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Properties

Label 27.27.2545040555...9689.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 7^{18}\cdot 19^{18}$
Root discriminant $112.74$
Ramified primes $3, 7, 19$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_3^3$ (as 27T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1145024, -17057376, 52146132, 123185675, -419177382, -171844968, 1223353523, -364627638, -1484013339, 967374964, 827108970, -838172511, -183437513, 367102377, -16499340, -89943576, 17637390, 12493773, -3954764, -917742, 438771, 24482, -26013, 834, 774, -63, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 63*x^25 + 774*x^24 + 834*x^23 - 26013*x^22 + 24482*x^21 + 438771*x^20 - 917742*x^19 - 3954764*x^18 + 12493773*x^17 + 17637390*x^16 - 89943576*x^15 - 16499340*x^14 + 367102377*x^13 - 183437513*x^12 - 838172511*x^11 + 827108970*x^10 + 967374964*x^9 - 1484013339*x^8 - 364627638*x^7 + 1223353523*x^6 - 171844968*x^5 - 419177382*x^4 + 123185675*x^3 + 52146132*x^2 - 17057376*x - 1145024)
 
gp: K = bnfinit(x^27 - 9*x^26 - 63*x^25 + 774*x^24 + 834*x^23 - 26013*x^22 + 24482*x^21 + 438771*x^20 - 917742*x^19 - 3954764*x^18 + 12493773*x^17 + 17637390*x^16 - 89943576*x^15 - 16499340*x^14 + 367102377*x^13 - 183437513*x^12 - 838172511*x^11 + 827108970*x^10 + 967374964*x^9 - 1484013339*x^8 - 364627638*x^7 + 1223353523*x^6 - 171844968*x^5 - 419177382*x^4 + 123185675*x^3 + 52146132*x^2 - 17057376*x - 1145024, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 63 x^{25} + 774 x^{24} + 834 x^{23} - 26013 x^{22} + 24482 x^{21} + 438771 x^{20} - 917742 x^{19} - 3954764 x^{18} + 12493773 x^{17} + 17637390 x^{16} - 89943576 x^{15} - 16499340 x^{14} + 367102377 x^{13} - 183437513 x^{12} - 838172511 x^{11} + 827108970 x^{10} + 967374964 x^{9} - 1484013339 x^{8} - 364627638 x^{7} + 1223353523 x^{6} - 171844968 x^{5} - 419177382 x^{4} + 123185675 x^{3} + 52146132 x^{2} - 17057376 x - 1145024 \)

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:  \(25450405558360134867067668541820239712260815470994619689=3^{36}\cdot 7^{18}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $112.74$
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$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(961,·)$, $\chi_{1197}(520,·)$, $\chi_{1197}(1033,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(463,·)$, $\chi_{1197}(400,·)$, $\chi_{1197}(121,·)$, $\chi_{1197}(277,·)$, $\chi_{1197}(919,·)$, $\chi_{1197}(856,·)$, $\chi_{1197}(634,·)$, $\chi_{1197}(862,·)$, $\chi_{1197}(799,·)$, $\chi_{1197}(163,·)$, $\chi_{1197}(676,·)$, $\chi_{1197}(106,·)$, $\chi_{1197}(235,·)$, $\chi_{1197}(172,·)$, $\chi_{1197}(904,·)$, $\chi_{1197}(562,·)$, $\chi_{1197}(1075,·)$, $\chi_{1197}(457,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(58,·)$, $\chi_{1197}(571,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{15} - \frac{1}{4} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{15} + \frac{1}{6} a^{9} + \frac{1}{4} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{12} a^{19} - \frac{1}{12} a^{16} + \frac{1}{6} a^{10} + \frac{1}{4} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{20} - \frac{1}{24} a^{19} - \frac{1}{24} a^{18} + \frac{1}{12} a^{17} + \frac{1}{24} a^{16} - \frac{1}{12} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{6} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{9} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{7}{24} a^{5} + \frac{5}{24} a^{4} + \frac{1}{3} a^{3} + \frac{11}{24} a^{2} + \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{24} a^{21} - \frac{1}{24} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{16} + \frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{3} a^{6} + \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{7}{24} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{24} a^{22} - \frac{1}{24} a^{19} - \frac{1}{24} a^{18} - \frac{1}{8} a^{17} - \frac{1}{12} a^{15} + \frac{1}{12} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{9} + \frac{1}{8} a^{8} - \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{3}{8} a^{5} - \frac{1}{24} a^{4} + \frac{11}{24} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{48} a^{23} - \frac{1}{48} a^{22} + \frac{1}{48} a^{19} - \frac{1}{48} a^{18} + \frac{5}{48} a^{17} - \frac{1}{16} a^{16} + \frac{1}{12} a^{15} - \frac{1}{12} a^{14} + \frac{5}{24} a^{13} + \frac{1}{8} a^{12} + \frac{1}{24} a^{11} - \frac{11}{48} a^{9} + \frac{1}{48} a^{8} + \frac{5}{12} a^{7} + \frac{1}{8} a^{6} - \frac{5}{16} a^{5} - \frac{17}{48} a^{4} - \frac{13}{48} a^{3} + \frac{5}{48} a^{2} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{6096} a^{24} + \frac{13}{2032} a^{23} + \frac{29}{1524} a^{22} + \frac{1}{1016} a^{21} + \frac{33}{2032} a^{20} + \frac{193}{6096} a^{19} - \frac{251}{6096} a^{18} + \frac{253}{2032} a^{17} + \frac{11}{254} a^{16} - \frac{11}{254} a^{15} - \frac{147}{1016} a^{14} - \frac{691}{3048} a^{13} + \frac{517}{3048} a^{12} - \frac{97}{508} a^{11} + \frac{471}{2032} a^{10} + \frac{161}{6096} a^{9} + \frac{83}{508} a^{8} + \frac{647}{1524} a^{7} + \frac{73}{2032} a^{6} - \frac{409}{2032} a^{5} - \frac{833}{6096} a^{4} - \frac{2213}{6096} a^{3} + \frac{17}{127} a^{2} + \frac{9}{127} a + \frac{190}{381}$, $\frac{1}{30480} a^{25} - \frac{131}{15240} a^{23} + \frac{181}{30480} a^{22} - \frac{389}{30480} a^{21} + \frac{71}{15240} a^{20} - \frac{539}{30480} a^{19} + \frac{7}{30480} a^{18} + \frac{1}{40} a^{17} + \frac{3791}{30480} a^{16} - \frac{373}{15240} a^{15} - \frac{413}{1905} a^{14} - \frac{2887}{15240} a^{13} + \frac{359}{7620} a^{12} + \frac{109}{10160} a^{11} - \frac{1057}{15240} a^{10} + \frac{47}{1524} a^{9} - \frac{6919}{30480} a^{8} - \frac{1907}{30480} a^{7} - \frac{967}{3810} a^{6} - \frac{2129}{30480} a^{5} + \frac{10081}{30480} a^{4} - \frac{1381}{3810} a^{3} - \frac{431}{10160} a^{2} + \frac{117}{508} a - \frac{806}{1905}$, $\frac{1}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{26} + \frac{6328700473352100685524858829058371989258497936032443498610049753599924393}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{25} + \frac{2692833215910590915086340445048451466579960870627341162880161286194918929}{38425190286735527427092150394228509529153310475632625279904994124243129296700} a^{24} + \frac{192222662007582242415806262058394207683030482516368469264954841104092610049}{92220456688165265825021160946148422869967945141518300671771985898183510312080} a^{23} - \frac{2485443267299266168521751981799617021348789596170934597577676984254392192263}{230551141720413164562552902365371057174919862853795751679429964745458775780200} a^{22} + \frac{1388456660468552170317943062448447644691889562097811530388849809365415772807}{92220456688165265825021160946148422869967945141518300671771985898183510312080} a^{21} + \frac{321016179724811903623265207476714183100072978680734126154847267551305558909}{153700761146942109708368601576914038116613241902530501119619976496972517186800} a^{20} + \frac{64909336739777640115720783318301912286245972714349744819654885237757209229}{9222045668816526582502116094614842286996794514151830067177198589818351031208} a^{19} + \frac{17662524235919692147947854206202519652454919227694388411501384780425039372633}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{18} - \frac{48285311970241545612639651593611447844650249465428419195676372445649410312573}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{17} - \frac{16879550704757359708177910340560492513925682002032871440026099096509766637041}{153700761146942109708368601576914038116613241902530501119619976496972517186800} a^{16} - \frac{9160927273462760237037894887802320057993561892336957287092349226394734294951}{76850380573471054854184300788457019058306620951265250559809988248486258593400} a^{15} + \frac{8974588980291162311695524595687797812985488657882109391567478583536311559227}{76850380573471054854184300788457019058306620951265250559809988248486258593400} a^{14} + \frac{5304236182874865734134898642948093980964954898935109248323371163374240914939}{76850380573471054854184300788457019058306620951265250559809988248486258593400} a^{13} + \frac{19046142120783688816432770930666624036017080552817784811378334861563547330839}{92220456688165265825021160946148422869967945141518300671771985898183510312080} a^{12} + \frac{6173656199626510018506929442145376436917994586597397117575386107697496947877}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{11} + \frac{17296658540656211962903521792489442149834248897945978771935667043859650671567}{115275570860206582281276451182685528587459931426897875839714982372729387890100} a^{10} + \frac{16030739881858119498258760256751950377208973427279821238501370491652896278977}{153700761146942109708368601576914038116613241902530501119619976496972517186800} a^{9} + \frac{1581122769945230127679838871494043877253542242287338355285405191029808754636}{28818892715051645570319112795671382146864982856724468959928745593182346972525} a^{8} + \frac{114131307286544376921401498452045816261212014207865721510924191507121098915563}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{7} - \frac{114427052590414345368073801620501912410334798311249773220798290975519444050737}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{6} + \frac{21735327187702181873494042134508784625624021333304601479558529513200964078179}{76850380573471054854184300788457019058306620951265250559809988248486258593400} a^{5} - \frac{5225366210846473093000208687155705266656589721628396861960722540663965528779}{92220456688165265825021160946148422869967945141518300671771985898183510312080} a^{4} - \frac{33621630682311529880283476438105982211339644031020968883984814515351822503347}{461102283440826329125105804730742114349839725707591503358859929490917551560400} a^{3} - \frac{73969262095903779643939455663492739003247788561076561906961904677947209523023}{153700761146942109708368601576914038116613241902530501119619976496972517186800} a^{2} - \frac{2922385819238584990755946404588598565295919831030180671870282986165120886941}{28818892715051645570319112795671382146864982856724468959928745593182346972525} a + \frac{1931509170825896247466309763522596530106643871027200446050775709410483051044}{9606297571683881856773037598557127382288327618908156319976248531060782324175}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 8214137232374256000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^3$ (as 27T4):

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 $C_3^3$
Character table for $C_3^3$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.1432809.2, 3.3.17689.1, 3.3.1432809.4, 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), 3.3.1432809.3, 3.3.17689.2, 3.3.1432809.1, 3.3.361.1, 3.3.29241.1, 3.3.29241.2, 9.9.2941473244627851129.12, 9.9.62523502209.1, 9.9.2941473244627851129.5, 9.9.25002110044521.1, 9.9.2941473244627851129.9, 9.9.2941473244627851129.8, 9.9.2941473244627851129.4, 9.9.2941473244627851129.11, 9.9.2941473244627851129.7, 9.9.5534900853769.1, 9.9.2941473244627851129.6, 9.9.2941473244627851129.10, 9.9.2941473244627851129.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$