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Properties

Label 27.1.476...207.1
Degree $27$
Signature $[1, 13]$
Discriminant $-4.767\times 10^{41}$
Root discriminant $34.97$
Ramified prime $1607$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 11*x^25 + 17*x^24 + 38*x^23 + x^22 + 157*x^21 + 384*x^20 + 124*x^19 - 139*x^18 + 318*x^17 + 1284*x^16 + 1757*x^15 - 73*x^14 - 3289*x^13 - 3942*x^12 - 2202*x^11 - 2459*x^10 - 3813*x^9 - 1971*x^8 + 2174*x^7 + 7281*x^6 + 9348*x^5 + 4444*x^4 - 209*x^3 - 325*x^2 + 132*x + 1)
 
gp: K = bnfinit(x^27 - 3*x^26 - 11*x^25 + 17*x^24 + 38*x^23 + x^22 + 157*x^21 + 384*x^20 + 124*x^19 - 139*x^18 + 318*x^17 + 1284*x^16 + 1757*x^15 - 73*x^14 - 3289*x^13 - 3942*x^12 - 2202*x^11 - 2459*x^10 - 3813*x^9 - 1971*x^8 + 2174*x^7 + 7281*x^6 + 9348*x^5 + 4444*x^4 - 209*x^3 - 325*x^2 + 132*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 132, -325, -209, 4444, 9348, 7281, 2174, -1971, -3813, -2459, -2202, -3942, -3289, -73, 1757, 1284, 318, -139, 124, 384, 157, 1, 38, 17, -11, -3, 1]);
 

\( x^{27} - 3 x^{26} - 11 x^{25} + 17 x^{24} + 38 x^{23} + x^{22} + 157 x^{21} + 384 x^{20} + 124 x^{19} - 139 x^{18} + 318 x^{17} + 1284 x^{16} + 1757 x^{15} - 73 x^{14} - 3289 x^{13} - 3942 x^{12} - 2202 x^{11} - 2459 x^{10} - 3813 x^{9} - 1971 x^{8} + 2174 x^{7} + 7281 x^{6} + 9348 x^{5} + 4444 x^{4} - 209 x^{3} - 325 x^{2} + 132 x + 1 \)

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

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-476657463863730234951616960570180249031207\)\(\medspace = -\,1607^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.97$
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:  $1607$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{17} + \frac{2}{5} a^{16} + \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{20} - \frac{1}{5} a^{17} + \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{1}{5} a^{8} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{35} a^{21} + \frac{3}{35} a^{20} + \frac{3}{35} a^{19} - \frac{2}{35} a^{18} + \frac{6}{35} a^{17} - \frac{12}{35} a^{16} - \frac{1}{7} a^{15} + \frac{13}{35} a^{14} + \frac{1}{7} a^{13} + \frac{13}{35} a^{12} - \frac{2}{7} a^{10} + \frac{16}{35} a^{9} - \frac{17}{35} a^{8} - \frac{1}{5} a^{7} - \frac{17}{35} a^{6} + \frac{1}{35} a^{5} + \frac{13}{35} a^{4} - \frac{1}{7} a^{3} - \frac{1}{5} a^{2} + \frac{13}{35}$, $\frac{1}{665} a^{22} + \frac{4}{665} a^{21} + \frac{11}{133} a^{20} - \frac{48}{665} a^{19} - \frac{17}{665} a^{18} - \frac{11}{133} a^{17} - \frac{206}{665} a^{16} - \frac{11}{35} a^{15} - \frac{16}{133} a^{14} - \frac{192}{665} a^{13} + \frac{62}{665} a^{12} + \frac{33}{133} a^{11} + \frac{181}{665} a^{10} + \frac{11}{35} a^{9} - \frac{23}{133} a^{8} + \frac{102}{665} a^{7} - \frac{2}{665} a^{6} - \frac{6}{19} a^{5} - \frac{216}{665} a^{4} + \frac{226}{665} a^{3} + \frac{5}{19} a^{2} + \frac{328}{665} a - \frac{253}{665}$, $\frac{1}{7315} a^{23} - \frac{3}{7315} a^{22} - \frac{6}{1463} a^{21} + \frac{327}{7315} a^{20} + \frac{136}{1463} a^{19} - \frac{124}{1463} a^{18} - \frac{272}{1463} a^{17} - \frac{1142}{7315} a^{16} + \frac{2599}{7315} a^{15} + \frac{3484}{7315} a^{14} - \frac{19}{77} a^{13} - \frac{116}{665} a^{12} - \frac{2969}{7315} a^{11} - \frac{1818}{7315} a^{10} + \frac{699}{1463} a^{9} + \frac{116}{1045} a^{8} + \frac{43}{1463} a^{7} - \frac{138}{1463} a^{6} + \frac{3}{11} a^{5} - \frac{2062}{7315} a^{4} + \frac{164}{665} a^{3} - \frac{1296}{7315} a^{2} - \frac{27}{133} a + \frac{2759}{7315}$, $\frac{1}{7315} a^{24} + \frac{1}{1463} a^{22} - \frac{1}{1463} a^{21} - \frac{9}{665} a^{20} - \frac{69}{1045} a^{19} - \frac{206}{7315} a^{18} - \frac{81}{209} a^{17} + \frac{488}{1463} a^{16} - \frac{628}{1463} a^{15} + \frac{2619}{7315} a^{14} + \frac{3253}{7315} a^{13} + \frac{2201}{7315} a^{12} - \frac{9}{19} a^{11} + \frac{82}{209} a^{10} - \frac{15}{133} a^{9} + \frac{23}{95} a^{8} + \frac{1517}{7315} a^{7} + \frac{2554}{7315} a^{6} + \frac{316}{1463} a^{5} + \frac{75}{209} a^{4} + \frac{16}{77} a^{3} + \frac{864}{7315} a^{2} - \frac{3357}{7315} a + \frac{683}{1463}$, $\frac{1}{507331825} a^{25} + \frac{47}{2899039} a^{24} + \frac{2901}{101466365} a^{23} - \frac{170218}{507331825} a^{22} + \frac{2821334}{507331825} a^{21} + \frac{5269406}{101466365} a^{20} - \frac{15266079}{507331825} a^{19} + \frac{15305287}{507331825} a^{18} - \frac{212664664}{507331825} a^{17} - \frac{168498919}{507331825} a^{16} - \frac{21747853}{46121075} a^{15} - \frac{2193648}{39025525} a^{14} + \frac{151978142}{507331825} a^{13} + \frac{209800709}{507331825} a^{12} - \frac{45198969}{101466365} a^{11} + \frac{239972862}{507331825} a^{10} - \frac{16459513}{72475975} a^{9} - \frac{6495960}{20293273} a^{8} + \frac{176365536}{507331825} a^{7} - \frac{99661203}{507331825} a^{6} - \frac{37279094}{507331825} a^{5} - \frac{230978304}{507331825} a^{4} + \frac{1509022}{4192825} a^{3} + \frac{1405846}{5575075} a^{2} - \frac{9701359}{46121075} a + \frac{7477044}{507331825}$, $\frac{1}{428868841741781426109186287225} a^{26} - \frac{278297844777010632302}{428868841741781426109186287225} a^{25} + \frac{1145844218385667626085717}{17154753669671257044367451489} a^{24} - \frac{7732945108632891182717323}{428868841741781426109186287225} a^{23} - \frac{12502488363078348647664205}{17154753669671257044367451489} a^{22} - \frac{1164579325629159958520926148}{428868841741781426109186287225} a^{21} + \frac{31568016733704365912024356941}{428868841741781426109186287225} a^{20} + \frac{6518369974081796127077360641}{85773768348356285221837257445} a^{19} + \frac{3209065575087253957326213934}{32989910903213955854552791325} a^{18} - \frac{172270955373800794227453244316}{428868841741781426109186287225} a^{17} + \frac{1256456649155263786239352428}{7797615304396025929257932495} a^{16} - \frac{116214233108560885714586995518}{428868841741781426109186287225} a^{15} - \frac{26006292905467005913468189497}{85773768348356285221837257445} a^{14} + \frac{370290406967653022151098592}{6597982180642791170910558265} a^{13} + \frac{210048499361139290941718622687}{428868841741781426109186287225} a^{12} + \frac{86533574042789474012902511832}{428868841741781426109186287225} a^{11} + \frac{24667610858795351838145847698}{85773768348356285221837257445} a^{10} + \frac{126070723644710921600311363747}{428868841741781426109186287225} a^{9} - \frac{6662030357205247806536133837}{61266977391683060872740898175} a^{8} - \frac{39746888767483753286531721}{88153924304579943701785465} a^{7} - \frac{198071170610519074238339961833}{428868841741781426109186287225} a^{6} - \frac{121820910773846789839053296716}{428868841741781426109186287225} a^{5} + \frac{259441432797730746640577686}{7797615304396025929257932495} a^{4} - \frac{161364442280294508345099747798}{428868841741781426109186287225} a^{3} + \frac{635951452125229850686432827}{5569725217425732806612808925} a^{2} + \frac{2470702217691593758520790713}{22572044302199022426799278275} a + \frac{360071919525117303031542702}{3544370592907284513299060225}$

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:  $13$
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:  \( 16112744313.230444 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 16112744313.230444 \cdot 1}{2\sqrt{476657463863730234951616960570180249031207}}\approx 0.555142664056882$ (assuming GRH)

Galois group

$D_{27}$ (as 27T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 15 conjugacy class representatives for $D_{27}$
Character table for $D_{27}$

Intermediate fields

3.1.1607.1, 9.1.6669042837601.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ $27$

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
1607Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1607.2t1.a.a$1$ $ 1607 $ \(\Q(\sqrt{-1607}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1607.3t2.a.a$2$ $ 1607 $ 3.1.1607.1 $S_3$ (as 3T2) $1$ $0$
* 2.1607.9t3.a.c$2$ $ 1607 $ 9.1.6669042837601.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1607.9t3.a.a$2$ $ 1607 $ 9.1.6669042837601.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1607.9t3.a.b$2$ $ 1607 $ 9.1.6669042837601.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1607.27t8.a.c$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.g$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.i$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.a$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.f$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.e$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.h$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.d$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1607.27t8.a.b$2$ $ 1607 $ 27.1.476657463863730234951616960570180249031207.1 $D_{27}$ (as 27T8) $1$ $0$

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 *.