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Properties

Label 27.1.321...296.1
Degree $27$
Signature $[1, 13]$
Discriminant $-3.216\times 10^{39}$
Root discriminant $29.06$
Ramified primes $2, 419$
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 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16)
 
gp: K = bnfinit(x^27 - 10*x^26 + 38*x^25 - 52*x^24 - 84*x^23 + 458*x^22 - 715*x^21 + 40*x^20 + 1948*x^19 - 3756*x^18 + 1448*x^17 + 4952*x^16 - 7109*x^15 + 94*x^14 + 4978*x^13 - 2352*x^12 - 1300*x^11 + 2422*x^10 + 639*x^9 - 3848*x^8 - 28*x^7 + 1728*x^6 + 572*x^5 - 368*x^4 - 244*x^3 + 48*x^2 + 32*x - 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-16, 32, 48, -244, -368, 572, 1728, -28, -3848, 639, 2422, -1300, -2352, 4978, 94, -7109, 4952, 1448, -3756, 1948, 40, -715, 458, -84, -52, 38, -10, 1]);
 

\( x^{27} - 10 x^{26} + 38 x^{25} - 52 x^{24} - 84 x^{23} + 458 x^{22} - 715 x^{21} + 40 x^{20} + 1948 x^{19} - 3756 x^{18} + 1448 x^{17} + 4952 x^{16} - 7109 x^{15} + 94 x^{14} + 4978 x^{13} - 2352 x^{12} - 1300 x^{11} + 2422 x^{10} + 639 x^{9} - 3848 x^{8} - 28 x^{7} + 1728 x^{6} + 572 x^{5} - 368 x^{4} - 244 x^{3} + 48 x^{2} + 32 x - 16 \)

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:  \(-3216045767164746225347277064349747511296\)\(\medspace = -\,2^{18}\cdot 419^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $29.06$
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:  $2, 419$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{17} + \frac{1}{4} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{18} + \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{19} + \frac{1}{4} a^{10} + \frac{1}{4} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{23} - \frac{1}{4} a^{17} + \frac{1}{4} a^{11} - \frac{1}{2} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{304} a^{24} + \frac{17}{152} a^{23} - \frac{5}{76} a^{22} - \frac{17}{152} a^{21} + \frac{1}{76} a^{20} - \frac{37}{152} a^{19} + \frac{5}{304} a^{18} - \frac{7}{76} a^{17} - \frac{29}{152} a^{16} - \frac{13}{152} a^{15} - \frac{15}{76} a^{14} + \frac{5}{38} a^{13} - \frac{17}{304} a^{12} + \frac{73}{152} a^{11} - \frac{25}{76} a^{10} + \frac{15}{152} a^{9} + \frac{35}{76} a^{8} + \frac{33}{152} a^{7} + \frac{135}{304} a^{6} + \frac{13}{76} a^{5} + \frac{41}{152} a^{4} - \frac{3}{8} a^{3} + \frac{1}{19} a^{2} + \frac{15}{38} a + \frac{1}{38}$, $\frac{1}{1401136} a^{25} + \frac{455}{700568} a^{24} - \frac{1303}{31844} a^{23} - \frac{31653}{700568} a^{22} + \frac{490}{87571} a^{21} + \frac{38727}{700568} a^{20} + \frac{309557}{1401136} a^{19} - \frac{58515}{350284} a^{18} + \frac{3453}{36872} a^{17} - \frac{25721}{700568} a^{16} + \frac{1283}{350284} a^{15} - \frac{28909}{175142} a^{14} - \frac{7993}{1401136} a^{13} - \frac{91809}{700568} a^{12} - \frac{140001}{350284} a^{11} - \frac{29649}{700568} a^{10} - \frac{7996}{87571} a^{9} + \frac{42277}{700568} a^{8} - \frac{261553}{1401136} a^{7} + \frac{5893}{31844} a^{6} + \frac{19397}{700568} a^{5} - \frac{108921}{700568} a^{4} + \frac{27647}{175142} a^{3} + \frac{3059}{9218} a^{2} + \frac{72079}{175142} a + \frac{43492}{87571}$, $\frac{1}{2021524412970411423609490736936320265936} a^{26} - \frac{149310243908651546984597866318147}{1010762206485205711804745368468160132968} a^{25} + \frac{725769167919854291470636155805252771}{1010762206485205711804745368468160132968} a^{24} + \frac{47159158428516729468405631054011477041}{1010762206485205711804745368468160132968} a^{23} + \frac{7724272003002890784964152017943195079}{252690551621301427951186342117040033242} a^{22} - \frac{65775096526084604723170077839903511177}{1010762206485205711804745368468160132968} a^{21} + \frac{143468462002591006408784737812140442093}{2021524412970411423609490736936320265936} a^{20} + \frac{4954247751847883050954572655544806571}{505381103242602855902372684234080066484} a^{19} + \frac{258751511852258854052035575699407981}{5003773299431711444577947368654258084} a^{18} + \frac{143118338224460513990917079080827119439}{1010762206485205711804745368468160132968} a^{17} - \frac{9323170223199933286420699700432911620}{126345275810650713975593171058520016621} a^{16} + \frac{83025686381738109237952915488961816243}{505381103242602855902372684234080066484} a^{15} + \frac{489907161932065286925531333266738902599}{2021524412970411423609490736936320265936} a^{14} + \frac{22131320261993499663612294814657108215}{91887473316836882891340488042560012088} a^{13} + \frac{49661116057341538440807270642118487553}{1010762206485205711804745368468160132968} a^{12} - \frac{165438157122931003320093835635911329671}{1010762206485205711804745368468160132968} a^{11} - \frac{8965459137545940156654743853601782}{18670795893401908375290848390500963} a^{10} + \frac{219021078536998816490958699604240696725}{1010762206485205711804745368468160132968} a^{9} - \frac{622998458212255281532552028590751329585}{2021524412970411423609490736936320265936} a^{8} - \frac{99445563326765160201459000665274986737}{505381103242602855902372684234080066484} a^{7} + \frac{21696212947767430036921459410573771195}{505381103242602855902372684234080066484} a^{6} - \frac{444166229560181567386952439199069432901}{1010762206485205711804745368468160132968} a^{5} - \frac{207804859798452263605731001716062553001}{505381103242602855902372684234080066484} a^{4} + \frac{182776170099630548036176837015269720821}{505381103242602855902372684234080066484} a^{3} - \frac{793461660443770418881586828649579289}{2501886649715855722288973684327129042} a^{2} + \frac{6528976213544883390158468582789390940}{126345275810650713975593171058520016621} a - \frac{19871357481321199085180341607179695161}{126345275810650713975593171058520016621}$

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:  \( 2806458409.197778 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 2806458409.197778 \cdot 1}{2\sqrt{3216045767164746225347277064349747511296}}\approx 1.17716079305573$ (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

3.1.419.1, 9.1.30821664721.1

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 $27$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $27$

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}$
419Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.419.2t1.a.a$1$ $ 419 $ \(\Q(\sqrt{-419}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.419.3t2.a.a$2$ $ 419 $ 3.1.419.1 $S_3$ (as 3T2) $1$ $0$
* 2.419.9t3.a.a$2$ $ 419 $ 9.1.30821664721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.419.9t3.a.b$2$ $ 419 $ 9.1.30821664721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.419.9t3.a.c$2$ $ 419 $ 9.1.30821664721.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1676.27t8.a.f$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.d$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.c$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.a$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.b$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.e$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.i$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.h$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1676.27t8.a.g$2$ $ 2^{2} \cdot 419 $ 27.1.3216045767164746225347277064349747511296.1 $D_{27}$ (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 *.