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Properties

Label 27.27.2904636282...3089.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 109^{26}$
Root discriminant $1343.58$
Ramified primes $3, 109$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1874520162296417917, 3063320788537767798, 12404409711245032941, 1194656375883666906, -10291095715445447166, -1964343944326117491, 3669235005570784794, 690099098631125814, -690714165440312325, -117578436400215843, 73180199137284495, 11800421536957974, -4515452537534952, -729737997409152, 164576945137689, 27613039133250, -3580587873270, -640020284352, 46242066805, 9085530519, -341276166, -77704356, 1302768, 382590, -1962, -981, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 981*x^25 - 1962*x^24 + 382590*x^23 + 1302768*x^22 - 77704356*x^21 - 341276166*x^20 + 9085530519*x^19 + 46242066805*x^18 - 640020284352*x^17 - 3580587873270*x^16 + 27613039133250*x^15 + 164576945137689*x^14 - 729737997409152*x^13 - 4515452537534952*x^12 + 11800421536957974*x^11 + 73180199137284495*x^10 - 117578436400215843*x^9 - 690714165440312325*x^8 + 690099098631125814*x^7 + 3669235005570784794*x^6 - 1964343944326117491*x^5 - 10291095715445447166*x^4 + 1194656375883666906*x^3 + 12404409711245032941*x^2 + 3063320788537767798*x - 1874520162296417917)
 
gp: K = bnfinit(x^27 - 981*x^25 - 1962*x^24 + 382590*x^23 + 1302768*x^22 - 77704356*x^21 - 341276166*x^20 + 9085530519*x^19 + 46242066805*x^18 - 640020284352*x^17 - 3580587873270*x^16 + 27613039133250*x^15 + 164576945137689*x^14 - 729737997409152*x^13 - 4515452537534952*x^12 + 11800421536957974*x^11 + 73180199137284495*x^10 - 117578436400215843*x^9 - 690714165440312325*x^8 + 690099098631125814*x^7 + 3669235005570784794*x^6 - 1964343944326117491*x^5 - 10291095715445447166*x^4 + 1194656375883666906*x^3 + 12404409711245032941*x^2 + 3063320788537767798*x - 1874520162296417917, 1)
 

Normalized defining polynomial

\( x^{27} - 981 x^{25} - 1962 x^{24} + 382590 x^{23} + 1302768 x^{22} - 77704356 x^{21} - 341276166 x^{20} + 9085530519 x^{19} + 46242066805 x^{18} - 640020284352 x^{17} - 3580587873270 x^{16} + 27613039133250 x^{15} + 164576945137689 x^{14} - 729737997409152 x^{13} - 4515452537534952 x^{12} + 11800421536957974 x^{11} + 73180199137284495 x^{10} - 117578436400215843 x^{9} - 690714165440312325 x^{8} + 690099098631125814 x^{7} + 3669235005570784794 x^{6} - 1964343944326117491 x^{5} - 10291095715445447166 x^{4} + 1194656375883666906 x^{3} + 12404409711245032941 x^{2} + 3063320788537767798 x - 1874520162296417917 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2904636282635430627257935704503025621208529312293717639293800815805291320812817603089=3^{66}\cdot 109^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1343.58$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2943=3^{3}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{2943}(1,·)$, $\chi_{2943}(1858,·)$, $\chi_{2943}(772,·)$, $\chi_{2943}(2503,·)$, $\chi_{2943}(2311,·)$, $\chi_{2943}(910,·)$, $\chi_{2943}(2134,·)$, $\chi_{2943}(25,·)$, $\chi_{2943}(1498,·)$, $\chi_{2943}(1135,·)$, $\chi_{2943}(1117,·)$, $\chi_{2943}(1438,·)$, $\chi_{2943}(2305,·)$, $\chi_{2943}(1888,·)$, $\chi_{2943}(112,·)$, $\chi_{2943}(2149,·)$, $\chi_{2943}(2086,·)$, $\chi_{2943}(2791,·)$, $\chi_{2943}(1642,·)$, $\chi_{2943}(2119,·)$, $\chi_{2943}(1708,·)$, $\chi_{2943}(751,·)$, $\chi_{2943}(2800,·)$, $\chi_{2943}(625,·)$, $\chi_{2943}(376,·)$, $\chi_{2943}(634,·)$, $\chi_{2943}(571,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{241} a^{25} + \frac{113}{241} a^{24} + \frac{60}{241} a^{23} - \frac{7}{241} a^{22} + \frac{95}{241} a^{21} - \frac{32}{241} a^{20} + \frac{51}{241} a^{19} + \frac{116}{241} a^{18} + \frac{70}{241} a^{17} - \frac{77}{241} a^{16} + \frac{57}{241} a^{15} + \frac{106}{241} a^{14} - \frac{8}{241} a^{13} + \frac{1}{241} a^{12} + \frac{29}{241} a^{11} + \frac{85}{241} a^{10} - \frac{110}{241} a^{9} - \frac{11}{241} a^{8} + \frac{83}{241} a^{7} + \frac{80}{241} a^{6} - \frac{62}{241} a^{5} + \frac{7}{241} a^{4} + \frac{7}{241} a^{3} + \frac{72}{241} a^{2} + \frac{34}{241} a + \frac{95}{241}$, $\frac{1}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{26} + \frac{290028825183230139429259292579345754792291381742737515422410203338397045049025169444975648554300166535843556131287220910619367011398828583565593389752117064158643988585517968133199980427902915987065766401220369327773561276573144941832569782245193003}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{25} + \frac{43006629673054629798486428073997772023965210862194047813585054744851228568782596409698103824134340959365586975143090115608763804407663934880522748908454608399323724438086788762675011935888285970865374001421291346603268114400354807754842138759713842822}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{24} - \frac{35504117802690933604115255139933187247814092802363889516344694030508130044821554417796075973432395875507016538551610329653017781657802889445818916158218603619913172987223753794019866618260805186486944055338723025537262159415626777913208139598803048545}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{23} + \frac{21184996660953411431428998027988337280847987970076435127904728823804251993675436840093306682390138119456345566075770064742467014511124946301592415238470191719606058916337697888403756693004458695537346843846992520311370892729278252040097782987360313244}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{22} - \frac{14200605450864233825514106504102935041785778395390738185230484317821349486472860191735366220105954915741112529285817198442319713386290010873189291236972684847414199925199227934656742346233224116974691977661976483499275177531685871849521387960180584751}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{21} - \frac{75688070210162850036194872197614050111336547812492502178814945936038286541378688382508910821953091608561598046754486841830475163732279515406650108900959672965837834478810632063818235396380004038902247221880213911769179892457119472386003254503133146604}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{20} + \frac{78079953718509049984258550306331066456393689287326848275854816319829721789586435279566960524890385971982117002022476539072124368350163565628918551965017901602410455741503442837081247711956999179825610669827492576289320215255061455743649086634198445510}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{19} + \frac{100642615656338866366430006534718896505303468724849465418998931522098347706419531651477686758603004385624171841988736998133576525356723064490781927267029271389782363638983464883771699128280924360834183883379738332388318457202571941751389860731993585123}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{18} + \frac{1490136760262294039425250260409329090750405449190852593689471832305478868630477884937891779970958026972163221213646727878262947374494173732860783840591218790019912657086637219275422448304697930208491137640458923767174215486065985565510810007140674170}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{17} - \frac{53204890445057214700654442800583403764287121545525867478427566057528523648157790628510510833661751715024102235939616111666382361283041464139669500805651987036682895838622729311800786656991954932415380154778070745157036761290467798884036299347040707027}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{16} - \frac{58855076529672346180405805573528784583592496676821248239559937965802516379850432435919274835779765089393321152727289915481302908299586545188714034604476548306951485634814285822817471434467151105416301463001155281749348312198412773021426839521634704465}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{15} + \frac{65658671561054678379655594259281193000034992082392698752261501860200217960469402869771125358025861472059411089667916099582192541743836265663291683741443496320586395413989908882324588652579562709269357926394722674818123287133462517860245906036801196759}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{14} - \frac{114810551575136358917471080034581263876435623052085737090173190621571494372868339601055265228067421532216440139955754005585959965836963541102799227781627601706697511761304867753038182170023090065115021360172123669858532158034066238293118392911726099357}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{13} - \frac{108728345846677118388140196311234423935227741533073353997035567442295392183488049835106720082411279378047022114707310394113942341329592827514756492793858121227118240347435543309043293033202684368432926756508483987240941804325510412415291175020610802726}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{12} - \frac{66165884482000932264420814146096368210396893352359963558836209572780511056514146608150163904351355987382468758140460337988635116758463123944072273687638284108464744230729106705127249929992179557399969382311191104165554825831990344364053424079818449845}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{11} - \frac{26085626775164841850956505509802672432239496912602865395470815671271793138115747065253997845330970026092989072321472946258333477045266314665245792235024463704315955845794428281689687299344738928449057808289029706431857350601093187460638323090361777584}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{10} - \frac{87547321727218672681344677986970815702138907846947723657748964289709304292735462232182863484159238177247246075443778129432279743642139353965963754378363471086770597123105941066726548760843007556649017393929001953768856296253778475758901097480236454166}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{9} - \frac{26263541579776630518488200319148742104748792129557372364728015803594135170279053848784615026689765240330255357296109731085922396970051822171826441691680826358724653946731783136907518377721724349497269822949815493114778013047939865094378005629422908796}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{8} + \frac{104564847701507863901528899245409639238717222356960769741528338194302016391827401343387430007124971183244260413355677192470512856789615189540923459697745857154503736052059853338579858944127328139865798847684299219668949786844359518594190149913265688407}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{7} + \frac{33530480683999085596332189701976073945549230744041095466665802360621726406110547998834235751443644725592755577706813346478583221129718642295671470631590813551911204582135857562519179464515700790819655694715971081469293059427596812170200622010558260}{394607767015913915957246351162620802752352751708348335620342644084032461051309037968787211042250190090990276014354149556959946837618346046279453629448922088470055934381795525152842796085090567477557902689863348373514553786806914367372164243910666301} a^{6} - \frac{96784001774922453937318253572260297591659239339879523029338150658155679546936333538184867728935330847824946961269785725368475632662502700626920607354667923115130178441141881337664805968141739176453558583578081884536028754533341252543085349378956794659}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{5} - \frac{42183944666659692823021629795188429124665170056028466634748831291840580393638217362206703617221519949044929212087719708133302690132466597260060937565799844747392582440258383693171796785001466846579305933447298957240792448291096732110245411331521038623}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{4} - \frac{65097858360029020158469811135470829293673546552551462104728754086355380041236552448312263483688107777369701153479950828952256536742380120521307023007908321131724691754895337322813913960412285654621776568847241598824947191149659332105660904877584367882}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{3} - \frac{64588104336157334669251237080964090855944233307153181509967674390796123664053760586943673300477459533768075719527957093194052972611392654014679861579911424705100368308033949570789984683130285743406854166493202181452301272535848560251007104716837405956}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a^{2} - \frac{58385241115752962761965504324283132731346607454261373350005977592299266620401254326738902689857080308551498469935346698592074531520120699935708473225353370571213448194756534617303922319897864217245580062108990916292267787608280651664672238376707521645}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513} a - \frac{73837333594540556927784728134524823681760773935670632684845655850656384009768954831710308093761675839897210299143329898544877721345077098758713087704041143086508242168632313217495426364572667803335776905778760300187689633533459800690186640944119335762}{241894561180755230481792013262686552087192236797217529735270040823511898624452440274866560368899366525777039196799093678416447411460046126369305074852189240232144287776040656918692634000160517863742994348886232552964421471312638507199136681517238442513}$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $26$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.11881.1, 9.9.10589294828624773798161.4

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ $27$ $27$

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
3Data not computed
109Data not computed