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Properties

Label 27.27.3217863581...5889.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 19^{18}$
Root discriminant $104.42$
Ramified primes $3, 19$
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![-105461, -411264, 3255948, 5399355, -28758384, -21421602, 115889073, 22047264, -240759207, 44503222, 261338922, -120638421, -138866430, 100880532, 30109320, -38801514, 647631, 7506369, -1345159, -738450, 229473, 31872, -17460, 27, 636, -45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 45*x^25 + 636*x^24 + 27*x^23 - 17460*x^22 + 31872*x^21 + 229473*x^20 - 738450*x^19 - 1345159*x^18 + 7506369*x^17 + 647631*x^16 - 38801514*x^15 + 30109320*x^14 + 100880532*x^13 - 138866430*x^12 - 120638421*x^11 + 261338922*x^10 + 44503222*x^9 - 240759207*x^8 + 22047264*x^7 + 115889073*x^6 - 21421602*x^5 - 28758384*x^4 + 5399355*x^3 + 3255948*x^2 - 411264*x - 105461)
 
gp: K = bnfinit(x^27 - 9*x^26 - 45*x^25 + 636*x^24 + 27*x^23 - 17460*x^22 + 31872*x^21 + 229473*x^20 - 738450*x^19 - 1345159*x^18 + 7506369*x^17 + 647631*x^16 - 38801514*x^15 + 30109320*x^14 + 100880532*x^13 - 138866430*x^12 - 120638421*x^11 + 261338922*x^10 + 44503222*x^9 - 240759207*x^8 + 22047264*x^7 + 115889073*x^6 - 21421602*x^5 - 28758384*x^4 + 5399355*x^3 + 3255948*x^2 - 411264*x - 105461, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 45 x^{25} + 636 x^{24} + 27 x^{23} - 17460 x^{22} + 31872 x^{21} + 229473 x^{20} - 738450 x^{19} - 1345159 x^{18} + 7506369 x^{17} + 647631 x^{16} - 38801514 x^{15} + 30109320 x^{14} + 100880532 x^{13} - 138866430 x^{12} - 120638421 x^{11} + 261338922 x^{10} + 44503222 x^{9} - 240759207 x^{8} + 22047264 x^{7} + 115889073 x^{6} - 21421602 x^{5} - 28758384 x^{4} + 5399355 x^{3} + 3255948 x^{2} - 411264 x - 105461 \)

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:  \(3217863581710817038235175421508758764893959268723195889=3^{66}\cdot 19^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.42$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(513=3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{513}(448,·)$, $\chi_{513}(1,·)$, $\chi_{513}(235,·)$, $\chi_{513}(7,·)$, $\chi_{513}(64,·)$, $\chi_{513}(457,·)$, $\chi_{513}(334,·)$, $\chi_{513}(463,·)$, $\chi_{513}(400,·)$, $\chi_{513}(277,·)$, $\chi_{513}(406,·)$, $\chi_{513}(343,·)$, $\chi_{513}(220,·)$, $\chi_{513}(349,·)$, $\chi_{513}(286,·)$, $\chi_{513}(163,·)$, $\chi_{513}(292,·)$, $\chi_{513}(229,·)$, $\chi_{513}(106,·)$, $\chi_{513}(391,·)$, $\chi_{513}(172,·)$, $\chi_{513}(49,·)$, $\chi_{513}(178,·)$, $\chi_{513}(115,·)$, $\chi_{513}(121,·)$, $\chi_{513}(505,·)$, $\chi_{513}(58,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{17} a^{24} - \frac{8}{17} a^{23} - \frac{2}{17} a^{22} + \frac{1}{17} a^{21} - \frac{8}{17} a^{20} - \frac{1}{17} a^{19} - \frac{8}{17} a^{18} - \frac{3}{17} a^{17} - \frac{3}{17} a^{16} - \frac{5}{17} a^{15} - \frac{8}{17} a^{14} + \frac{3}{17} a^{13} + \frac{6}{17} a^{12} - \frac{5}{17} a^{11} - \frac{1}{17} a^{10} + \frac{8}{17} a^{8} + \frac{8}{17} a^{7} + \frac{1}{17} a^{6} - \frac{2}{17} a^{5} - \frac{2}{17} a^{4} - \frac{1}{17} a^{3} + \frac{3}{17} a^{2} + \frac{6}{17}$, $\frac{1}{2771} a^{25} + \frac{6}{2771} a^{24} + \frac{651}{2771} a^{23} - \frac{1285}{2771} a^{22} - \frac{946}{2771} a^{21} - \frac{1065}{2771} a^{20} + \frac{896}{2771} a^{19} - \frac{30}{2771} a^{18} - \frac{487}{2771} a^{17} - \frac{1118}{2771} a^{16} - \frac{180}{2771} a^{15} - \frac{466}{2771} a^{14} + \frac{133}{2771} a^{13} - \frac{244}{2771} a^{12} - \frac{3}{2771} a^{11} + \frac{700}{2771} a^{10} - \frac{774}{2771} a^{9} - \frac{1223}{2771} a^{8} + \frac{1269}{2771} a^{7} - \frac{396}{2771} a^{6} - \frac{1186}{2771} a^{5} - \frac{1219}{2771} a^{4} + \frac{1247}{2771} a^{3} - \frac{332}{2771} a^{2} - \frac{96}{2771} a + \frac{5}{17}$, $\frac{1}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{26} - \frac{21652408514035241556965620353659214046965672934874425142775794706505711538531}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{25} - \frac{15733136826991925562975680426309417495232430443878145180986688122971367677880194}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{24} + \frac{444978875302578699483951811629755046717289771279486048448327686237747112549544073}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{23} + \frac{2757111758684672328793457393206491866734843804200497276623106828453754016623622}{106770781899043402933352938247398071457648161768202177859455150136142433346490949} a^{22} + \frac{482648774689029301109510294500409246249046178416268149466850047379818979484284627}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{21} + \frac{162099145332215271913049690936185827769057198819749234741740163887021263321932590}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{20} - \frac{261376383146347838232970393110641415002981940698988630663053244362681834429029578}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{19} + \frac{197478763819534019758629934074390755110716112894171980903172663219247444615538935}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{18} + \frac{727321625793393487545636474356998240645492676182584835023157151171352280562014404}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{17} + \frac{311104180907471523254061562005803487192393909841662727196948624628672591555117704}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{16} - \frac{422755160098614679812291806476880718897722459900741850270461895406489192955347829}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{15} - \frac{41899619706448175343796363978400153015278116509704045073722149220184465174072190}{106770781899043402933352938247398071457648161768202177859455150136142433346490949} a^{14} + \frac{892225577009644792399706193010286837442906625363639843403224643041166241421164303}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{13} - \frac{631288385677401312466719386336753913336371373403525554115875762116200297546176620}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{12} - \frac{498304805663822692048679676935837937001551008720889902576102150530975346714751889}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{11} - \frac{36283759661216183297433863140763380378993561009446195115868232790125845630594195}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{10} - \frac{737968088908223746854220159530891550619810032452650705783711770485820661948871934}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{9} + \frac{26160612885372726543391817200967469172862712382051830194591006838247810458033193}{106770781899043402933352938247398071457648161768202177859455150136142433346490949} a^{8} - \frac{63248929067739711229662991420237931327354551907156559550795923987869792284053715}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{7} - \frac{93504532177285643512834113527161886626501133012363990594964120815839195015727545}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{6} + \frac{26527043932765964812630875504672580424787294351336449275504163157805317537243032}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{5} - \frac{597941550456394281572664823557293968749793374490314126112323441008375481984148646}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a^{4} + \frac{656168800078360871881260838425812897169235426431420903580824794049824126491911}{16963582170876054671654205142109973969906717290275112370193808900134779129816319} a^{3} + \frac{14780707778370620071293025657751785088471241555466008881880644207968202381812515}{106770781899043402933352938247398071457648161768202177859455150136142433346490949} a^{2} + \frac{630570139498402452493141981705701466538389343068949222216270460318372978959702614}{1815103292283737849866999950205767214780018750059437023610737552314421366890346133} a - \frac{3314452848789428718608812866575118627074824728641880088912210217783601329843078}{11135603020145630980779140798808387820736311350057895850372623020333873416505191}$

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:  \( 4236617885729581600 \) (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

3.3.29241.2, \(\Q(\zeta_{9})^+\), 3.3.361.1, 3.3.29241.1, 9.9.25002110044521.1, 9.9.1476349596018920529.1, \(\Q(\zeta_{27})^+\), 9.9.1476349596018920529.2

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$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}$