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Properties

Label 27.27.7806116202...4321.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 43^{18}$
Root discriminant $180.00$
Ramified primes $3, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2809696843, 9431461404, 44317649232, -14975802189, -162267197085, -81038691729, 186039613671, 167444847135, -77931984402, -120399207328, 3486053466, 42360771069, 6189476667, -8535335625, -2036139291, 1076463318, 315512154, -90158310, -28705768, 5247171, 1616472, -218118, -55647, 6363, 1077, -117, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 117*x^25 + 1077*x^24 + 6363*x^23 - 55647*x^22 - 218118*x^21 + 1616472*x^20 + 5247171*x^19 - 28705768*x^18 - 90158310*x^17 + 315512154*x^16 + 1076463318*x^15 - 2036139291*x^14 - 8535335625*x^13 + 6189476667*x^12 + 42360771069*x^11 + 3486053466*x^10 - 120399207328*x^9 - 77931984402*x^8 + 167444847135*x^7 + 186039613671*x^6 - 81038691729*x^5 - 162267197085*x^4 - 14975802189*x^3 + 44317649232*x^2 + 9431461404*x - 2809696843)
 
gp: K = bnfinit(x^27 - 9*x^26 - 117*x^25 + 1077*x^24 + 6363*x^23 - 55647*x^22 - 218118*x^21 + 1616472*x^20 + 5247171*x^19 - 28705768*x^18 - 90158310*x^17 + 315512154*x^16 + 1076463318*x^15 - 2036139291*x^14 - 8535335625*x^13 + 6189476667*x^12 + 42360771069*x^11 + 3486053466*x^10 - 120399207328*x^9 - 77931984402*x^8 + 167444847135*x^7 + 186039613671*x^6 - 81038691729*x^5 - 162267197085*x^4 - 14975802189*x^3 + 44317649232*x^2 + 9431461404*x - 2809696843, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 117 x^{25} + 1077 x^{24} + 6363 x^{23} - 55647 x^{22} - 218118 x^{21} + 1616472 x^{20} + 5247171 x^{19} - 28705768 x^{18} - 90158310 x^{17} + 315512154 x^{16} + 1076463318 x^{15} - 2036139291 x^{14} - 8535335625 x^{13} + 6189476667 x^{12} + 42360771069 x^{11} + 3486053466 x^{10} - 120399207328 x^{9} - 77931984402 x^{8} + 167444847135 x^{7} + 186039613671 x^{6} - 81038691729 x^{5} - 162267197085 x^{4} - 14975802189 x^{3} + 44317649232 x^{2} + 9431461404 x - 2809696843 \)

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:  \(7806116202371923778076833319704235358304327213604702954864321=3^{66}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $180.00$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1161=3^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1161}(1,·)$, $\chi_{1161}(130,·)$, $\chi_{1161}(259,·)$, $\chi_{1161}(388,·)$, $\chi_{1161}(517,·)$, $\chi_{1161}(646,·)$, $\chi_{1161}(775,·)$, $\chi_{1161}(904,·)$, $\chi_{1161}(1033,·)$, $\chi_{1161}(79,·)$, $\chi_{1161}(208,·)$, $\chi_{1161}(337,·)$, $\chi_{1161}(466,·)$, $\chi_{1161}(595,·)$, $\chi_{1161}(724,·)$, $\chi_{1161}(853,·)$, $\chi_{1161}(982,·)$, $\chi_{1161}(1111,·)$, $\chi_{1161}(49,·)$, $\chi_{1161}(178,·)$, $\chi_{1161}(307,·)$, $\chi_{1161}(436,·)$, $\chi_{1161}(565,·)$, $\chi_{1161}(694,·)$, $\chi_{1161}(823,·)$, $\chi_{1161}(952,·)$, $\chi_{1161}(1081,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{34} a^{24} - \frac{4}{17} a^{23} + \frac{4}{17} a^{22} + \frac{1}{34} a^{21} + \frac{3}{17} a^{20} + \frac{7}{34} a^{19} - \frac{5}{34} a^{18} - \frac{2}{17} a^{16} - \frac{8}{17} a^{15} + \frac{13}{34} a^{14} + \frac{13}{34} a^{13} + \frac{8}{17} a^{12} - \frac{1}{17} a^{11} - \frac{7}{34} a^{10} + \frac{15}{34} a^{9} + \frac{3}{17} a^{8} + \frac{9}{34} a^{7} - \frac{5}{17} a^{6} + \frac{3}{34} a^{5} + \frac{11}{34} a^{4} + \frac{7}{34} a^{3} + \frac{9}{34} a^{2} + \frac{1}{34} a + \frac{6}{17}$, $\frac{1}{34} a^{25} - \frac{5}{34} a^{23} - \frac{3}{34} a^{22} - \frac{3}{34} a^{21} + \frac{2}{17} a^{20} - \frac{3}{17} a^{18} - \frac{2}{17} a^{17} - \frac{7}{17} a^{16} - \frac{13}{34} a^{15} + \frac{15}{34} a^{14} + \frac{1}{34} a^{13} + \frac{7}{34} a^{12} - \frac{3}{17} a^{11} + \frac{5}{17} a^{10} - \frac{5}{17} a^{9} - \frac{11}{34} a^{8} - \frac{3}{17} a^{7} + \frac{4}{17} a^{6} + \frac{1}{34} a^{5} - \frac{7}{34} a^{4} + \frac{7}{17} a^{3} - \frac{6}{17} a^{2} - \frac{7}{17} a - \frac{3}{17}$, $\frac{1}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{26} - \frac{821504579536017541084559666575985666265093677936266373264594230182898500145012626247625965434363416943209855192711526560843}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{25} - \frac{318431336889713359214164729020801121903732848611609099419467368865696811893317485012869995486306438455350684111853446228209}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{24} - \frac{1065733142322131013107503625717940992062737046155091458530650931525224862925256925425209447031389454913975542615913916916485}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{23} + \frac{8852300292070943966306856508335237217033686704052870639016180027653586778516038880802786935128028239622359638507838634753911}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{22} - \frac{5479634411411225900874547293212074832919904643977479542694208147632223967917835178950994117838064551495815005856753724413764}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{21} - \frac{4956591379298471049163611440986605050885266629119615108457194274036985079922021037508022012481709751241638701457741918651214}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{20} + \frac{5304809927099986784298288720053953420887870439938786676974329461517606464302621254936709696370722927122255650243724640952725}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{19} - \frac{6328453299800892278846445238176006226427534747757985769642273511143692565452830432790411398379098745176344935213778548966431}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{18} + \frac{7685326542921797298472482214079714893437103953494263136209039959176673080746495879323692978755697833447517528775429776283439}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{17} + \frac{22329399241630944567585968420035569054864434464557606203843625506651480985035200991154281528344739907858969572856053952679355}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{16} + \frac{5466462624959268661449159445284199794626020504177064802326481759195949631996350423445559656560852524112937085312538431162102}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{15} + \frac{4581249274920725169026876019644266898710860341912271152421380322846262710763356012486958749043183482184336897991968604452309}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{14} - \frac{107536541677281512326053784831019234081065882694577243752013650848175267014531040035311756043734727381905511835640374635233}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{13} + \frac{22800153998355725542058046606890496403963875379919971447546235896790930362515769477644317562808725310368543482140608676833743}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{12} - \frac{1900728750683355940258647971334732332764610982504941220306586587740857249132784528165231835904993470678601041137264934955035}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{11} - \frac{12843396670409243312020973973580453331192747614362878800795866565142288586317987586131469121141120439234816850436728091269845}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{10} - \frac{12046800193397209409830344396380081304042150550023562116485986012054523210539804836794572003370041043128096152600452233343553}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{9} + \frac{1637545865688335198402546131272437399197582171462796646013285316856571891751335284236143944710488654015578806645877614290664}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{8} + \frac{21503051427172043457542826082236104465741920399653196396936464174204462608303909518182827668680710670241834181159232548613277}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{7} - \frac{1166630504596372072402248270303851401759186933012516137031774382386653360957549881051439230496509522460027007895759713519178}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{6} + \frac{23244503929554890612958706418023608643134830424128204361977740211091474494534499111275424733249868638041973762989710092374231}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a^{5} - \frac{36916706690503814656528520787525805326769860277824346852956124923548540311264934953457463941388070949399273264237644610430}{106817777268701344857398879884445508896929972768432651966899779908976611067792783147868623231228143437086714918644334562043} a^{4} + \frac{6261790268052564405121135625025239624814674858069169248189602206479054812282661040672867575836023161972428414620571185753925}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{3} - \frac{3923569301310536482829564051707811580740368782414441229382257033772929669911513753470107172823899803366197874460506919754950}{28733982085280661766640298688915841893274162674708383379096040795514708377236258666776659649200370584576326313115325997189567} a^{2} + \frac{4336014001467219024171677799764657209763774666999213381459962234433784939051090678444383735140985169907599563961290611717645}{57467964170561323533280597377831683786548325349416766758192081591029416754472517333553319298400741169152652626230651994379134} a + \frac{164391841461838086410107917638509077886706204296833546612782982455183044252130851363505082464537730833672740663577553259829}{537083777294965640497949508204034427911666592050623988394318519542330997705350629285545040171969543636940678749819177517562}$

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:  \( 5685564564612443000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.149769.1, \(\Q(\zeta_{9})^+\), 3.3.1849.1, 3.3.149769.2, 9.9.3359431500123609.1, 9.9.198371070650798987841.2, 9.9.198371070650798987841.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
43Data not computed