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Properties

Label 27.27.1108203351...0121.1
Degree $27$
Signature $[27, 0]$
Discriminant $379^{26}$
Root discriminant $304.18$
Ramified prime $379$
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![743723, 46387161, -576163073, -1573415186, 11999475590, 18626907295, -75331044168, -57176007802, 181478462936, -8689190338, -119083238707, 31181666728, 33279114690, -12462752110, -4722897643, 2241789500, 379632386, -227967713, -18161357, 14278105, 525111, -563442, -9406, 13618, 117, -182, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 182*x^25 + 117*x^24 + 13618*x^23 - 9406*x^22 - 563442*x^21 + 525111*x^20 + 14278105*x^19 - 18161357*x^18 - 227967713*x^17 + 379632386*x^16 + 2241789500*x^15 - 4722897643*x^14 - 12462752110*x^13 + 33279114690*x^12 + 31181666728*x^11 - 119083238707*x^10 - 8689190338*x^9 + 181478462936*x^8 - 57176007802*x^7 - 75331044168*x^6 + 18626907295*x^5 + 11999475590*x^4 - 1573415186*x^3 - 576163073*x^2 + 46387161*x + 743723)
 
gp: K = bnfinit(x^27 - x^26 - 182*x^25 + 117*x^24 + 13618*x^23 - 9406*x^22 - 563442*x^21 + 525111*x^20 + 14278105*x^19 - 18161357*x^18 - 227967713*x^17 + 379632386*x^16 + 2241789500*x^15 - 4722897643*x^14 - 12462752110*x^13 + 33279114690*x^12 + 31181666728*x^11 - 119083238707*x^10 - 8689190338*x^9 + 181478462936*x^8 - 57176007802*x^7 - 75331044168*x^6 + 18626907295*x^5 + 11999475590*x^4 - 1573415186*x^3 - 576163073*x^2 + 46387161*x + 743723, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 182 x^{25} + 117 x^{24} + 13618 x^{23} - 9406 x^{22} - 563442 x^{21} + 525111 x^{20} + 14278105 x^{19} - 18161357 x^{18} - 227967713 x^{17} + 379632386 x^{16} + 2241789500 x^{15} - 4722897643 x^{14} - 12462752110 x^{13} + 33279114690 x^{12} + 31181666728 x^{11} - 119083238707 x^{10} - 8689190338 x^{9} + 181478462936 x^{8} - 57176007802 x^{7} - 75331044168 x^{6} + 18626907295 x^{5} + 11999475590 x^{4} - 1573415186 x^{3} - 576163073 x^{2} + 46387161 x + 743723 \)

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:  \(11082033513026159843197131857404812003970296978813351756002427720121=379^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $304.18$
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:  $379$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(379\)
Dirichlet character group:    $\lbrace$$\chi_{379}(1,·)$, $\chi_{379}(107,·)$, $\chi_{379}(197,·)$, $\chi_{379}(193,·)$, $\chi_{379}(87,·)$, $\chi_{379}(268,·)$, $\chi_{379}(271,·)$, $\chi_{379}(115,·)$, $\chi_{379}(339,·)$, $\chi_{379}(84,·)$, $\chi_{379}(213,·)$, $\chi_{379}(151,·)$, $\chi_{379}(24,·)$, $\chi_{379}(79,·)$, $\chi_{379}(294,·)$, $\chi_{379}(234,·)$, $\chi_{379}(327,·)$, $\chi_{379}(239,·)$, $\chi_{379}(368,·)$, $\chi_{379}(177,·)$, $\chi_{379}(51,·)$, $\chi_{379}(180,·)$, $\chi_{379}(310,·)$, $\chi_{379}(121,·)$, $\chi_{379}(185,·)$, $\chi_{379}(251,·)$, $\chi_{379}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{1644343} a^{25} + \frac{285392}{1644343} a^{24} + \frac{418997}{1644343} a^{23} + \frac{168997}{1644343} a^{22} - \frac{351984}{1644343} a^{21} + \frac{650417}{1644343} a^{20} - \frac{350047}{1644343} a^{19} - \frac{366452}{1644343} a^{18} - \frac{612969}{1644343} a^{17} + \frac{229079}{1644343} a^{16} + \frac{752260}{1644343} a^{15} + \frac{803280}{1644343} a^{14} - \frac{528441}{1644343} a^{13} - \frac{424053}{1644343} a^{12} - \frac{5151}{1644343} a^{11} - \frac{556278}{1644343} a^{10} - \frac{616106}{1644343} a^{9} - \frac{95058}{1644343} a^{8} - \frac{544618}{1644343} a^{7} - \frac{345408}{1644343} a^{6} - \frac{380487}{1644343} a^{5} - \frac{357566}{1644343} a^{4} - \frac{326459}{1644343} a^{3} + \frac{54545}{1644343} a^{2} + \frac{724040}{1644343} a + \frac{643601}{1644343}$, $\frac{1}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{26} + \frac{1458397269910909849747390366438531924255575060102568753334347016097421150985988201231662060786694132637215}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{25} - \frac{4485456882438848765965477690852605773108049511806486823909770129283128614798266131830658998394324082944725359566}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{24} - \frac{981906382867142106721983216630624907108841123251578671411940260157825856848394155337815189194432335325403270498}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{23} + \frac{1402576220449004816792015674430260906459553374772271199386788635112419363996040957647735486445051047058640137608}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{22} - \frac{4258117810354197935566416252542477117299858692538532123155297121709026634726225229315592934466509854221696312814}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{21} - \frac{888322801472192033792290769694127537973080140613905180408176974965825612370957008697426447266015393669514575423}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{20} + \frac{4359255797419501543381279787178190485876213324476866710754903483530978429208040359744773466661790010282848741727}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{19} - \frac{4572042581248360416646472764751202267670092706007995568953680193550818451543478473243634157089813449177828859254}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{18} + \frac{3154677079537835402431520133127397153724334084892852936167790327805422667101707278628449141043508929316542135511}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{17} + \frac{424109828959056352973431724913671558824679341186619110025592222511116863279010591213607087533084236448778737393}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{16} - \frac{2423409373972688667233513505947177939268256165224898132645583596300721529396450715549421670212208721109033765944}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{15} - \frac{4421796752777721433929243654083180494041775004669596913995566488409543082946444748239903599816746148692826131334}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{14} - \frac{3993637041254019275253424750403839466823815253754221276615635167150323830620121181706780802073352055503970408868}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{13} - \frac{1092125242001768321098346903671239145687863859844353735462554330516238728016089064187042141368213199798983880423}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{12} - \frac{3808308654804855422463336136053717126968753331401268807424207551253938716244707478806255261662431879936401321949}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{11} - \frac{3424335587944851688575719590706302897305058062724940438338171011462619862535136175769969753798411446707697515384}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{10} + \frac{1055849552766577625247988025922645366805187851764510211654343003886938467060214578830532187759080360532542568055}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{9} + \frac{2587124334656382138093622744440077819548277041097226938269935466342268339939932871306161099155343099049893304466}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{8} + \frac{816385945169506239812159514588476024498869652215808478115962084521238961492871013944680172576162917299029792367}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{7} + \frac{1336304842062600098811910300888216715283568766678365328514650341321858043963674626418051939137048643912240463156}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{6} + \frac{4664283476887947199230172986509290404476482830086150685201355300980617009004611243270933521251251305117041946811}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{5} - \frac{2900281402481214447330182187375236353052529047053300692418874503762111566923379354947623265288438643052482919175}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{4} + \frac{1117718367655391570262353783795855133816925227415604464883339665578284621175818745482482639056500151234228750805}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{3} - \frac{96955927132638882897400472107786759933182928303846487894856586648459478452605481924491354262024836876610013752}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a^{2} - \frac{1862194079134591859378432557218636011403307825296095404046160331880806344308747563278490877376044464166926291198}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037} a + \frac{757010107300000319633699348141949699176317415061983696588490843909263306367878930894449846162616180156255254566}{9574948526058979840283468959867935119927547199892452315774801562515778484324983430401990606361135526553747196037}$

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:  \( 6288288434413969000000000 \) (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

3.3.143641.1, 9.9.425709831200577608161.1

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 $27$ $27$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ $27$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
379Data not computed