Properties

Label 27.9.656...256.1
Degree $27$
Signature $[9, 9]$
Discriminant $-6.562\times 10^{38}$
Root discriminant $27.39$
Ramified primes $2, 3$
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^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*x - 1)
 
gp: K = bnfinit(x^27 - 9*x^26 + 27*x^25 - 24*x^24 - 9*x^23 - 63*x^22 + 273*x^21 + 144*x^20 - 2376*x^19 + 4050*x^18 - 396*x^17 - 3465*x^16 - 141*x^15 + 441*x^14 + 17019*x^13 - 33225*x^12 + 12852*x^11 + 17379*x^10 - 14277*x^9 + 1188*x^8 + 1143*x^7 - 501*x^6 + 891*x^5 + 216*x^4 + 129*x^3 + 18*x^2 + 9*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 9, 18, 129, 216, 891, -501, 1143, 1188, -14277, 17379, 12852, -33225, 17019, 441, -141, -3465, -396, 4050, -2376, 144, 273, -63, -9, -24, 27, -9, 1]);
 

\( x^{27} - 9 x^{26} + 27 x^{25} - 24 x^{24} - 9 x^{23} - 63 x^{22} + 273 x^{21} + 144 x^{20} - 2376 x^{19} + 4050 x^{18} - 396 x^{17} - 3465 x^{16} - 141 x^{15} + 441 x^{14} + 17019 x^{13} - 33225 x^{12} + 12852 x^{11} + 17379 x^{10} - 14277 x^{9} + 1188 x^{8} + 1143 x^{7} - 501 x^{6} + 891 x^{5} + 216 x^{4} + 129 x^{3} + 18 x^{2} + 9 x - 1 \)

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:  \(-656187196700948326335269302720488800256\)\(\medspace = -\,2^{18}\cdot 3^{70}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.39$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{53} a^{21} - \frac{22}{53} a^{20} + \frac{15}{53} a^{19} - \frac{22}{53} a^{18} + \frac{18}{53} a^{17} + \frac{8}{53} a^{16} - \frac{11}{53} a^{15} + \frac{3}{53} a^{14} + \frac{26}{53} a^{13} - \frac{6}{53} a^{12} - \frac{16}{53} a^{11} + \frac{13}{53} a^{10} - \frac{8}{53} a^{9} + \frac{8}{53} a^{8} + \frac{7}{53} a^{7} - \frac{8}{53} a^{6} - \frac{13}{53} a^{5} + \frac{7}{53} a^{4} + \frac{24}{53} a^{3} + \frac{25}{53} a^{2} - \frac{26}{53} a + \frac{10}{53}$, $\frac{1}{53} a^{22} + \frac{8}{53} a^{20} - \frac{10}{53} a^{19} + \frac{11}{53} a^{18} - \frac{20}{53} a^{17} + \frac{6}{53} a^{16} + \frac{26}{53} a^{15} - \frac{14}{53} a^{14} - \frac{17}{53} a^{13} + \frac{11}{53} a^{12} - \frac{21}{53} a^{11} + \frac{13}{53} a^{10} - \frac{9}{53} a^{9} + \frac{24}{53} a^{8} - \frac{13}{53} a^{7} + \frac{23}{53} a^{6} - \frac{14}{53} a^{5} + \frac{19}{53} a^{4} + \frac{23}{53} a^{3} - \frac{6}{53} a^{2} + \frac{21}{53} a + \frac{8}{53}$, $\frac{1}{53} a^{23} + \frac{7}{53} a^{20} - \frac{3}{53} a^{19} - \frac{3}{53} a^{18} + \frac{21}{53} a^{17} + \frac{15}{53} a^{16} + \frac{21}{53} a^{15} + \frac{12}{53} a^{14} + \frac{15}{53} a^{13} - \frac{26}{53} a^{12} - \frac{18}{53} a^{11} - \frac{7}{53} a^{10} - \frac{18}{53} a^{9} - \frac{24}{53} a^{8} + \frac{20}{53} a^{7} - \frac{3}{53} a^{6} + \frac{17}{53} a^{5} + \frac{20}{53} a^{4} + \frac{14}{53} a^{3} - \frac{20}{53} a^{2} + \frac{4}{53} a + \frac{26}{53}$, $\frac{1}{53} a^{24} - \frac{8}{53} a^{20} - \frac{2}{53} a^{19} + \frac{16}{53} a^{18} - \frac{5}{53} a^{17} + \frac{18}{53} a^{16} - \frac{17}{53} a^{15} - \frac{6}{53} a^{14} + \frac{4}{53} a^{13} + \frac{24}{53} a^{12} - \frac{1}{53} a^{11} - \frac{3}{53} a^{10} - \frac{21}{53} a^{9} + \frac{17}{53} a^{8} + \frac{1}{53} a^{7} + \frac{20}{53} a^{6} + \frac{5}{53} a^{5} + \frac{18}{53} a^{4} + \frac{24}{53} a^{3} - \frac{12}{53} a^{2} - \frac{4}{53} a - \frac{17}{53}$, $\frac{1}{2809} a^{25} + \frac{15}{2809} a^{24} + \frac{22}{2809} a^{23} + \frac{11}{2809} a^{22} + \frac{16}{2809} a^{21} - \frac{408}{2809} a^{20} + \frac{1071}{2809} a^{19} + \frac{345}{2809} a^{18} + \frac{299}{2809} a^{17} + \frac{1159}{2809} a^{16} + \frac{488}{2809} a^{15} - \frac{911}{2809} a^{14} - \frac{209}{2809} a^{13} - \frac{1243}{2809} a^{12} + \frac{1356}{2809} a^{11} + \frac{1401}{2809} a^{10} + \frac{1400}{2809} a^{9} - \frac{1088}{2809} a^{8} + \frac{606}{2809} a^{7} + \frac{1201}{2809} a^{6} - \frac{264}{2809} a^{5} - \frac{638}{2809} a^{4} - \frac{635}{2809} a^{3} + \frac{334}{2809} a^{2} + \frac{254}{2809} a + \frac{645}{2809}$, $\frac{1}{65896664597735887461174969209118266060657749} a^{26} + \frac{143008972600660223213184923875890341471}{1243333294296903537003301305832420114352033} a^{25} + \frac{98937767298118770315430508659985381857149}{65896664597735887461174969209118266060657749} a^{24} - \frac{445532342683893184439900723383960564633012}{65896664597735887461174969209118266060657749} a^{23} - \frac{370377643251772573642631165520404183715600}{65896664597735887461174969209118266060657749} a^{22} - \frac{184782597662731615130752156656480000594559}{65896664597735887461174969209118266060657749} a^{21} - \frac{19972362014499540137427781715882055294349142}{65896664597735887461174969209118266060657749} a^{20} + \frac{21158661811177703206793217786197890303077456}{65896664597735887461174969209118266060657749} a^{19} + \frac{477055264801843864777087344572346487536862}{1243333294296903537003301305832420114352033} a^{18} - \frac{23816509907664820400619379380657056721507789}{65896664597735887461174969209118266060657749} a^{17} + \frac{2279712015678631052538098868324032275647356}{65896664597735887461174969209118266060657749} a^{16} + \frac{14618558009363228329025863262523700485184090}{65896664597735887461174969209118266060657749} a^{15} - \frac{540775973146362679678705069877684941231121}{65896664597735887461174969209118266060657749} a^{14} + \frac{12403503436229389244632511377905379026230282}{65896664597735887461174969209118266060657749} a^{13} - \frac{20592207325656271277278953040624292953002164}{65896664597735887461174969209118266060657749} a^{12} + \frac{80958367482002198677584767932072255149575}{65896664597735887461174969209118266060657749} a^{11} + \frac{30972106181239185759135492603905787831573871}{65896664597735887461174969209118266060657749} a^{10} + \frac{19296328746221887759117750662763255271689148}{65896664597735887461174969209118266060657749} a^{9} + \frac{12903279159641427379172911364572608816914591}{65896664597735887461174969209118266060657749} a^{8} + \frac{5690276212733927178315941151705724780989691}{65896664597735887461174969209118266060657749} a^{7} - \frac{16484525559635234065278062839830749190093268}{65896664597735887461174969209118266060657749} a^{6} + \frac{10978967509218101585242264652913898052598589}{65896664597735887461174969209118266060657749} a^{5} - \frac{21341645168593815592705501522990010568651821}{65896664597735887461174969209118266060657749} a^{4} - \frac{22675843309400403978385517982030689205691526}{65896664597735887461174969209118266060657749} a^{3} - \frac{17512844860034837174804323447672564887307014}{65896664597735887461174969209118266060657749} a^{2} + \frac{29032518862234978030086022539451317341569053}{65896664597735887461174969209118266060657749} a + \frac{24969359196603763875369549931725961414814802}{65896664597735887461174969209118266060657749}$

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:  \( 1816757010.3045344 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{9}\cdot 1816757010.3045344 \cdot 1}{2\sqrt{656187196700948326335269302720488800256}}\approx 0.277103332705435$ (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

\(\Q(\zeta_{9})^+\), 3.1.324.1, \(\Q(\zeta_{27})^+\), 9.3.2754990144.1

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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ $18{,}\,{\href{/LocalNumberField/7.9.0.1}{9} }$ $18{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ $18{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $18{,}\,{\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $18{,}\,{\href{/LocalNumberField/43.9.0.1}{9} }$ $18{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ $18{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$

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
2Data not computed
3Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.36.6t1.b.a$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.36.6t1.b.b$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.27.9t1.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) $0$ $1$
1.108.18t1.a.a$1$ $ 2^{2} \cdot 3^{3}$ 18.0.258151783382020583032356864.7 $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.108.18t1.a.b$1$ $ 2^{2} \cdot 3^{3}$ 18.0.258151783382020583032356864.7 $C_{18}$ (as 18T1) $0$ $-1$
1.108.18t1.a.c$1$ $ 2^{2} \cdot 3^{3}$ 18.0.258151783382020583032356864.7 $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.108.18t1.a.d$1$ $ 2^{2} \cdot 3^{3}$ 18.0.258151783382020583032356864.7 $C_{18}$ (as 18T1) $0$ $-1$
* 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) $0$ $1$
1.108.18t1.a.e$1$ $ 2^{2} \cdot 3^{3}$ 18.0.258151783382020583032356864.7 $C_{18}$ (as 18T1) $0$ $-1$
1.108.18t1.a.f$1$ $ 2^{2} \cdot 3^{3}$ 18.0.258151783382020583032356864.7 $C_{18}$ (as 18T1) $0$ $-1$
* 2.324.3t2.b.a$2$ $ 2^{2} \cdot 3^{4}$ 3.1.324.1 $S_3$ (as 3T2) $1$ $0$
* 2.324.6t5.d.a$2$ $ 2^{2} \cdot 3^{4}$ 6.0.419904.3 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.324.6t5.d.b$2$ $ 2^{2} \cdot 3^{4}$ 6.0.419904.3 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2916.18t16.a.a$2$ $ 2^{2} \cdot 3^{6}$ 27.9.656187196700948326335269302720488800256.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.a.b$2$ $ 2^{2} \cdot 3^{6}$ 27.9.656187196700948326335269302720488800256.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.a.c$2$ $ 2^{2} \cdot 3^{6}$ 27.9.656187196700948326335269302720488800256.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.a.d$2$ $ 2^{2} \cdot 3^{6}$ 27.9.656187196700948326335269302720488800256.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.a.e$2$ $ 2^{2} \cdot 3^{6}$ 27.9.656187196700948326335269302720488800256.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.a.f$2$ $ 2^{2} \cdot 3^{6}$ 27.9.656187196700948326335269302720488800256.1 $C_9\times S_3$ (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 *.