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Properties

Label 27.9.162...375.1
Degree $27$
Signature $[9, 9]$
Discriminant $-1.630\times 10^{39}$
Root discriminant $28.33$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_9\times S_3$ (as 27T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^25 + 54*x^23 - 63*x^22 - 267*x^21 + 513*x^20 + 468*x^19 - 981*x^18 - 972*x^17 - 1107*x^16 + 9846*x^15 - 4860*x^14 - 24624*x^13 + 41403*x^12 - 9909*x^11 - 43317*x^10 + 67377*x^9 - 45441*x^8 - 4905*x^7 + 46368*x^6 - 47520*x^5 + 19584*x^4 + 1146*x^3 - 3483*x^2 + 594*x + 107)
 
gp: K = bnfinit(x^27 - 9*x^25 + 54*x^23 - 63*x^22 - 267*x^21 + 513*x^20 + 468*x^19 - 981*x^18 - 972*x^17 - 1107*x^16 + 9846*x^15 - 4860*x^14 - 24624*x^13 + 41403*x^12 - 9909*x^11 - 43317*x^10 + 67377*x^9 - 45441*x^8 - 4905*x^7 + 46368*x^6 - 47520*x^5 + 19584*x^4 + 1146*x^3 - 3483*x^2 + 594*x + 107, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![107, 594, -3483, 1146, 19584, -47520, 46368, -4905, -45441, 67377, -43317, -9909, 41403, -24624, -4860, 9846, -1107, -972, -981, 468, 513, -267, -63, 54, 0, -9, 0, 1]);
 

\( x^{27} - 9 x^{25} + 54 x^{23} - 63 x^{22} - 267 x^{21} + 513 x^{20} + 468 x^{19} - 981 x^{18} - 972 x^{17} - 1107 x^{16} + 9846 x^{15} - 4860 x^{14} - 24624 x^{13} + 41403 x^{12} - 9909 x^{11} - 43317 x^{10} + 67377 x^{9} - 45441 x^{8} - 4905 x^{7} + 46368 x^{6} - 47520 x^{5} + 19584 x^{4} + 1146 x^{3} - 3483 x^{2} + 594 x + 107 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[9, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1629658531896641667523158845107974609375\)\(\medspace = -\,3^{69}\cdot 5^{9}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $28.33$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $9$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{29263810769911153046352889049466687356254874979839618} a^{26} + \frac{3419550329809412100880078169440009637214349803917056}{14631905384955576523176444524733343678127437489919809} a^{25} - \frac{2892624569425075524006339116503772507303244459906005}{14631905384955576523176444524733343678127437489919809} a^{24} + \frac{1726836209005783767240250356164171668536551921176999}{29263810769911153046352889049466687356254874979839618} a^{23} + \frac{4178775804803567919164203985543911742617299816248643}{29263810769911153046352889049466687356254874979839618} a^{22} + \frac{296160998104678823710257757609644828162161956208527}{29263810769911153046352889049466687356254874979839618} a^{21} - \frac{874745324455829679374943934805293566085750136428379}{29263810769911153046352889049466687356254874979839618} a^{20} + \frac{2741346013697942828644213103460334809263173790707268}{14631905384955576523176444524733343678127437489919809} a^{19} + \frac{3294680994200639642734725690102078100953253411135604}{14631905384955576523176444524733343678127437489919809} a^{18} + \frac{4772576373462896971772979056012890941455821967325978}{14631905384955576523176444524733343678127437489919809} a^{17} + \frac{6603539202718690322459573745740364891415815153420057}{29263810769911153046352889049466687356254874979839618} a^{16} + \frac{3387446350092365861575450016791555378421980120866244}{14631905384955576523176444524733343678127437489919809} a^{15} - \frac{10247430567154000755426291522006859365991615321672345}{29263810769911153046352889049466687356254874979839618} a^{14} - \frac{14271935746968336440094101569245943657751398375098783}{29263810769911153046352889049466687356254874979839618} a^{13} - \frac{1172658153356232313508158720486346525885381870852431}{14631905384955576523176444524733343678127437489919809} a^{12} - \frac{10504118014573427210986397984904277803357338502995883}{29263810769911153046352889049466687356254874979839618} a^{11} + \frac{11479076854800872950744103875848556992599464310389021}{29263810769911153046352889049466687356254874979839618} a^{10} - \frac{6239268676769101192492088501209074012937355453862325}{29263810769911153046352889049466687356254874979839618} a^{9} - \frac{7235190757258376429296280770016964135803707843916607}{29263810769911153046352889049466687356254874979839618} a^{8} + \frac{9067373601357130814120087736413392942777374591863563}{29263810769911153046352889049466687356254874979839618} a^{7} - \frac{5220173301891205313706987805229276017888684889636756}{14631905384955576523176444524733343678127437489919809} a^{6} - \frac{4952830302607840292039676049581550051330320816885637}{29263810769911153046352889049466687356254874979839618} a^{5} + \frac{3537759033787507736198931889313954435716808732012290}{14631905384955576523176444524733343678127437489919809} a^{4} - \frac{1608666662883179873754072355649383721794580194608110}{14631905384955576523176444524733343678127437489919809} a^{3} + \frac{11702908843437837050799236432376631631822647079190721}{29263810769911153046352889049466687356254874979839618} a^{2} - \frac{6109599231982349451426351220820722163109110060433765}{14631905384955576523176444524733343678127437489919809} a - \frac{4827344544412023449872620756210775991941492214695481}{29263810769911153046352889049466687356254874979839618}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2153441387.4033856 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{9}\cdot 2153441387.4033856 \cdot 1}{2\sqrt{1629658531896641667523158845107974609375}}\approx 0.208421985481075$ (assuming GRH)

Galois group

$C_9\times S_3$ (as 27T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_9\times S_3$
Character table for $C_9\times S_3$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.1.135.1, \(\Q(\zeta_{27})^+\), 9.3.1793613375.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed

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 R $18{,}\,{\href{/LocalNumberField/7.9.0.1}{9} }$ $18{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }$ $18{,}\,{\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $18{,}\,{\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ $18{,}\,{\href{/LocalNumberField/41.9.0.1}{9} }$ $18{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ $18{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.15.2t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\sqrt{-15}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.45.6t1.b.a$1$ $ 3^{2} \cdot 5 $ 6.0.2460375.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.b.b$1$ $ 3^{2} \cdot 5 $ 6.0.2460375.1 $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.135.18t1.b.a$1$ $ 3^{3} \cdot 5 $ 18.0.5770142004982097067662109375.1 $C_{18}$ (as 18T1) $0$ $-1$
1.135.18t1.b.b$1$ $ 3^{3} \cdot 5 $ 18.0.5770142004982097067662109375.1 $C_{18}$ (as 18T1) $0$ $-1$
1.135.18t1.b.c$1$ $ 3^{3} \cdot 5 $ 18.0.5770142004982097067662109375.1 $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.a$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.135.18t1.b.d$1$ $ 3^{3} \cdot 5 $ 18.0.5770142004982097067662109375.1 $C_{18}$ (as 18T1) $0$ $-1$
1.135.18t1.b.e$1$ $ 3^{3} \cdot 5 $ 18.0.5770142004982097067662109375.1 $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.e$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.135.18t1.b.f$1$ $ 3^{3} \cdot 5 $ 18.0.5770142004982097067662109375.1 $C_{18}$ (as 18T1) $0$ $-1$
* 2.135.3t2.b.a$2$ $ 3^{3} \cdot 5 $ 3.1.135.1 $S_3$ (as 3T2) $1$ $0$
* 2.405.6t5.a.a$2$ $ 3^{4} \cdot 5 $ 6.0.2460375.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.405.6t5.a.b$2$ $ 3^{4} \cdot 5 $ 6.0.2460375.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3645.18t16.a.a$2$ $ 3^{6} \cdot 5 $ 27.9.1629658531896641667523158845107974609375.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.3645.18t16.a.b$2$ $ 3^{6} \cdot 5 $ 27.9.1629658531896641667523158845107974609375.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.3645.18t16.a.c$2$ $ 3^{6} \cdot 5 $ 27.9.1629658531896641667523158845107974609375.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.3645.18t16.a.d$2$ $ 3^{6} \cdot 5 $ 27.9.1629658531896641667523158845107974609375.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.3645.18t16.a.e$2$ $ 3^{6} \cdot 5 $ 27.9.1629658531896641667523158845107974609375.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.3645.18t16.a.f$2$ $ 3^{6} \cdot 5 $ 27.9.1629658531896641667523158845107974609375.1 $C_9\times S_3$ (as 27T12) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.