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Properties

Label 27.1.149...191.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.491\times 10^{40}$
Root discriminant $30.75$
Ramified prime $1231$
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 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1)
 
gp: K = bnfinit(x^27 - 4*x^26 + 14*x^25 - 10*x^24 + 37*x^22 - 3*x^21 - x^20 + 139*x^19 - 26*x^18 + 76*x^17 + 144*x^16 - 117*x^15 - 101*x^14 + 12*x^13 - 340*x^12 + x^11 + 205*x^10 + 307*x^9 + 774*x^8 + 909*x^7 + 892*x^6 + 864*x^5 + 521*x^4 + 327*x^3 + 86*x^2 + 71*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 71, 86, 327, 521, 864, 892, 909, 774, 307, 205, 1, -340, 12, -101, -117, 144, 76, -26, 139, -1, -3, 37, 0, -10, 14, -4, 1]);
 

\( x^{27} - 4 x^{26} + 14 x^{25} - 10 x^{24} + 37 x^{22} - 3 x^{21} - x^{20} + 139 x^{19} - 26 x^{18} + 76 x^{17} + 144 x^{16} - 117 x^{15} - 101 x^{14} + 12 x^{13} - 340 x^{12} + x^{11} + 205 x^{10} + 307 x^{9} + 774 x^{8} + 909 x^{7} + 892 x^{6} + 864 x^{5} + 521 x^{4} + 327 x^{3} + 86 x^{2} + 71 x + 1 \)

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:  \(-14905779350658261917360296447434047234191\)\(\medspace = -\,1231^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $30.75$
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:  $1231$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{27} a^{21} - \frac{1}{27} a^{19} + \frac{1}{27} a^{18} + \frac{1}{27} a^{17} + \frac{1}{27} a^{16} - \frac{1}{9} a^{15} - \frac{2}{27} a^{13} - \frac{1}{9} a^{12} + \frac{2}{27} a^{11} + \frac{1}{27} a^{10} + \frac{4}{27} a^{9} + \frac{4}{27} a^{8} - \frac{2}{9} a^{7} + \frac{1}{3} a^{6} - \frac{8}{27} a^{5} + \frac{4}{9} a^{4} - \frac{10}{27} a^{3} + \frac{7}{27} a^{2} + \frac{13}{27} a + \frac{4}{27}$, $\frac{1}{27} a^{22} - \frac{1}{27} a^{20} + \frac{1}{27} a^{19} + \frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{2}{27} a^{14} - \frac{1}{9} a^{13} + \frac{2}{27} a^{12} + \frac{1}{27} a^{11} + \frac{4}{27} a^{10} + \frac{4}{27} a^{9} - \frac{1}{9} a^{8} + \frac{1}{3} a^{7} - \frac{8}{27} a^{6} + \frac{4}{9} a^{5} - \frac{10}{27} a^{4} + \frac{7}{27} a^{3} + \frac{13}{27} a^{2} + \frac{4}{27} a - \frac{2}{9}$, $\frac{1}{81} a^{23} + \frac{1}{81} a^{22} + \frac{1}{81} a^{21} + \frac{1}{27} a^{20} + \frac{4}{81} a^{18} + \frac{2}{81} a^{16} - \frac{8}{81} a^{15} - \frac{5}{81} a^{14} + \frac{4}{81} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{10}{81} a^{10} - \frac{1}{27} a^{9} + \frac{5}{81} a^{8} - \frac{38}{81} a^{7} - \frac{5}{81} a^{6} + \frac{4}{81} a^{5} + \frac{11}{27} a^{4} - \frac{4}{9} a^{3} - \frac{23}{81} a^{2} + \frac{4}{27} a + \frac{38}{81}$, $\frac{1}{135999} a^{24} + \frac{206}{135999} a^{23} + \frac{1538}{135999} a^{22} - \frac{1697}{135999} a^{21} - \frac{2246}{45333} a^{20} + \frac{3988}{135999} a^{19} - \frac{3137}{135999} a^{18} - \frac{742}{135999} a^{17} - \frac{121}{15111} a^{16} + \frac{128}{135999} a^{15} + \frac{11759}{135999} a^{14} - \frac{6080}{135999} a^{13} - \frac{1181}{15111} a^{12} - \frac{16958}{135999} a^{11} + \frac{2437}{135999} a^{10} + \frac{13640}{135999} a^{9} + \frac{53}{15111} a^{8} - \frac{51472}{135999} a^{7} - \frac{157}{1863} a^{6} - \frac{485}{135999} a^{5} - \frac{14119}{45333} a^{4} + \frac{29305}{135999} a^{3} + \frac{5695}{135999} a^{2} - \frac{52588}{135999} a - \frac{56035}{135999}$, $\frac{1}{21623841} a^{25} + \frac{58}{21623841} a^{24} + \frac{93617}{21623841} a^{23} - \frac{41273}{21623841} a^{22} + \frac{205801}{21623841} a^{21} - \frac{509888}{21623841} a^{20} + \frac{70070}{2402649} a^{19} - \frac{217667}{7207947} a^{18} + \frac{647686}{21623841} a^{17} + \frac{2937}{88987} a^{16} - \frac{3541480}{21623841} a^{15} + \frac{2405755}{21623841} a^{14} + \frac{100081}{21623841} a^{13} - \frac{84328}{940167} a^{12} + \frac{16042}{313389} a^{11} + \frac{835195}{7207947} a^{10} - \frac{5197}{407997} a^{9} - \frac{701096}{7207947} a^{8} - \frac{2521333}{21623841} a^{7} + \frac{9580327}{21623841} a^{6} - \frac{7559657}{21623841} a^{5} - \frac{9563372}{21623841} a^{4} - \frac{2451215}{7207947} a^{3} + \frac{2971090}{7207947} a^{2} - \frac{1035866}{7207947} a - \frac{9106297}{21623841}$, $\frac{1}{717044643038151} a^{26} + \frac{15099116}{717044643038151} a^{25} - \frac{445493129}{239014881012717} a^{24} - \frac{525974495980}{239014881012717} a^{23} - \frac{6273715611433}{717044643038151} a^{22} + \frac{7554204232715}{717044643038151} a^{21} + \frac{38591987442955}{717044643038151} a^{20} + \frac{32820573286}{611291255787} a^{19} - \frac{37328321562947}{717044643038151} a^{18} + \frac{357275968862}{8639092084797} a^{17} - \frac{376548944054}{42179096649303} a^{16} - \frac{1200727106698}{26557209001413} a^{15} - \frac{1565854272901}{9822529356687} a^{14} - \frac{71411258304379}{717044643038151} a^{13} + \frac{53009882666989}{717044643038151} a^{12} + \frac{12441069640637}{79671627004239} a^{11} + \frac{9892065981073}{717044643038151} a^{10} + \frac{28139744111119}{717044643038151} a^{9} + \frac{5385579118484}{717044643038151} a^{8} - \frac{10536512001476}{239014881012717} a^{7} - \frac{276749630863258}{717044643038151} a^{6} - \frac{98538605571637}{717044643038151} a^{5} + \frac{208887455610868}{717044643038151} a^{4} - \frac{45066103184386}{239014881012717} a^{3} + \frac{87820955548717}{239014881012717} a^{2} - \frac{344841463323733}{717044643038151} a + \frac{42099399472001}{717044643038151}$

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:  \( 5105816638.575763 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 5105816638.575763 \cdot 1}{2\sqrt{14905779350658261917360296447434047234191}}\approx 0.994777969725537$ (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

3.1.1231.1, 9.1.2296318960321.1

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 $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1231Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1231.2t1.a.a$1$ $ 1231 $ \(\Q(\sqrt{-1231}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1231.3t2.a.a$2$ $ 1231 $ 3.1.1231.1 $S_3$ (as 3T2) $1$ $0$
* 2.1231.9t3.a.a$2$ $ 1231 $ 9.1.2296318960321.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1231.9t3.a.b$2$ $ 1231 $ 9.1.2296318960321.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1231.9t3.a.c$2$ $ 1231 $ 9.1.2296318960321.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1231.27t8.a.e$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.f$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.a$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.h$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.c$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.i$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.g$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.d$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1231.27t8.a.b$2$ $ 1231 $ 27.1.14905779350658261917360296447434047234191.1 $D_{27}$ (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 *.