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Properties

Label 32.0.72313627753...2736.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 3^{16}\cdot 13^{24}$
Root discriminant $79.77$
Ramified primes $2, 3, 13$
Class number $1800$ (GRH)
Class group $[15, 120]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![796849, -6403360, 27524944, -84510352, 211310100, -190278160, -76133580, 101306868, 162618454, -119345528, -154416048, 138776892, 48726322, -68988532, 372598, 12413912, -2969721, 1288480, 508272, -1131864, 198866, 247944, -106514, -43172, 29657, 7696, -5254, -1028, 531, 88, -30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 30*x^30 + 88*x^29 + 531*x^28 - 1028*x^27 - 5254*x^26 + 7696*x^25 + 29657*x^24 - 43172*x^23 - 106514*x^22 + 247944*x^21 + 198866*x^20 - 1131864*x^19 + 508272*x^18 + 1288480*x^17 - 2969721*x^16 + 12413912*x^15 + 372598*x^14 - 68988532*x^13 + 48726322*x^12 + 138776892*x^11 - 154416048*x^10 - 119345528*x^9 + 162618454*x^8 + 101306868*x^7 - 76133580*x^6 - 190278160*x^5 + 211310100*x^4 - 84510352*x^3 + 27524944*x^2 - 6403360*x + 796849)
 
gp: K = bnfinit(x^32 - 4*x^31 - 30*x^30 + 88*x^29 + 531*x^28 - 1028*x^27 - 5254*x^26 + 7696*x^25 + 29657*x^24 - 43172*x^23 - 106514*x^22 + 247944*x^21 + 198866*x^20 - 1131864*x^19 + 508272*x^18 + 1288480*x^17 - 2969721*x^16 + 12413912*x^15 + 372598*x^14 - 68988532*x^13 + 48726322*x^12 + 138776892*x^11 - 154416048*x^10 - 119345528*x^9 + 162618454*x^8 + 101306868*x^7 - 76133580*x^6 - 190278160*x^5 + 211310100*x^4 - 84510352*x^3 + 27524944*x^2 - 6403360*x + 796849, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 30 x^{30} + 88 x^{29} + 531 x^{28} - 1028 x^{27} - 5254 x^{26} + 7696 x^{25} + 29657 x^{24} - 43172 x^{23} - 106514 x^{22} + 247944 x^{21} + 198866 x^{20} - 1131864 x^{19} + 508272 x^{18} + 1288480 x^{17} - 2969721 x^{16} + 12413912 x^{15} + 372598 x^{14} - 68988532 x^{13} + 48726322 x^{12} + 138776892 x^{11} - 154416048 x^{10} - 119345528 x^{9} + 162618454 x^{8} + 101306868 x^{7} - 76133580 x^{6} - 190278160 x^{5} + 211310100 x^{4} - 84510352 x^{3} + 27524944 x^{2} - 6403360 x + 796849 \)

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:  \(7231362775399344187879888625220455562201364498198121976692736=2^{88}\cdot 3^{16}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.77$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(624=2^{4}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{624}(1,·)$, $\chi_{624}(5,·)$, $\chi_{624}(385,·)$, $\chi_{624}(521,·)$, $\chi_{624}(209,·)$, $\chi_{624}(493,·)$, $\chi_{624}(25,·)$, $\chi_{624}(281,·)$, $\chi_{624}(157,·)$, $\chi_{624}(389,·)$, $\chi_{624}(545,·)$, $\chi_{624}(421,·)$, $\chi_{624}(541,·)$, $\chi_{624}(53,·)$, $\chi_{624}(265,·)$, $\chi_{624}(313,·)$, $\chi_{624}(317,·)$, $\chi_{624}(437,·)$, $\chi_{624}(577,·)$, $\chi_{624}(181,·)$, $\chi_{624}(161,·)$, $\chi_{624}(73,·)$, $\chi_{624}(77,·)$, $\chi_{624}(109,·)$, $\chi_{624}(593,·)$, $\chi_{624}(469,·)$, $\chi_{624}(473,·)$, $\chi_{624}(229,·)$, $\chi_{624}(337,·)$, $\chi_{624}(233,·)$, $\chi_{624}(365,·)$, $\chi_{624}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{18} + \frac{1}{3} a^{16} - \frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{19} + \frac{1}{3} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{3} a^{22} + \frac{1}{3} a^{16} - \frac{1}{3} a^{14} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{23} + \frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{24} + \frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{25} + \frac{1}{3} a^{19} - \frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{26} + \frac{1}{3} a^{16} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{27} + \frac{1}{3} a^{17} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{216855} a^{28} + \frac{11416}{216855} a^{27} + \frac{637}{72285} a^{26} - \frac{2338}{14457} a^{25} + \frac{4534}{43371} a^{24} + \frac{1588}{14457} a^{23} - \frac{7427}{216855} a^{22} + \frac{17081}{216855} a^{21} + \frac{1079}{14457} a^{20} + \frac{25597}{216855} a^{19} - \frac{38}{24095} a^{18} - \frac{35324}{72285} a^{17} + \frac{7354}{24095} a^{16} + \frac{103816}{216855} a^{15} + \frac{108362}{216855} a^{14} - \frac{6814}{72285} a^{13} - \frac{22267}{72285} a^{12} - \frac{40801}{216855} a^{11} + \frac{34159}{216855} a^{10} - \frac{3602}{24095} a^{9} - \frac{37652}{216855} a^{8} - \frac{12199}{216855} a^{7} + \frac{528}{4819} a^{6} + \frac{14636}{43371} a^{5} + \frac{5167}{72285} a^{4} + \frac{5638}{14457} a^{3} - \frac{53714}{216855} a^{2} + \frac{23096}{216855} a - \frac{89422}{216855}$, $\frac{1}{216855} a^{29} + \frac{22}{711} a^{27} - \frac{6992}{72285} a^{26} - \frac{965}{43371} a^{25} + \frac{680}{43371} a^{24} - \frac{377}{216855} a^{23} + \frac{13408}{216855} a^{22} - \frac{27866}{216855} a^{21} + \frac{18097}{216855} a^{20} + \frac{32561}{216855} a^{19} - \frac{729}{4819} a^{18} + \frac{26926}{72285} a^{17} - \frac{19462}{43371} a^{16} - \frac{87089}{216855} a^{15} - \frac{67829}{216855} a^{14} + \frac{35887}{72285} a^{13} + \frac{4990}{43371} a^{12} - \frac{11690}{43371} a^{11} + \frac{58298}{216855} a^{10} - \frac{52964}{216855} a^{9} + \frac{88708}{216855} a^{8} - \frac{77936}{216855} a^{7} - \frac{20446}{43371} a^{6} - \frac{9634}{216855} a^{5} + \frac{2183}{72285} a^{4} + \frac{5911}{216855} a^{3} + \frac{20150}{43371} a^{2} - \frac{19226}{72285} a + \frac{32782}{216855}$, $\frac{1}{1347745195377084864096337868500749730104015} a^{30} + \frac{1443259981067296772966830746806412842}{1347745195377084864096337868500749730104015} a^{29} - \frac{2716055761343560311874120526043654652}{1347745195377084864096337868500749730104015} a^{28} - \frac{12524411841318985013554003355649232655423}{1347745195377084864096337868500749730104015} a^{27} - \frac{37439102041275436930511267256431862994579}{1347745195377084864096337868500749730104015} a^{26} - \frac{12655749299574385791192528295202689560695}{269549039075416972819267573700149946020803} a^{25} - \frac{66022126407927044868297966870689707403209}{449248398459028288032112622833583243368005} a^{24} - \frac{47822247604991502041095259756732357097332}{449248398459028288032112622833583243368005} a^{23} - \frac{11969117338008913808577908529501468204019}{149749466153009429344037540944527747789335} a^{22} + \frac{1339430024427589396276753282444248941071}{449248398459028288032112622833583243368005} a^{21} - \frac{36489664592113749962778304142462245821163}{269549039075416972819267573700149946020803} a^{20} - \frac{592608220310496878781755612259302218160177}{1347745195377084864096337868500749730104015} a^{19} - \frac{86339756710642438140137692316939501276336}{449248398459028288032112622833583243368005} a^{18} - \frac{66293977863453020988808553719536553090867}{269549039075416972819267573700149946020803} a^{17} - \frac{203302285396646784389982775225506452106212}{449248398459028288032112622833583243368005} a^{16} + \frac{47139377268968333846582476529710902236909}{149749466153009429344037540944527747789335} a^{15} - \frac{187131310816091948245085486155908283577806}{1347745195377084864096337868500749730104015} a^{14} + \frac{573039074851096065089862469249720975302566}{1347745195377084864096337868500749730104015} a^{13} + \frac{217733519150444492393444169032840631610454}{449248398459028288032112622833583243368005} a^{12} + \frac{10020490498932735149250612577118646857233}{29949893230601885868807508188905549557867} a^{11} - \frac{622276151566113343809207938292161353867006}{1347745195377084864096337868500749730104015} a^{10} - \frac{39573691834497727511386836852101585174526}{149749466153009429344037540944527747789335} a^{9} + \frac{57158047991270753619086600338303445418386}{149749466153009429344037540944527747789335} a^{8} + \frac{153426232563031647483958049042209878324812}{449248398459028288032112622833583243368005} a^{7} - \frac{8457343074277077359292731744528868308099}{1347745195377084864096337868500749730104015} a^{6} + \frac{273993469511684373957573281629785744341821}{1347745195377084864096337868500749730104015} a^{5} - \frac{315399785751964920557139895241375059620458}{1347745195377084864096337868500749730104015} a^{4} + \frac{193878725243636001411314092622185736435549}{449248398459028288032112622833583243368005} a^{3} + \frac{74269933499458929050136984806474810953300}{269549039075416972819267573700149946020803} a^{2} + \frac{529250084531443135423959662099363556888679}{1347745195377084864096337868500749730104015} a + \frac{612669361917001243019611493472047109659798}{1347745195377084864096337868500749730104015}$, $\frac{1}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{31} + \frac{23269445664500444194534452326146894614629713313}{137649147196493194994277989140513064827123964306697524789988014873579302882935505559322935} a^{30} + \frac{459487331162095046157583645631262118243951366522798330767231302696065008614557228763000}{309462812727156000986135775185701472344340096554317375232851055038780988741415603598469822467} a^{29} + \frac{60219787827434275449158672223054389363334399828752162214685009628870705037101376243848}{515771354545260001643559625309502453907233494257195625388085091731301647902359339330783037445} a^{28} - \frac{14281488979935654358027163853695089713078349622498247555011695045768402550246370203484684797}{171923784848420000547853208436500817969077831419065208462695030577100549300786446443594345815} a^{27} + \frac{17072857907151260294275116886308613358012015894589518226066632657588215174910958857372150459}{309462812727156000986135775185701472344340096554317375232851055038780988741415603598469822467} a^{26} + \frac{2460597376361947733369976577367806374635748093117787030322954049310036169106312385494757967}{19586253970073164619375681973778574199008866870526416153977914875872214477304785037877836865} a^{25} - \frac{49403168662433482558436900966854298435583448124216724921546043246730932838425389566292019186}{515771354545260001643559625309502453907233494257195625388085091731301647902359339330783037445} a^{24} + \frac{23294653078986206864825681750028634676522622875988197730046158501789441630866992576734887092}{171923784848420000547853208436500817969077831419065208462695030577100549300786446443594345815} a^{23} - \frac{58930393792781247478351678579516818875867626913003614049790391254250765621445651473125711212}{515771354545260001643559625309502453907233494257195625388085091731301647902359339330783037445} a^{22} - \frac{170421527268575105474213161516663979532801508880014573217772067755525459730812740618847715928}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{21} - \frac{118773436603528109634919943282995632829823040250205014793966411177812093470063621368453789752}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{20} + \frac{99461086121658822049403364812208001872886729246200536178286604332505354577600648769663559053}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{19} - \frac{621081905619656867934893237819277055294555820310128999804409809071537948316245414854193806716}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{18} + \frac{628241742514694581819789093695155053185291654705596979206910800141489497723410731251410478471}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{17} + \frac{41394384112517944621537193058751760127955419658432556260931485645737605066034819060146215056}{171923784848420000547853208436500817969077831419065208462695030577100549300786446443594345815} a^{16} + \frac{257328686763868311548375255235731186916557973998502389867425410712628009395666098996016184324}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{15} - \frac{198082641346840647484529403945576636171006600212069243723412325041363072938901381926179306082}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{14} - \frac{4454442221705552815356337621956911437769256323334104447735779309536209126032351079824084568}{309462812727156000986135775185701472344340096554317375232851055038780988741415603598469822467} a^{13} - \frac{28491946413447006716657216033987960549347933927105182834311066214019003803343133284521660053}{171923784848420000547853208436500817969077831419065208462695030577100549300786446443594345815} a^{12} + \frac{29906613449224713425243080720725654794575344825748282589741100931828922563735723680158663977}{309462812727156000986135775185701472344340096554317375232851055038780988741415603598469822467} a^{11} - \frac{102402624744975895228505768151823715886883523679249914774606180368815613800627513063887042179}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{10} + \frac{66317980210898955158188719300768232796780572895350923399685596372330563533290821031998063711}{515771354545260001643559625309502453907233494257195625388085091731301647902359339330783037445} a^{9} + \frac{2234573936897734359202817607583889211527684642474904639835912750422060883123072090281807844}{515771354545260001643559625309502453907233494257195625388085091731301647902359339330783037445} a^{8} + \frac{407468306185449563412326614803922652212577189432609993095784229400148541171157255973763897376}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{7} - \frac{257104276753198831793368728948732519337797619164519359879076437476234942413278378748297241498}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{6} - \frac{440758065516137534812509148850504453283488237776677236280575225092542317358913903173434754232}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{5} + \frac{60150813826727620392183207380116192628763247492563484894058325756571279588212797118017835898}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{4} + \frac{26062541582320004811573020362814723359255651432931570487899902660960103300281077196357155122}{1547314063635780004930678875928507361721700482771586876164255275193904943707078017992349112335} a^{3} + \frac{1706320778698002161722988818038399767888697886913034365843989185797507935725379043427774019}{6528751323357721539791893991259524733002955623508805384659304958624071492434928345959278955} a^{2} + \frac{20144827769994808104924455553173356935707845834435225569964409822534081396773077367148809577}{515771354545260001643559625309502453907233494257195625388085091731301647902359339330783037445} a + \frac{127832512604538629257014781048195635487677798866353097799378938510995127355836587003765733284}{309462812727156000986135775185701472344340096554317375232851055038780988741415603598469822467}$

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

Class group and class number

$C_{15}\times C_{120}$, which has order $1800$ (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{394931953917134471018985621754596480880270185032}{707014925521692241255833606606329961249677767873323063} a^{31} + \frac{1202503181232027639235957204426270644517865662}{566064792251154716778089356770480353282368108785687} a^{30} + \frac{12171037358062045622622600923532671545903673405336}{707014925521692241255833606606329961249677767873323063} a^{29} - \frac{32468335536414472550605966495562563050467158322905}{707014925521692241255833606606329961249677767873323063} a^{28} - \frac{216943262096275415023580124554801077502557937836768}{707014925521692241255833606606329961249677767873323063} a^{27} + \frac{121935121220408892106850451666620532777616541320350}{235671641840564080418611202202109987083225922624441021} a^{26} + \frac{27372493670038980141943963933912177047430526766728}{8949556019261927104504222868434556471514908454092697} a^{25} - \frac{881176350870431951924083928918539599983220907074097}{235671641840564080418611202202109987083225922624441021} a^{24} - \frac{4131304382283306269351399638749131092963915453923612}{235671641840564080418611202202109987083225922624441021} a^{23} + \frac{4941749198105775971913767957184702532816770811103008}{235671641840564080418611202202109987083225922624441021} a^{22} + \frac{45956031319053034996239053781849696313338057711177568}{707014925521692241255833606606329961249677767873323063} a^{21} - \frac{29996109987318600883181864242348858822739104492669685}{235671641840564080418611202202109987083225922624441021} a^{20} - \frac{99952256039024233099112440275051796407403196471198356}{707014925521692241255833606606329961249677767873323063} a^{19} + \frac{433749721393063353744134871334300464675870020984768962}{707014925521692241255833606606329961249677767873323063} a^{18} - \frac{107327147532189871235795271136037947981064228766034592}{707014925521692241255833606606329961249677767873323063} a^{17} - \frac{187410529812650265529556404321821231983919246724461085}{235671641840564080418611202202109987083225922624441021} a^{16} + \frac{355702755628589043646861347875360679667126936485111704}{235671641840564080418611202202109987083225922624441021} a^{15} - \frac{1548302605163060198788778084153422086473125331724223192}{235671641840564080418611202202109987083225922624441021} a^{14} - \frac{1120935340725684873616892704334960085594709221173468884}{707014925521692241255833606606329961249677767873323063} a^{13} + \frac{9122078554758496873134855416454312953924692038346229278}{235671641840564080418611202202109987083225922624441021} a^{12} - \frac{13813838172804736219825367679651486958832635678910186324}{707014925521692241255833606606329961249677767873323063} a^{11} - \frac{59688493966438965831757139493941296952080033473856660112}{707014925521692241255833606606329961249677767873323063} a^{10} + \frac{50055294510368730563625176532646277305430225606381379568}{707014925521692241255833606606329961249677767873323063} a^{9} + \frac{20603283315404308689321892422945806963597369119744611895}{235671641840564080418611202202109987083225922624441021} a^{8} - \frac{55029120212153812537212147384465877515821698196912790700}{707014925521692241255833606606329961249677767873323063} a^{7} - \frac{18748793023204757032718154657777010559465988936376541822}{235671641840564080418611202202109987083225922624441021} a^{6} + \frac{21918527007010355821977532448617028220498688556762729228}{707014925521692241255833606606329961249677767873323063} a^{5} + \frac{84602803633821426934747138335595128204043262458232426888}{707014925521692241255833606606329961249677767873323063} a^{4} - \frac{22316227705752167050689870825950473730367972613933424260}{235671641840564080418611202202109987083225922624441021} a^{3} + \frac{173598640749765875491073058328177992666803920937386292}{8949556019261927104504222868434556471514908454092697} a^{2} - \frac{4401305290129025200875071423657427885875775032012334208}{707014925521692241255833606606329961249677767873323063} a + \frac{1052743357728579016735921236295813722094266882000265873}{707014925521692241255833606606329961249677767873323063} \) (order $6$)
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:  \( 298974970196971.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-78}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, \(\Q(\sqrt{-6}, \sqrt{13})\), \(\Q(\sqrt{-6}, \sqrt{26})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-39})\), 4.4.346112.1, 4.0.3115008.1, \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{26})\), 4.0.2197.1, 4.4.1265472.2, 4.0.140608.2, 4.4.19773.1, 4.0.4499456.2, 4.4.40495104.2, 4.0.4499456.1, 4.4.40495104.1, 8.0.339738624.2, 8.0.9475854336.2, 8.0.9703274840064.7, 8.8.119793516544.1, 8.0.9703274840064.5, 8.0.9703274840064.4, 8.0.9703274840064.1, 8.0.1601419382784.3, 8.0.1601419382784.1, 8.0.1639853447970816.40, 8.0.1639853447970816.63, 8.0.19770609664.2, 8.8.1601419382784.1, 8.0.20245104295936.1, 8.8.1639853447970816.2, 8.0.390971529.1, 8.0.1601419382784.5, 8.0.1639853447970816.65, 8.0.1639853447970816.1, 16.0.94153542621819044779524096.1, 16.0.2564544039556287515590656.1, 16.0.2689119330821773737947987705856.2, 16.0.409864247953326282266116096.2, 16.16.2689119330821773737947987705856.1, 16.0.2689119330821773737947987705856.4, 16.0.2689119330821773737947987705856.6

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed