338216960662113250022641451323327530708} a^{2} - \frac{4604011627821318415369777360144623271}{78640361857872730746270769958786559828} a + \frac{31611498417093555246618517013869989463}{90601840845963405002176032703326910352}$, $\frac{1}{297009233226261160973438429063424914304629712} a^{31} - \frac{1}{297009233226261160973438429063424914304629712} a^{30} + \frac{1325}{37126154153282645121679803632928114288078714} a^{29} - \frac{2768131512029003841300687205438304153}{148504616613130580486719214531712457152314856} a^{28} - \frac{45288717437163229770419781164131072153807}{12375384717760881707226601210976038096026238} a^{27} + \frac{419258456481173176495563965548514964582515}{297009233226261160973438429063424914304629712} a^{26} + \frac{4262402268450825752876391523473442946850415}{297009233226261160973438429063424914304629712} a^{25} - \frac{3936386250885602063567724025828880456052551}{148504616613130580486719214531712457152314856} a^{24} - \frac{536994882107132211599464308224881778874655}{148504616613130580486719214531712457152314856} a^{23} - \frac{1767054806224727524707962648901906675264283}{148504616613130580486719214531712457152314856} a^{22} + \frac{535751460523684585669223766137906445175571}{74252308306565290243359607265856228576157428} a^{21} + \frac{2576570133393424951074357375413712535515373}{74252308306565290243359607265856228576157428} a^{20} - \frac{4521452559198431570940974510038825053176629}{18563077076641322560839901816464057144039357} a^{19} + \frac{291385786014355967452147719525839285569711}{3159672693896395329504664138972605471325848} a^{18} + \frac{3908107574227849301930274968508255370406949}{24750769435521763414453202421952076192052476} a^{17} - \frac{19377259640120407768883315523241257877939493}{297009233226261160973438429063424914304629712} a^{16} + \frac{31262094359913453093370899821680058233320779}{297009233226261160973438429063424914304629712} a^{15} + \frac{22665702781108435021076475426951969257171393}{148504616613130580486719214531712457152314856} a^{14} - \frac{34685096264806281611689121890251216104516621}{148504616613130580486719214531712457152314856} a^{13} - \frac{29048408249320856283987408072654311237442185}{148504616613130580486719214531712457152314856} a^{12} + \frac{10262712657185445854515243244249050882776235}{74252308306565290243359607265856228576157428} a^{11} + \frac{10785460780223634282257681857027801566377531}{74252308306565290243359607265856228576157428} a^{10} - \frac{221679209982086783817387505273981425046129}{37126154153282645121679803632928114288078714} a^{9} - \frac{6256008463470799848201444357424290717229019}{148504616613130580486719214531712457152314856} a^{8} + \frac{10319125234964227601354135308126998952612511}{24750769435521763414453202421952076192052476} a^{7} - \frac{46972086727709697472468421842629689403314125}{297009233226261160973438429063424914304629712} a^{6} + \frac{24370696000523129787061950823674635148090631}{297009233226261160973438429063424914304629712} a^{5} - \frac{45098242691252125051376717840129124663342299}{148504616613130580486719214531712457152314856} a^{4} + \frac{12482059331117437930630268481149358894160997}{148504616613130580486719214531712457152314856} a^{3} - \frac{491562252672593499828976243447553264064769}{1441792394302238645502128296424392787886552} a^{2} - \frac{802584104010084266988555640748126754847}{9331989607134230715224131368442671766256} a + \frac{5261567506854777508042770619429370901}{90601840845963405002176032703326910352}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{170}$, which has order $340$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1039151276270944823838889819}{1203068177395698825422879106717911056} a^{31} + \frac{8064756586939837499637951607}{902301133046774119067159330038433292} a^{30} - \frac{12461931746980336310541533773}{1804602266093548238134318660076866584} a^{29} - \frac{17396117517038132423218311941}{225575283261693529766789832509608323} a^{28} + \frac{176695271590170921130774659583}{1804602266093548238134318660076866584} a^{27} + \frac{2826648114224618390026721411377}{3609204532187096476268637320153733168} a^{26} + \frac{884064090625919267016040290817}{1804602266093548238134318660076866584} a^{25} - \frac{5652493634347510293697571404367}{902301133046774119067159330038433292} a^{24} - \frac{397035918821433982972341858691}{300767044348924706355719776679477764} a^{23} + \frac{102107628074613438235010623134491}{1804602266093548238134318660076866584} a^{22} + \frac{17864291312581593544270661191151}{601534088697849412711439553358955528} a^{21} + \frac{12893360517849937112456404853207}{902301133046774119067159330038433292} a^{20} + \frac{48281351652474553164912872076877}{1804602266093548238134318660076866584} a^{19} - \frac{134736818499477954644995922170297}{902301133046774119067159330038433292} a^{18} - \frac{415285987206728729031929575825639}{1804602266093548238134318660076866584} a^{17} + \frac{432539798206142217234929006662553}{3609204532187096476268637320153733168} a^{16} + \frac{1084398263454372182276924432531249}{1804602266093548238134318660076866584} a^{15} - \frac{1119421961297492066533052980854187}{902301133046774119067159330038433292} a^{14} + \frac{88105659355893947987715579454043}{150383522174462353177859888339738882} a^{13} + \frac{33710898182431579384241503523686891}{1804602266093548238134318660076866584} a^{12} + \frac{3014894320847122576524942116712853}{601534088697849412711439553358955528} a^{11} - \frac{782554236794165453889670194146627}{225575283261693529766789832509608323} a^{10} + \frac{111443376791582284385479389873629}{17520410350422798428488530680357928} a^{9} - \frac{13279392231810042610290043726412915}{902301133046774119067159330038433292} a^{8} - \frac{16518891404646627525416381980274681}{1804602266093548238134318660076866584} a^{7} - \frac{6059090492783585251183127268373523}{3609204532187096476268637320153733168} a^{6} + \frac{51020655541480544393107919283882775}{1804602266093548238134318660076866584} a^{5} - \frac{18681534311255358218659077757683805}{451150566523387059533579665019216646} a^{4} + \frac{48588612762459624410084896095961}{2920068391737133071414755113392988} a^{3} + \frac{1264133152472634979278010945352145365}{1804602266093548238134318660076866584} a^{2} + \frac{5410550853363003838818109640999}{113400714242218760054941946151184} a - \frac{2735179778542849480340132202569}{28350178560554690013735486537796} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 324074432504351.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), 4.0.614125.2, \(\Q(\sqrt{5}, \sqrt{17})\), 4.0.614125.1, 4.4.122825.1, 4.4.4913.1, 4.0.36125.1, \(\Q(\zeta_{5})\), 8.0.377149515625.1, 8.8.15085980625.1, 8.0.1305015625.1, 8.8.519334883015625.1, 8.8.519334883015625.2, 8.0.33237432513.1, 8.0.20773395320625.1, 16.0.142241757136172119140625.1, 16.16.269708720716852904093994140625.1, 16.0.431533953146964646550390625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed
17Data not computed