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Properties

Label 27.1.363...239.1
Degree $27$
Signature $[1, 13]$
Discriminant $-3.640\times 10^{42}$
Root discriminant $37.70$
Ramified prime $1879$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 2*x^26 + x^25 + 18*x^24 + 107*x^23 + 180*x^22 + 332*x^21 + 290*x^20 + 295*x^19 - 179*x^18 - 772*x^17 - 1774*x^16 - 2092*x^15 - 2320*x^14 - 811*x^13 + 489*x^12 + 3551*x^11 + 5424*x^10 + 7617*x^9 + 6962*x^8 + 6380*x^7 + 2824*x^6 + 2291*x^5 - 521*x^4 + 623*x^3 - 221*x^2 + 207*x - 1)
 
gp: K = bnfinit(x^27 - 2*x^26 + x^25 + 18*x^24 + 107*x^23 + 180*x^22 + 332*x^21 + 290*x^20 + 295*x^19 - 179*x^18 - 772*x^17 - 1774*x^16 - 2092*x^15 - 2320*x^14 - 811*x^13 + 489*x^12 + 3551*x^11 + 5424*x^10 + 7617*x^9 + 6962*x^8 + 6380*x^7 + 2824*x^6 + 2291*x^5 - 521*x^4 + 623*x^3 - 221*x^2 + 207*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 207, -221, 623, -521, 2291, 2824, 6380, 6962, 7617, 5424, 3551, 489, -811, -2320, -2092, -1774, -772, -179, 295, 290, 332, 180, 107, 18, 1, -2, 1]);
 

\( x^{27} - 2 x^{26} + x^{25} + 18 x^{24} + 107 x^{23} + 180 x^{22} + 332 x^{21} + 290 x^{20} + 295 x^{19} - 179 x^{18} - 772 x^{17} - 1774 x^{16} - 2092 x^{15} - 2320 x^{14} - 811 x^{13} + 489 x^{12} + 3551 x^{11} + 5424 x^{10} + 7617 x^{9} + 6962 x^{8} + 6380 x^{7} + 2824 x^{6} + 2291 x^{5} - 521 x^{4} + 623 x^{3} - 221 x^{2} + 207 x - 1 \)

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:  \(-3639553781467035763182087002112051895733239\)\(\medspace = -\,1879^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.70$
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:  $1879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{9} a^{5} - \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{4}{9} a^{2} - \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{16} + \frac{1}{27} a^{15} - \frac{4}{27} a^{12} - \frac{4}{27} a^{10} - \frac{2}{27} a^{9} - \frac{1}{9} a^{8} + \frac{5}{27} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} - \frac{5}{27} a^{4} + \frac{1}{9} a^{3} + \frac{8}{27} a - \frac{7}{27}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{17} + \frac{1}{27} a^{16} - \frac{1}{27} a^{13} + \frac{2}{27} a^{11} + \frac{4}{27} a^{10} + \frac{2}{27} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{8}{27} a^{5} - \frac{2}{9} a^{4} + \frac{1}{9} a^{3} + \frac{11}{27} a^{2} - \frac{10}{27} a + \frac{4}{9}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} - \frac{1}{27} a^{14} - \frac{1}{9} a^{12} + \frac{4}{27} a^{11} + \frac{4}{27} a^{10} + \frac{4}{27} a^{9} - \frac{1}{9} a^{8} - \frac{11}{27} a^{7} + \frac{7}{27} a^{6} - \frac{1}{3} a^{5} - \frac{10}{27} a^{4} + \frac{8}{27} a^{3} - \frac{10}{27} a^{2} + \frac{4}{27} a - \frac{11}{27}$, $\frac{1}{81} a^{21} - \frac{1}{81} a^{19} + \frac{4}{81} a^{17} + \frac{1}{27} a^{16} + \frac{4}{81} a^{15} - \frac{1}{27} a^{14} + \frac{1}{81} a^{13} - \frac{4}{81} a^{12} + \frac{2}{81} a^{11} + \frac{4}{81} a^{10} - \frac{10}{81} a^{9} - \frac{4}{81} a^{8} + \frac{29}{81} a^{7} - \frac{1}{27} a^{6} - \frac{35}{81} a^{5} + \frac{40}{81} a^{4} + \frac{29}{81} a^{3} - \frac{16}{81} a^{2} + \frac{7}{27} a + \frac{28}{81}$, $\frac{1}{1053} a^{22} - \frac{1}{351} a^{21} + \frac{14}{1053} a^{20} + \frac{2}{351} a^{19} + \frac{7}{1053} a^{18} + \frac{4}{117} a^{17} + \frac{22}{1053} a^{16} + \frac{2}{117} a^{15} + \frac{49}{1053} a^{14} + \frac{53}{1053} a^{13} - \frac{34}{1053} a^{12} + \frac{82}{1053} a^{11} + \frac{47}{1053} a^{10} + \frac{53}{1053} a^{9} - \frac{4}{81} a^{8} - \frac{146}{351} a^{7} + \frac{475}{1053} a^{6} - \frac{176}{1053} a^{5} + \frac{419}{1053} a^{4} + \frac{503}{1053} a^{3} + \frac{2}{27} a^{2} - \frac{278}{1053} a + \frac{32}{117}$, $\frac{1}{660231} a^{23} - \frac{14}{220077} a^{22} + \frac{1036}{220077} a^{21} - \frac{46}{24453} a^{20} - \frac{2446}{220077} a^{19} + \frac{727}{220077} a^{18} - \frac{31633}{660231} a^{17} - \frac{137}{220077} a^{16} + \frac{12971}{660231} a^{15} - \frac{9268}{660231} a^{14} + \frac{4907}{220077} a^{13} - \frac{35428}{220077} a^{12} + \frac{3077}{60021} a^{11} - \frac{22949}{220077} a^{10} - \frac{4742}{50787} a^{9} + \frac{41201}{660231} a^{8} + \frac{97589}{220077} a^{7} + \frac{160855}{660231} a^{6} - \frac{92767}{220077} a^{5} + \frac{1106}{2717} a^{4} - \frac{13834}{50787} a^{3} + \frac{9134}{24453} a^{2} - \frac{73471}{220077} a - \frac{16976}{50787}$, $\frac{1}{660231} a^{24} + \frac{10}{73359} a^{22} + \frac{80}{20007} a^{21} - \frac{137}{24453} a^{20} - \frac{1328}{73359} a^{19} + \frac{2285}{660231} a^{18} + \frac{3998}{73359} a^{17} - \frac{7426}{660231} a^{16} + \frac{15731}{660231} a^{15} + \frac{734}{16929} a^{14} - \frac{71}{16929} a^{13} - \frac{2207}{660231} a^{12} + \frac{28102}{220077} a^{11} + \frac{60769}{660231} a^{10} - \frac{38677}{660231} a^{9} - \frac{2048}{24453} a^{8} + \frac{176647}{660231} a^{7} - \frac{1321}{24453} a^{6} - \frac{3008}{24453} a^{5} + \frac{181892}{660231} a^{4} - \frac{67403}{220077} a^{3} + \frac{1822}{16929} a^{2} + \frac{260530}{660231} a - \frac{11281}{73359}$, $\frac{1}{2748541653} a^{25} - \frac{379}{2748541653} a^{24} + \frac{37}{249867423} a^{23} - \frac{70184}{305393517} a^{22} - \frac{4644260}{916180551} a^{21} - \frac{53558}{23491809} a^{20} - \frac{2391217}{2748541653} a^{19} - \frac{37820729}{2748541653} a^{18} + \frac{14361475}{916180551} a^{17} + \frac{40660544}{916180551} a^{16} + \frac{61609466}{2748541653} a^{15} + \frac{95881726}{2748541653} a^{14} + \frac{15152659}{2748541653} a^{13} - \frac{294418369}{2748541653} a^{12} - \frac{4819130}{48220029} a^{11} + \frac{161806582}{2748541653} a^{10} + \frac{39508234}{916180551} a^{9} + \frac{25391468}{2748541653} a^{8} - \frac{281013985}{2748541653} a^{7} + \frac{1281820217}{2748541653} a^{6} + \frac{732597692}{2748541653} a^{5} - \frac{792823778}{2748541653} a^{4} - \frac{903329036}{2748541653} a^{3} + \frac{516456796}{2748541653} a^{2} - \frac{355118539}{2748541653} a + \frac{1359261317}{2748541653}$, $\frac{1}{20415664415361501} a^{26} - \frac{77512}{618656497435197} a^{25} - \frac{372163378}{887637583276587} a^{24} + \frac{11356514027}{20415664415361501} a^{23} + \frac{2064472311619}{6805221471787167} a^{22} - \frac{1439099449343}{618656497435197} a^{21} - \frac{7475350304912}{887637583276587} a^{20} - \frac{50445961554746}{6805221471787167} a^{19} - \frac{161504644084115}{20415664415361501} a^{18} + \frac{53867607631555}{6805221471787167} a^{17} - \frac{207808209734284}{20415664415361501} a^{16} - \frac{170618308883543}{6805221471787167} a^{15} + \frac{42868602822712}{887637583276587} a^{14} - \frac{6383639394283}{295879194425529} a^{13} - \frac{105453720071077}{1074508653440079} a^{12} - \frac{2684484427036436}{20415664415361501} a^{11} - \frac{463213092146975}{20415664415361501} a^{10} - \frac{1560623540623783}{20415664415361501} a^{9} - \frac{99420778868012}{1570435724258577} a^{8} + \frac{305705949961427}{1855969492305591} a^{7} - \frac{10172989146730250}{20415664415361501} a^{6} - \frac{3392601989309746}{6805221471787167} a^{5} - \frac{2572942807864126}{20415664415361501} a^{4} - \frac{6022331046167413}{20415664415361501} a^{3} + \frac{92268633458152}{206218832478399} a^{2} + \frac{412509601318531}{20415664415361501} a + \frac{2159165400573521}{20415664415361501}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 25801652404.76311 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 25801652404.76311 \cdot 4}{2\sqrt{3639553781467035763182087002112051895733239}}\approx 1.28683163686933$ (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

3.1.1879.1, 9.1.12465425870881.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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
1879Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1879.2t1.a.a$1$ $ 1879 $ \(\Q(\sqrt{-1879}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1879.3t2.a.a$2$ $ 1879 $ 3.1.1879.1 $S_3$ (as 3T2) $1$ $0$
* 2.1879.9t3.a.b$2$ $ 1879 $ 9.1.12465425870881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1879.9t3.a.c$2$ $ 1879 $ 9.1.12465425870881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1879.9t3.a.a$2$ $ 1879 $ 9.1.12465425870881.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.1879.27t8.a.c$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.h$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.i$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.b$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.d$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.f$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.g$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.e$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.1879.27t8.a.a$2$ $ 1879 $ 27.1.3639553781467035763182087002112051895733239.1 $D_{27}$ (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 *.