Properties

Label 27.1.252...624.1
Degree $27$
Signature $[1, 13]$
Discriminant $-2.527\times 10^{40}$
Root discriminant $31.36$
Ramified primes $2, 491$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 + 45*x^25 - 154*x^24 + 444*x^23 - 1140*x^22 + 2734*x^21 - 5824*x^20 + 11115*x^19 - 18567*x^18 + 29531*x^17 - 43306*x^16 + 56712*x^15 - 56676*x^14 + 52570*x^13 - 50872*x^12 + 59685*x^11 - 40721*x^10 + 10893*x^9 + 10490*x^8 - 13378*x^7 + 7126*x^6 - 3604*x^5 - 204*x^4 + 1348*x^3 - 376*x^2 + 8*x + 8)
 
gp: K = bnfinit(x^27 - 9*x^26 + 45*x^25 - 154*x^24 + 444*x^23 - 1140*x^22 + 2734*x^21 - 5824*x^20 + 11115*x^19 - 18567*x^18 + 29531*x^17 - 43306*x^16 + 56712*x^15 - 56676*x^14 + 52570*x^13 - 50872*x^12 + 59685*x^11 - 40721*x^10 + 10893*x^9 + 10490*x^8 - 13378*x^7 + 7126*x^6 - 3604*x^5 - 204*x^4 + 1348*x^3 - 376*x^2 + 8*x + 8, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 8, -376, 1348, -204, -3604, 7126, -13378, 10490, 10893, -40721, 59685, -50872, 52570, -56676, 56712, -43306, 29531, -18567, 11115, -5824, 2734, -1140, 444, -154, 45, -9, 1]);
 

\( x^{27} - 9 x^{26} + 45 x^{25} - 154 x^{24} + 444 x^{23} - 1140 x^{22} + 2734 x^{21} - 5824 x^{20} + 11115 x^{19} - 18567 x^{18} + 29531 x^{17} - 43306 x^{16} + 56712 x^{15} - 56676 x^{14} + 52570 x^{13} - 50872 x^{12} + 59685 x^{11} - 40721 x^{10} + 10893 x^{9} + 10490 x^{8} - 13378 x^{7} + 7126 x^{6} - 3604 x^{5} - 204 x^{4} + 1348 x^{3} - 376 x^{2} + 8 x + 8 \)

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:  \(-25269655401421846003597046184882515738624\)\(\medspace = -\,2^{18}\cdot 491^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.36$
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:  $2, 491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{56} a^{21} - \frac{3}{56} a^{20} - \frac{3}{28} a^{19} + \frac{1}{56} a^{18} - \frac{5}{56} a^{17} + \frac{1}{14} a^{16} - \frac{3}{56} a^{15} + \frac{1}{56} a^{14} + \frac{9}{56} a^{13} - \frac{3}{56} a^{11} - \frac{3}{56} a^{10} + \frac{3}{14} a^{9} + \frac{19}{56} a^{8} + \frac{11}{56} a^{7} - \frac{3}{7} a^{6} + \frac{13}{28} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{56} a^{22} - \frac{1}{56} a^{20} - \frac{3}{56} a^{19} - \frac{1}{28} a^{18} + \frac{3}{56} a^{17} - \frac{5}{56} a^{16} + \frac{3}{28} a^{15} + \frac{3}{14} a^{14} - \frac{1}{56} a^{13} + \frac{11}{56} a^{12} + \frac{1}{28} a^{11} + \frac{3}{56} a^{10} - \frac{1}{56} a^{9} - \frac{2}{7} a^{8} + \frac{9}{56} a^{7} + \frac{5}{28} a^{6} + \frac{1}{14} a^{5} - \frac{1}{14} a^{4} - \frac{5}{14} a^{3} + \frac{3}{14} a + \frac{3}{7}$, $\frac{1}{56} a^{23} - \frac{3}{28} a^{20} + \frac{3}{28} a^{19} + \frac{1}{14} a^{18} + \frac{1}{14} a^{17} - \frac{1}{14} a^{16} - \frac{5}{56} a^{15} - \frac{1}{4} a^{14} + \frac{3}{28} a^{13} - \frac{3}{14} a^{12} - \frac{1}{4} a^{11} + \frac{5}{28} a^{10} + \frac{5}{28} a^{9} + \frac{1}{4} a^{8} - \frac{1}{8} a^{7} + \frac{11}{28} a^{6} - \frac{3}{28} a^{5} - \frac{3}{7} a^{4} - \frac{5}{14} a^{3} - \frac{1}{14} a^{2} - \frac{3}{14} a + \frac{1}{7}$, $\frac{1}{56} a^{24} + \frac{1}{28} a^{20} - \frac{1}{14} a^{19} - \frac{1}{14} a^{18} - \frac{3}{28} a^{17} + \frac{5}{56} a^{16} - \frac{1}{14} a^{15} - \frac{1}{28} a^{14} - \frac{1}{7} a^{11} + \frac{3}{28} a^{10} - \frac{3}{14} a^{9} + \frac{23}{56} a^{8} + \frac{1}{14} a^{7} + \frac{9}{28} a^{6} + \frac{3}{28} a^{5} + \frac{3}{14} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{1064} a^{25} + \frac{3}{1064} a^{24} + \frac{1}{1064} a^{23} - \frac{1}{532} a^{22} - \frac{1}{532} a^{21} - \frac{65}{532} a^{20} + \frac{59}{532} a^{19} - \frac{9}{76} a^{18} - \frac{65}{1064} a^{17} + \frac{43}{1064} a^{16} - \frac{131}{1064} a^{15} - \frac{5}{266} a^{14} + \frac{4}{19} a^{13} - \frac{9}{38} a^{12} + \frac{25}{133} a^{11} - \frac{115}{532} a^{10} - \frac{11}{152} a^{9} - \frac{27}{1064} a^{8} - \frac{333}{1064} a^{7} - \frac{5}{532} a^{6} + \frac{187}{532} a^{5} + \frac{111}{266} a^{4} + \frac{1}{7} a^{3} - \frac{10}{133} a^{2} - \frac{131}{266} a - \frac{33}{133}$, $\frac{1}{11776982814661747081266946327127559268924438312} a^{26} - \frac{316868231461305449175974286252092511119085}{5888491407330873540633473163563779634462219156} a^{25} + \frac{61382801617095698974897292167332982311218981}{11776982814661747081266946327127559268924438312} a^{24} - \frac{10535307559286140418408792881954134515626697}{11776982814661747081266946327127559268924438312} a^{23} + \frac{3329279552757385844966614020942923865211101}{619841200771670899014049806690924172048654648} a^{22} + \frac{99674371239401048808495340437734068106245867}{11776982814661747081266946327127559268924438312} a^{21} + \frac{26716824157181641821041738682434248810911883}{5888491407330873540633473163563779634462219156} a^{20} + \frac{1245625346226780781731555430771537971825061325}{11776982814661747081266946327127559268924438312} a^{19} - \frac{129113143649594977332417517561336481889991145}{2944245703665436770316736581781889817231109578} a^{18} + \frac{53018472728769837867365616391032592531139919}{2944245703665436770316736581781889817231109578} a^{17} + \frac{261192685810207639513634605056850223077672419}{5888491407330873540633473163563779634462219156} a^{16} + \frac{21814862341740303895810125379433652316173021}{420606529095062395759533797397412831033015654} a^{15} + \frac{731044344954894058345109442440937115359611375}{11776982814661747081266946327127559268924438312} a^{14} - \frac{318437475916313967165092976416323583519244463}{1472122851832718385158368290890944908615554789} a^{13} - \frac{238361447514948634148153857845600688070716775}{11776982814661747081266946327127559268924438312} a^{12} - \frac{37406145930551071011012776898548682189256447}{222207222918146171344659364662784137149517704} a^{11} + \frac{2878710296940722089210584855719250691839872653}{11776982814661747081266946327127559268924438312} a^{10} - \frac{2250171475486346535330743260802453942681916893}{11776982814661747081266946327127559268924438312} a^{9} + \frac{42520079748617173657579414633440445671603405}{1472122851832718385158368290890944908615554789} a^{8} - \frac{269575535536111164981257198753648174906820143}{619841200771670899014049806690924172048654648} a^{7} + \frac{59482957249965527052606200968675205460277306}{210303264547531197879766898698706415516507827} a^{6} - \frac{80624115062329757010348335075130651727017291}{203051427838995639332188729778061366705593764} a^{5} + \frac{601259957826811502724778767845212367778868616}{1472122851832718385158368290890944908615554789} a^{4} + \frac{423145212745835136361953394550704004983307519}{1472122851832718385158368290890944908615554789} a^{3} - \frac{152877172728232903378661363850450563710762449}{2944245703665436770316736581781889817231109578} a^{2} - \frac{803450226880215235924617002878724965598278229}{2944245703665436770316736581781889817231109578} a + \frac{514751351474758337250688796698613940631388997}{1472122851832718385158368290890944908615554789}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 4486112564.467699 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 4486112564.467699 \cdot 2}{2\sqrt{25269655401421846003597046184882515738624}}\approx 1.34257572705331$ (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

3.1.491.1, 9.1.58120048561.1

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 R $27$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
491Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.491.2t1.a.a$1$ $ 491 $ \(\Q(\sqrt{-491}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.491.3t2.a.a$2$ $ 491 $ 3.1.491.1 $S_3$ (as 3T2) $1$ $0$
* 2.491.9t3.a.c$2$ $ 491 $ 9.1.58120048561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.491.9t3.a.a$2$ $ 491 $ 9.1.58120048561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.491.9t3.a.b$2$ $ 491 $ 9.1.58120048561.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1964.27t8.a.d$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.h$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.b$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.c$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.g$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.f$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.e$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.a$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1964.27t8.a.i$2$ $ 2^{2} \cdot 491 $ 27.1.25269655401421846003597046184882515738624.1 $D_{27}$ (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 *.