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Properties

Label 27.27.1197336751...1009.7
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 7^{18}\cdot 19^{24}$
Root discriminant $216.89$
Ramified primes $3, 7, 19$
Class number $3$ (GRH)
Class group $[3]$ (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![343950769, -2960472531, 5579986791, 11473645521, -32729761209, -22855569024, 69532031920, 36838185093, -70735013982, -40428696341, 35193844791, 23559312123, -8680107239, -7259510481, 1075653042, 1288690420, -54908529, -140166894, -1383413, 9627678, 322335, -417803, -16344, 11058, 362, -162, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 162*x^25 + 362*x^24 + 11058*x^23 - 16344*x^22 - 417803*x^21 + 322335*x^20 + 9627678*x^19 - 1383413*x^18 - 140166894*x^17 - 54908529*x^16 + 1288690420*x^15 + 1075653042*x^14 - 7259510481*x^13 - 8680107239*x^12 + 23559312123*x^11 + 35193844791*x^10 - 40428696341*x^9 - 70735013982*x^8 + 36838185093*x^7 + 69532031920*x^6 - 22855569024*x^5 - 32729761209*x^4 + 11473645521*x^3 + 5579986791*x^2 - 2960472531*x + 343950769)
 
gp: K = bnfinit(x^27 - 3*x^26 - 162*x^25 + 362*x^24 + 11058*x^23 - 16344*x^22 - 417803*x^21 + 322335*x^20 + 9627678*x^19 - 1383413*x^18 - 140166894*x^17 - 54908529*x^16 + 1288690420*x^15 + 1075653042*x^14 - 7259510481*x^13 - 8680107239*x^12 + 23559312123*x^11 + 35193844791*x^10 - 40428696341*x^9 - 70735013982*x^8 + 36838185093*x^7 + 69532031920*x^6 - 22855569024*x^5 - 32729761209*x^4 + 11473645521*x^3 + 5579986791*x^2 - 2960472531*x + 343950769, 1)
 

Normalized defining polynomial

\( x^{27} - 3 x^{26} - 162 x^{25} + 362 x^{24} + 11058 x^{23} - 16344 x^{22} - 417803 x^{21} + 322335 x^{20} + 9627678 x^{19} - 1383413 x^{18} - 140166894 x^{17} - 54908529 x^{16} + 1288690420 x^{15} + 1075653042 x^{14} - 7259510481 x^{13} - 8680107239 x^{12} + 23559312123 x^{11} + 35193844791 x^{10} - 40428696341 x^{9} - 70735013982 x^{8} + 36838185093 x^{7} + 69532031920 x^{6} - 22855569024 x^{5} - 32729761209 x^{4} + 11473645521 x^{3} + 5579986791 x^{2} - 2960472531 x + 343950769 \)

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:  \(1197336751300349460100016353165918520894496565611371829528951009=3^{36}\cdot 7^{18}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.89$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(130,·)$, $\chi_{1197}(709,·)$, $\chi_{1197}(904,·)$, $\chi_{1197}(142,·)$, $\chi_{1197}(463,·)$, $\chi_{1197}(400,·)$, $\chi_{1197}(529,·)$, $\chi_{1197}(1108,·)$, $\chi_{1197}(214,·)$, $\chi_{1197}(541,·)$, $\chi_{1197}(862,·)$, $\chi_{1197}(799,·)$, $\chi_{1197}(928,·)$, $\chi_{1197}(289,·)$, $\chi_{1197}(739,·)$, $\chi_{1197}(613,·)$, $\chi_{1197}(340,·)$, $\chi_{1197}(106,·)$, $\chi_{1197}(940,·)$, $\chi_{1197}(688,·)$, $\chi_{1197}(1138,·)$, $\chi_{1197}(1012,·)$, $\chi_{1197}(310,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(1087,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{1724943773} a^{24} - \frac{326864620}{1724943773} a^{23} + \frac{181726131}{1724943773} a^{22} + \frac{340317402}{1724943773} a^{21} - \frac{66739009}{1724943773} a^{20} - \frac{316117020}{1724943773} a^{19} - \frac{566481918}{1724943773} a^{18} + \frac{62545999}{1724943773} a^{17} + \frac{233192063}{1724943773} a^{16} + \frac{538243353}{1724943773} a^{15} + \frac{854606705}{1724943773} a^{14} - \frac{613708803}{1724943773} a^{13} - \frac{330145938}{1724943773} a^{12} + \frac{137142951}{1724943773} a^{11} - \frac{377074400}{1724943773} a^{10} + \frac{23439688}{1724943773} a^{9} + \frac{800621246}{1724943773} a^{8} - \frac{438416844}{1724943773} a^{7} - \frac{46505414}{246420539} a^{6} + \frac{456517779}{1724943773} a^{5} + \frac{409082908}{1724943773} a^{4} - \frac{703724836}{1724943773} a^{3} + \frac{627380458}{1724943773} a^{2} + \frac{554727844}{1724943773} a - \frac{418842314}{1724943773}$, $\frac{1}{1724943773} a^{25} + \frac{281311648}{1724943773} a^{23} + \frac{23745200}{246420539} a^{22} - \frac{430689697}{1724943773} a^{21} - \frac{97959109}{246420539} a^{20} - \frac{415413563}{1724943773} a^{19} + \frac{575142696}{1724943773} a^{18} - \frac{567399109}{1724943773} a^{17} - \frac{729850199}{1724943773} a^{16} + \frac{64687556}{246420539} a^{15} + \frac{848510548}{1724943773} a^{14} + \frac{178722739}{1724943773} a^{13} - \frac{90800302}{246420539} a^{12} + \frac{67960279}{1724943773} a^{11} - \frac{677958514}{1724943773} a^{10} + \frac{53942213}{246420539} a^{9} - \frac{487152520}{1724943773} a^{8} + \frac{19789927}{1724943773} a^{7} - \frac{450772129}{1724943773} a^{6} - \frac{112271444}{1724943773} a^{5} + \frac{715262721}{1724943773} a^{4} + \frac{629366875}{1724943773} a^{3} - \frac{421517919}{1724943773} a^{2} - \frac{64012167}{246420539} a - \frac{160387230}{1724943773}$, $\frac{1}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{26} + \frac{52008490893260366461191271968168689589408279364595962045570869836326739770382545668005439030259025079919}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{25} + \frac{11774488568276991756031753865971465899559959461663341718028600449289679654235536310309371402636565430288}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{24} + \frac{12465344001561023866783918010983393058226156598508665790147382471324080638953107420093816286181662234074504611287}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{23} - \frac{3058298807637725281342705490834020718800616377584273486743109642681004830406918100150668085754561503826778480072}{31303150249081649740796769212495031894287633742400805682573004493275850528960288824595968529321506923816198730489} a^{22} + \frac{10119311655446921303037078052052477236761518268731418114920029728568572572384933605390419001370751614451728277605}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{21} + \frac{9230338057691497531888299405057165552479174964855232021989964224238545311462907522511984327466481282293921620574}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{20} - \frac{33630588862242641786845984599814436086127139648381823938372656618791698303058825130997609573278154439398674187695}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{19} - \frac{57528832038638120090780279693693332898392123541091271308164580055958469660659252225511021891246341425274942279025}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{18} + \frac{51128578887000080912036636043320391261798542207718136198567649547850076229398887734423768130491690226240618163244}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{17} - \frac{89375275858953096365298782633283758036857068803631397403607629896501644877758387807298038625004169558536352629059}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{16} - \frac{75865588711916763434236587539443376958851631123723943423388281913651831443503098058916207764542325703550651841897}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{15} + \frac{9663752316063546011532745763120832273382795483427696442605899536168024161965652943908957406822208140900763371014}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{14} + \frac{4761502160279027470667471672282480749585271082226388912282506187211999548908112205719256701993976667932419697758}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{13} - \frac{13295731435511472474879303029187146776796372425126582338941251593201026372318320490155029967063314898586513589847}{31303150249081649740796769212495031894287633742400805682573004493275850528960288824595968529321506923816198730489} a^{12} + \frac{71987991214001711820402042393895724948027943157246423307049849416074315388406267063937692977786244391545005291182}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{11} + \frac{15199525141431224244721208287085856313486115691976956650518159868866570275415152892759704728626611212915045624643}{31303150249081649740796769212495031894287633742400805682573004493275850528960288824595968529321506923816198730489} a^{10} - \frac{57512866579972890662551725618196709267509124983405467581242718383520511752192708997114254547086982125157628957959}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{9} - \frac{56638121305772827348497896570244157605857402299183412348138096972320457379952681651386415652497335350170022347469}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{8} - \frac{382429258878667242821888021311076291826922029158106918164658196982999047429070635036592661078172163162479124216}{31303150249081649740796769212495031894287633742400805682573004493275850528960288824595968529321506923816198730489} a^{7} - \frac{56053009662338088758479947396163406523484158764296989939207928910589272625744466949252867568883587144493137598946}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{6} + \frac{73838922417356869837187154947781694730191969330929652040782230024956761831713720936080238335870148037696658105596}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{5} - \frac{29307001333631251304442631182501563410547271587576579315296956937256899374638234205376826909541224295708584890080}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{4} + \frac{51034024390621094698347455249346860244716549219386459703203553306065421669026041886823437591173784733890296485621}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{3} + \frac{38113600359424030553654214072568258012153751944211170065846831745757629032067466601063980561287891000597891411062}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a^{2} - \frac{8368029331089340888432837571019962783264880451131219058658196684475579211020251155121835518403065236268418781354}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423} a - \frac{57096508141183598308475212398956370302827321178028370179852560271510802057903091904681606627136384350042045904635}{219122051743571548185577384487465223260013436196805639778011031452930953702722021772171779705250548466713391113423}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 26038858912982828000000 \) (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

\(\Q(\zeta_{9})^+\), 3.3.361.1, 3.3.29241.1, 3.3.29241.2, 9.9.25002110044521.1, 9.9.1998099208210609.2, 9.9.1061871841310654257569.6, 9.9.1061871841310654257569.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$