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Properties

Label 32.0.72192748326...1136.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 17^{24}$
Root discriminant $122.84$
Ramified primes $2, 17$
Class number $10627200$ (GRH)
Class group $[2, 4, 180, 7380]$ (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![449086927, 118091240, 646464852, -1030498048, 4368859022, -781734536, 6109266600, 442166472, 3831594961, 1134683112, 905081912, 421923464, -11820722, -9443456, -28776348, -23901944, 539142, 1498456, 734604, 562864, 49206, -151880, 6216, -10632, 2917, 2456, 1096, -984, 30, 48, 12, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^31 + 12*x^30 + 48*x^29 + 30*x^28 - 984*x^27 + 1096*x^26 + 2456*x^25 + 2917*x^24 - 10632*x^23 + 6216*x^22 - 151880*x^21 + 49206*x^20 + 562864*x^19 + 734604*x^18 + 1498456*x^17 + 539142*x^16 - 23901944*x^15 - 28776348*x^14 - 9443456*x^13 - 11820722*x^12 + 421923464*x^11 + 905081912*x^10 + 1134683112*x^9 + 3831594961*x^8 + 442166472*x^7 + 6109266600*x^6 - 781734536*x^5 + 4368859022*x^4 - 1030498048*x^3 + 646464852*x^2 + 118091240*x + 449086927)
 
gp: K = bnfinit(x^32 - 8*x^31 + 12*x^30 + 48*x^29 + 30*x^28 - 984*x^27 + 1096*x^26 + 2456*x^25 + 2917*x^24 - 10632*x^23 + 6216*x^22 - 151880*x^21 + 49206*x^20 + 562864*x^19 + 734604*x^18 + 1498456*x^17 + 539142*x^16 - 23901944*x^15 - 28776348*x^14 - 9443456*x^13 - 11820722*x^12 + 421923464*x^11 + 905081912*x^10 + 1134683112*x^9 + 3831594961*x^8 + 442166472*x^7 + 6109266600*x^6 - 781734536*x^5 + 4368859022*x^4 - 1030498048*x^3 + 646464852*x^2 + 118091240*x + 449086927, 1)
 

Normalized defining polynomial

\( x^{32} - 8 x^{31} + 12 x^{30} + 48 x^{29} + 30 x^{28} - 984 x^{27} + 1096 x^{26} + 2456 x^{25} + 2917 x^{24} - 10632 x^{23} + 6216 x^{22} - 151880 x^{21} + 49206 x^{20} + 562864 x^{19} + 734604 x^{18} + 1498456 x^{17} + 539142 x^{16} - 23901944 x^{15} - 28776348 x^{14} - 9443456 x^{13} - 11820722 x^{12} + 421923464 x^{11} + 905081912 x^{10} + 1134683112 x^{9} + 3831594961 x^{8} + 442166472 x^{7} + 6109266600 x^{6} - 781734536 x^{5} + 4368859022 x^{4} - 1030498048 x^{3} + 646464852 x^{2} + 118091240 x + 449086927 \)

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:  \(7219274832693987941813796201388115200494782171469796274056657371136=2^{124}\cdot 17^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.84$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(544=2^{5}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{544}(1,·)$, $\chi_{544}(259,·)$, $\chi_{544}(137,·)$, $\chi_{544}(523,·)$, $\chi_{544}(115,·)$, $\chi_{544}(273,·)$, $\chi_{544}(387,·)$, $\chi_{544}(89,·)$, $\chi_{544}(409,·)$, $\chi_{544}(33,·)$, $\chi_{544}(35,·)$, $\chi_{544}(169,·)$, $\chi_{544}(171,·)$, $\chi_{544}(305,·)$, $\chi_{544}(307,·)$, $\chi_{544}(441,·)$, $\chi_{544}(443,·)$, $\chi_{544}(67,·)$, $\chi_{544}(353,·)$, $\chi_{544}(203,·)$, $\chi_{544}(81,·)$, $\chi_{544}(339,·)$, $\chi_{544}(217,·)$, $\chi_{544}(475,·)$, $\chi_{544}(395,·)$, $\chi_{544}(225,·)$, $\chi_{544}(123,·)$, $\chi_{544}(361,·)$, $\chi_{544}(497,·)$, $\chi_{544}(531,·)$, $\chi_{544}(489,·)$, $\chi_{544}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{16} a^{24} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{4} a^{20} - \frac{1}{4} a^{18} - \frac{1}{8} a^{16} - \frac{1}{2} a^{15} + \frac{3}{8} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{16}$, $\frac{1}{16} a^{25} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{4} a^{21} - \frac{1}{4} a^{19} - \frac{1}{8} a^{17} - \frac{1}{2} a^{16} + \frac{3}{8} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{16} a$, $\frac{1}{32} a^{26} - \frac{1}{32} a^{24} - \frac{1}{4} a^{23} + \frac{1}{8} a^{22} - \frac{1}{4} a^{21} + \frac{1}{16} a^{18} - \frac{1}{4} a^{17} - \frac{7}{16} a^{16} - \frac{1}{4} a^{15} + \frac{3}{16} a^{14} - \frac{1}{2} a^{13} - \frac{7}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{7}{32} a^{2} - \frac{1}{2} a - \frac{9}{32}$, $\frac{1}{32} a^{27} - \frac{1}{32} a^{25} + \frac{1}{8} a^{23} - \frac{1}{4} a^{22} + \frac{1}{16} a^{19} - \frac{1}{4} a^{18} - \frac{7}{16} a^{17} + \frac{1}{4} a^{16} + \frac{3}{16} a^{15} - \frac{1}{2} a^{14} - \frac{7}{16} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{7}{32} a^{3} - \frac{1}{2} a^{2} - \frac{9}{32} a + \frac{1}{4}$, $\frac{1}{1504} a^{28} - \frac{7}{1504} a^{27} + \frac{3}{376} a^{26} - \frac{17}{1504} a^{25} + \frac{41}{1504} a^{24} + \frac{21}{376} a^{23} + \frac{1}{8} a^{22} - \frac{3}{188} a^{21} + \frac{157}{752} a^{20} + \frac{341}{752} a^{19} - \frac{49}{376} a^{18} - \frac{199}{752} a^{17} + \frac{181}{376} a^{16} - \frac{281}{752} a^{15} + \frac{11}{94} a^{14} - \frac{171}{752} a^{13} + \frac{23}{752} a^{12} - \frac{14}{47} a^{11} - \frac{133}{376} a^{10} - \frac{145}{376} a^{9} + \frac{23}{376} a^{8} + \frac{14}{47} a^{7} - \frac{57}{376} a^{6} + \frac{173}{376} a^{5} + \frac{77}{1504} a^{4} + \frac{553}{1504} a^{3} - \frac{33}{376} a^{2} - \frac{673}{1504} a + \frac{11}{32}$, $\frac{1}{191008} a^{29} - \frac{21}{95504} a^{28} - \frac{441}{47752} a^{27} + \frac{2383}{191008} a^{26} - \frac{2889}{191008} a^{25} - \frac{1633}{191008} a^{24} + \frac{22295}{47752} a^{23} + \frac{6339}{47752} a^{22} + \frac{31597}{95504} a^{21} + \frac{901}{47752} a^{20} + \frac{20703}{47752} a^{19} - \frac{39069}{95504} a^{18} - \frac{6405}{23876} a^{17} - \frac{27897}{95504} a^{16} - \frac{3829}{23876} a^{15} + \frac{39425}{95504} a^{14} + \frac{42621}{95504} a^{13} - \frac{10711}{95504} a^{12} + \frac{19579}{47752} a^{11} - \frac{858}{5969} a^{10} - \frac{18543}{47752} a^{9} + \frac{13595}{47752} a^{8} - \frac{13377}{47752} a^{7} - \frac{246}{5969} a^{6} - \frac{76971}{191008} a^{5} + \frac{30889}{95504} a^{4} - \frac{17127}{47752} a^{3} - \frac{527}{1504} a^{2} + \frac{30699}{191008} a - \frac{255}{4064}$, $\frac{1}{2760798944513924095687206117512809249109249265140726993142500239328} a^{30} - \frac{347880843590582552545474088657033004220911513119862177709499}{2760798944513924095687206117512809249109249265140726993142500239328} a^{29} + \frac{669061938684228724400818789477889358033944814017604376749794443}{2760798944513924095687206117512809249109249265140726993142500239328} a^{28} - \frac{26476590568296423800922925231121623651787888923690555116041426839}{2760798944513924095687206117512809249109249265140726993142500239328} a^{27} - \frac{28250008039191456861802317397166285864738916463807807198122888809}{2760798944513924095687206117512809249109249265140726993142500239328} a^{26} + \frac{80541668326574498867836794877486894032789912132880159475427933}{345099868064240511960900764689101156138656158142590874142812529916} a^{25} + \frac{46181837816958985720426181512482134170343190035606101931837188765}{2760798944513924095687206117512809249109249265140726993142500239328} a^{24} + \frac{2331627430626184477526423499628621036867430751898738897635746303}{172549934032120255980450382344550578069328079071295437071406264958} a^{23} + \frac{638661543388065170401716990784345039544952859263909511334627019823}{1380399472256962047843603058756404624554624632570363496571250119664} a^{22} - \frac{429112823183053138676755986911733907392108313915573862487250267231}{1380399472256962047843603058756404624554624632570363496571250119664} a^{21} + \frac{8522081566569159210697749664176101997539992853802085393283194505}{1380399472256962047843603058756404624554624632570363496571250119664} a^{20} + \frac{483006650287518198252010649541396468512693150051585116116816518955}{1380399472256962047843603058756404624554624632570363496571250119664} a^{19} + \frac{122906052954795230248497845823404531952489131871319850359928973177}{690199736128481023921801529378202312277312316285181748285625059832} a^{18} + \frac{309846573375103360239036081019365028152775145652300395355217287663}{1380399472256962047843603058756404624554624632570363496571250119664} a^{17} - \frac{29531015748431763956017994692764500026382756397743957285266431101}{172549934032120255980450382344550578069328079071295437071406264958} a^{16} + \frac{86943158577568678773077762864227110846053650329799013036705111391}{1380399472256962047843603058756404624554624632570363496571250119664} a^{15} + \frac{275917749468618484050603411085587953242850457287422357975285415909}{1380399472256962047843603058756404624554624632570363496571250119664} a^{14} - \frac{458644474142864763359999199354835179448734830633894126099165071}{3671275192172771403839369837118097405730384661091392278115026914} a^{13} - \frac{534013651924823759863547664524326307887738894309041225862573843743}{1380399472256962047843603058756404624554624632570363496571250119664} a^{12} - \frac{89601200346765167329726011098193411442265088637539461543211660769}{690199736128481023921801529378202312277312316285181748285625059832} a^{11} - \frac{112810771140900382817612042143411049147379809719711265667217887907}{345099868064240511960900764689101156138656158142590874142812529916} a^{10} + \frac{19037444019985593076816230679733417441317566161226309823365351963}{172549934032120255980450382344550578069328079071295437071406264958} a^{9} + \frac{148361908572249643867878180817822249220596755916580897840874481797}{345099868064240511960900764689101156138656158142590874142812529916} a^{8} + \frac{306721802203510578765978744290386616200613822680155148233799322141}{690199736128481023921801529378202312277312316285181748285625059832} a^{7} + \frac{336318207543324452358417964390426552155288477852612773973158395225}{2760798944513924095687206117512809249109249265140726993142500239328} a^{6} + \frac{1300421145982514645507701913587750314911287632498548975810024587925}{2760798944513924095687206117512809249109249265140726993142500239328} a^{5} - \frac{1004371788932738815618312275148317044704587505534176278314020339169}{2760798944513924095687206117512809249109249265140726993142500239328} a^{4} + \frac{1162899583304214001907346367259688750728546644076147100701595337105}{2760798944513924095687206117512809249109249265140726993142500239328} a^{3} + \frac{176462920139534946812895809959028619262770345564834349640120486883}{2760798944513924095687206117512809249109249265140726993142500239328} a^{2} + \frac{282766874835167728500310519902785092474627038279994546744496293039}{690199736128481023921801529378202312277312316285181748285625059832} a + \frac{5721514681972059744330119541545343642763323600566519935335339299}{58740403074764342461429917393889558491686154577462276449840430624}$, $\frac{1}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{31} - \frac{9282994774323651575771}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{30} + \frac{57901762583230113509628756814156318883842786997250137593750031429843718207253528383}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{29} + \frac{7450311398993330936946258118540220536970278727847855019446584191000818102772763164639}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{28} + \frac{961056395377263027492986167227196780247439076881487578063078459312481113702459746627573}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{27} - \frac{108944563804688988801809059089187226464834598843288736986904391964804451503174820626229}{8145682091777913426948414046230672541552652324120387751413859797018701164895047199071676} a^{26} - \frac{252542841970441095094878343413621594199944649424860422256949448043247968034846676906029}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{25} - \frac{1816563621463275202318581909025692354470219113282047090466880245375364749336130917060739}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{24} - \frac{50285923601493495728271000021679416777336337495534953462608562597671588255452083803049}{136329407393772609656040402447375272661969076554316112994374222544246044600753928017936} a^{23} + \frac{3895942322091328273488639400605987576357137870706533419457280016565930608748482822948863}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{22} - \frac{1657072006927300446786617790818195200185333166479968038900414957756635243521782453797209}{4072841045888956713474207023115336270776326162060193875706929898509350582447523599535838} a^{21} - \frac{1735549930082365448747487147517633706465817190606139425308937087140557556454388997513355}{4072841045888956713474207023115336270776326162060193875706929898509350582447523599535838} a^{20} - \frac{54228906143457088553663524463531140475268142943875220194784547349822356491101799185665}{346624769862889933061634640265135001768197971239165436230377012639093666591278604215816} a^{19} - \frac{5216325572629000671355295470780028759164690088135059001063473680863402949678927985215731}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{18} + \frac{1390465626160320507048907448069941289023105630944141938039944372129753390106532679560307}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{17} + \frac{13957359881269429964108915493686862273522586130787619212125975791540491519447320278525105}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{16} - \frac{13586190385233312117027088949119862246370252578205073062406443732649996767744107940365953}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{15} + \frac{1865015165203034585331814690358985563760753789721273648416624813856291716981383201070375}{4072841045888956713474207023115336270776326162060193875706929898509350582447523599535838} a^{14} - \frac{5414154513174509987602240961405494813102770428584475525552753861089551940125562098263363}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{13} + \frac{7491100767581135511864094749258764616280862338040580990523968296167262786936869768366061}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{12} - \frac{4690568224288726951693131733398726060228801488795382371894829847363708791598511002657919}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{11} - \frac{1537652261523691871330267647684904353698084708228384273166475871732258287586162233865477}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{10} - \frac{2142679875203611266540059639919391450007480242457660491753436728542269727676593477175785}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{9} + \frac{2272664996710445230590611939140856701268619181113266934673448110161430849193220555373479}{8145682091777913426948414046230672541552652324120387751413859797018701164895047199071676} a^{8} - \frac{9145715761699869784081502670574151296035660652795048927119937138446537986862963796075435}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{7} - \frac{1779040899515424163017746041888156720222168210471851833225403901299932181117466131093575}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{6} + \frac{4685359771570651029032666893130964922094891606113605771593779162785953688510532632942687}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{5} - \frac{1415439114222074181117924984436820438115539272091182631365929084801740297980520656851991}{32582728367111653707793656184922690166210609296481551005655439188074804659580188796286704} a^{4} + \frac{20776138928030518260404945587849154695563613102623453538293747237335794045364294503115897}{65165456734223307415587312369845380332421218592963102011310878376149609319160377592573408} a^{3} - \frac{999433767697754579950413542680111789490527579739522363555010480589231775851984736209515}{16291364183555826853896828092461345083105304648240775502827719594037402329790094398143352} a^{2} + \frac{635683676567152524433915128951510827592369333707255596437944351497627110221234432474589}{2036420522944478356737103511557668135388163081030096937853464949254675291223761799767919} a + \frac{380713292763942056250095196756265012172634088984415536329357255424521475060821876872831}{1386499079451559732246538561060540007072791884956661744921508050556374666365114416863264}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{180}\times C_{7380}$, which has order $10627200$ (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:  \( -1 \) (order $2$)
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:  \( 38649094918357.02 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{34}) \), 4.4.4913.1, \(\Q(\sqrt{2}, \sqrt{17})\), 4.4.314432.1, \(\Q(\zeta_{16})^+\), 4.4.591872.2, 4.4.10061824.1, 4.4.10061824.2, 8.8.98867482624.1, 8.8.350312464384.1, 8.8.101240302206976.1, 8.0.2147483648.1, 8.0.179359981764608.32, 8.0.51835034729971712.19, 8.0.51835034729971712.50, 16.16.10249598790959829536343064576.1, 16.0.32170003058600514289521393664.1, 16.0.2686870825457373553975116320210944.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$