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Properties

Label 27.27.3537409234...3569.1
Degree $27$
Signature $[27, 0]$
Discriminant $433^{26}$
Root discriminant $345.81$
Ramified prime $433$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![8539831241, 26713087915, -136886280910, -410938972356, 298182116040, 1660652330971, 1049934817162, -978803617366, -1173563894813, 83530098447, 475857218485, 81012640811, -97647366531, -29627415002, 11204887862, 4721934257, -739754812, -429353878, 27033889, 24039830, -443292, -841786, -1088, 17870, 123, -208, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 208*x^25 + 123*x^24 + 17870*x^23 - 1088*x^22 - 841786*x^21 - 443292*x^20 + 24039830*x^19 + 27033889*x^18 - 429353878*x^17 - 739754812*x^16 + 4721934257*x^15 + 11204887862*x^14 - 29627415002*x^13 - 97647366531*x^12 + 81012640811*x^11 + 475857218485*x^10 + 83530098447*x^9 - 1173563894813*x^8 - 978803617366*x^7 + 1049934817162*x^6 + 1660652330971*x^5 + 298182116040*x^4 - 410938972356*x^3 - 136886280910*x^2 + 26713087915*x + 8539831241)
 
gp: K = bnfinit(x^27 - x^26 - 208*x^25 + 123*x^24 + 17870*x^23 - 1088*x^22 - 841786*x^21 - 443292*x^20 + 24039830*x^19 + 27033889*x^18 - 429353878*x^17 - 739754812*x^16 + 4721934257*x^15 + 11204887862*x^14 - 29627415002*x^13 - 97647366531*x^12 + 81012640811*x^11 + 475857218485*x^10 + 83530098447*x^9 - 1173563894813*x^8 - 978803617366*x^7 + 1049934817162*x^6 + 1660652330971*x^5 + 298182116040*x^4 - 410938972356*x^3 - 136886280910*x^2 + 26713087915*x + 8539831241, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 208 x^{25} + 123 x^{24} + 17870 x^{23} - 1088 x^{22} - 841786 x^{21} - 443292 x^{20} + 24039830 x^{19} + 27033889 x^{18} - 429353878 x^{17} - 739754812 x^{16} + 4721934257 x^{15} + 11204887862 x^{14} - 29627415002 x^{13} - 97647366531 x^{12} + 81012640811 x^{11} + 475857218485 x^{10} + 83530098447 x^{9} - 1173563894813 x^{8} - 978803617366 x^{7} + 1049934817162 x^{6} + 1660652330971 x^{5} + 298182116040 x^{4} - 410938972356 x^{3} - 136886280910 x^{2} + 26713087915 x + 8539831241 \)

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:  \(353740923422167929660095172127764982174048848841031999908387732793569=433^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $345.81$
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:  $433$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(433\)
Dirichlet character group:    $\lbrace$$\chi_{433}(256,·)$, $\chi_{433}(1,·)$, $\chi_{433}(66,·)$, $\chi_{433}(3,·)$, $\chi_{433}(22,·)$, $\chi_{433}(198,·)$, $\chi_{433}(385,·)$, $\chi_{433}(9,·)$, $\chi_{433}(139,·)$, $\chi_{433}(269,·)$, $\chi_{433}(78,·)$, $\chi_{433}(335,·)$, $\chi_{433}(17,·)$, $\chi_{433}(150,·)$, $\chi_{433}(153,·)$, $\chi_{433}(26,·)$, $\chi_{433}(27,·)$, $\chi_{433}(161,·)$, $\chi_{433}(243,·)$, $\chi_{433}(81,·)$, $\chi_{433}(296,·)$, $\chi_{433}(234,·)$, $\chi_{433}(289,·)$, $\chi_{433}(417,·)$, $\chi_{433}(50,·)$, $\chi_{433}(51,·)$, $\chi_{433}(374,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2135291} a^{24} - \frac{704476}{2135291} a^{23} + \frac{346125}{2135291} a^{22} + \frac{463440}{2135291} a^{21} - \frac{864558}{2135291} a^{20} + \frac{244407}{2135291} a^{19} + \frac{788597}{2135291} a^{18} - \frac{541421}{2135291} a^{17} + \frac{2048}{2135291} a^{16} + \frac{460659}{2135291} a^{15} + \frac{51533}{2135291} a^{14} + \frac{373452}{2135291} a^{13} - \frac{700078}{2135291} a^{12} - \frac{250810}{2135291} a^{11} - \frac{47639}{2135291} a^{10} - \frac{1410}{2135291} a^{9} - \frac{199004}{2135291} a^{8} + \frac{606027}{2135291} a^{7} + \frac{885865}{2135291} a^{6} - \frac{344309}{2135291} a^{5} - \frac{68000}{2135291} a^{4} + \frac{426136}{2135291} a^{3} - \frac{160459}{2135291} a^{2} - \frac{527257}{2135291} a + \frac{638409}{2135291}$, $\frac{1}{2135291} a^{25} + \frac{381060}{2135291} a^{23} - \frac{201514}{2135291} a^{22} - \frac{230436}{2135291} a^{21} - \frac{388816}{2135291} a^{20} + \frac{464544}{2135291} a^{19} - \frac{81883}{2135291} a^{18} + \frac{391818}{2135291} a^{17} - \frac{229209}{2135291} a^{16} - \frac{400254}{2135291} a^{15} - \frac{82422}{2135291} a^{14} + \frac{202255}{2135291} a^{13} - \frac{237668}{2135291} a^{12} - \frac{748822}{2135291} a^{11} - \frac{164927}{2135291} a^{10} - \frac{599849}{2135291} a^{9} - \frac{405272}{2135291} a^{8} + \frac{1834}{27029} a^{7} - \frac{536684}{2135291} a^{6} + \frac{886061}{2135291} a^{5} - \frac{823570}{2135291} a^{4} - \frac{272704}{2135291} a^{3} + \frac{128508}{2135291} a^{2} + \frac{1011400}{2135291} a + \frac{287100}{2135291}$, $\frac{1}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{26} - \frac{35577112218332469501327992680217845915113897655971228352717104762409374720500803341410934068408652366356599533601765540737717}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{25} - \frac{65067846406455104352873886145960439871113891645849065482031594788842050577631670582700788786108238095666585998442533875879284}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{24} + \frac{245237825147650452686467175932647734822197417448175459137373511427322703956510210933810357700573051375009786913615431197004782686210}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{23} + \frac{288653845559679537068353589850801784100049923127595043833336700208834710978081535952018600440192940390876693935954944201762017739713}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{22} - \frac{32815250264126155889880509300825404084051986237026995765607651545419179110792226113357671025113800548045291295621728580148978131081}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{21} + \frac{68935919933512973200098116017507144412779061802696347688501995512953863038340646376833484352805328741037688808578085336909863688337}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{20} - \frac{99531318102376628368956880180962696604885801649093214117652353342695602885172454161898509143586377160005329193652110069878484082395}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{19} + \frac{156501289524891510507899639327106581303951737622175392060550678702581368790762546954367891813190292626623746376470899463598759785095}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{18} + \frac{297212147544093138645267149085333846436930707634666227684680399902502933070538018919708671391112713375469843781331604005217813180357}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{17} - \frac{386568190598630213156125090564523586698333386770460838820307625248047518322863667825787344734508117235787983881271690506100808093304}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{16} - \frac{362545666485566799458707493216446734234321720116072910617095235661047270170035016755017766607424483185439630182879437629277842408335}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{15} - \frac{141699381832656059595223036209736439608443744464008011106957834547686847498890295869614468067279277926806167466999448545944758489803}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{14} + \frac{209099445191380698064488658294934483529717412806012195961810962070439328622435958812798534899050787883246977627597208268349873560071}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{13} + \frac{10756111803545970535342443424650893234024732725654546163001907594213554592061767275528181181173169810234542124276085523450317835840}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{12} - \frac{185010790476733329739900473276389988515515520239877581670935405301181494132497267935922119544788591866571840876837426927513860346360}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{11} + \frac{208715512921697018404527056142919322974163111544769289513751470049376611490942967999128948608927895495023610610415662848898739010888}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{10} - \frac{347876836416831004650290027722614667954100736418481238482817166510272123098260054334852520221134098487389986229015600348191317060460}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{9} + \frac{317057299574508609401108804366489519950862700290418450563578965075401813686941445421007611742725199901553256497858078451865399761178}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{8} - \frac{227877367137401181358469368490282342073565365207454473491853377386052745066332146395243658996251999292194398775340865406679824374709}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{7} + \frac{360918333739906641855594826120621688089614507943587054700019813302998913144144111940886062404240191514051477086089612921770882491180}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{6} - \frac{395599819122313255568053917779369615408512428348542651100668842296895616916617501899800187684291318983884893643370406081842369735074}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{5} - \frac{112351615978401765139222796823578021769499863437086755436898780651160098275137533485343502086019014050638114566638298331516162123364}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{4} + \frac{262614459939887920854825235048071722514298343405683415520389639920859631603502804796363534853663179599535972760708575364981524377301}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{3} + \frac{41425015390449174379329193851994134174578789399049369125496253836459901510238781722070659298491050833348848971030704528726881905148}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a^{2} - \frac{177365099902706033327298645590938410861316138725946170035598773697488269549745992756428017302704016152823847559404977878883580713740}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437} a + \frac{97555838858431151957263999703091900015191610057977300907382639783350897661475348560306410015117069696790097399036735204266607195035}{867458145867242626855781726093988973668186217719758583640497658382996298780927365723119546104989012491301441616086413260029615276437}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $26$
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:  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:  \( 54890269246609620000000000 \) (assuming GRH)
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

3.3.187489.1, 9.9.1235671900522439264641.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ $27$ $27$ $27$ $27$ $27$ $27$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ $27$ $27$

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
433Data not computed