Properties

Label 27.1.219...911.1
Degree $27$
Signature $[1, 13]$
Discriminant $-2.199\times 10^{46}$
Root discriminant $52.04$
Ramified prime $3671$
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 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933)
 
gp: K = bnfinit(x^27 - 8*x^26 + 46*x^25 - 158*x^24 + 412*x^23 - 651*x^22 + 948*x^21 - 1384*x^20 + 7521*x^19 - 22249*x^18 + 61004*x^17 - 97499*x^16 + 110127*x^15 + 98440*x^14 - 785239*x^13 + 2068135*x^12 - 3012058*x^11 + 444932*x^10 + 9236470*x^9 - 28098698*x^8 + 53493798*x^7 - 76988059*x^6 + 87553666*x^5 - 79021445*x^4 + 56695661*x^3 - 30768538*x^2 + 10754601*x - 1715933, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1715933, 10754601, -30768538, 56695661, -79021445, 87553666, -76988059, 53493798, -28098698, 9236470, 444932, -3012058, 2068135, -785239, 98440, 110127, -97499, 61004, -22249, 7521, -1384, 948, -651, 412, -158, 46, -8, 1]);
 

\( x^{27} - 8 x^{26} + 46 x^{25} - 158 x^{24} + 412 x^{23} - 651 x^{22} + 948 x^{21} - 1384 x^{20} + 7521 x^{19} - 22249 x^{18} + 61004 x^{17} - 97499 x^{16} + 110127 x^{15} + 98440 x^{14} - 785239 x^{13} + 2068135 x^{12} - 3012058 x^{11} + 444932 x^{10} + 9236470 x^{9} - 28098698 x^{8} + 53493798 x^{7} - 76988059 x^{6} + 87553666 x^{5} - 79021445 x^{4} + 56695661 x^{3} - 30768538 x^{2} + 10754601 x - 1715933 \)

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:  \(-21988570612019400506053514781537922706637489911\)\(\medspace = -\,3671^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $52.04$
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:  $3671$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{1417} a^{24} - \frac{111}{1417} a^{23} + \frac{505}{1417} a^{22} - \frac{28}{109} a^{21} - \frac{437}{1417} a^{20} - \frac{444}{1417} a^{19} - \frac{45}{109} a^{18} + \frac{30}{109} a^{17} + \frac{326}{1417} a^{16} + \frac{424}{1417} a^{15} + \frac{196}{1417} a^{14} + \frac{54}{1417} a^{13} - \frac{349}{1417} a^{12} - \frac{190}{1417} a^{11} + \frac{215}{1417} a^{10} + \frac{605}{1417} a^{9} - \frac{600}{1417} a^{8} - \frac{183}{1417} a^{7} - \frac{461}{1417} a^{6} + \frac{454}{1417} a^{5} + \frac{561}{1417} a^{4} - \frac{617}{1417} a^{3} - \frac{437}{1417} a^{2} + \frac{404}{1417} a - \frac{447}{1417}$, $\frac{1}{58097} a^{25} + \frac{1}{58097} a^{24} + \frac{24915}{58097} a^{23} - \frac{1901}{58097} a^{22} + \frac{12641}{58097} a^{21} + \frac{7292}{58097} a^{20} - \frac{6386}{58097} a^{19} + \frac{222}{4469} a^{18} - \frac{1338}{58097} a^{17} + \frac{7179}{58097} a^{16} - \frac{1019}{4469} a^{15} - \frac{19087}{58097} a^{14} - \frac{9888}{58097} a^{13} + \frac{28738}{58097} a^{12} - \frac{19648}{58097} a^{11} - \frac{27744}{58097} a^{10} - \frac{16443}{58097} a^{9} + \frac{50}{1417} a^{8} + \frac{7383}{58097} a^{7} + \frac{14004}{58097} a^{6} + \frac{15984}{58097} a^{5} + \frac{135}{1417} a^{4} + \frac{19730}{58097} a^{3} + \frac{10974}{58097} a^{2} + \frac{125}{1417} a - \frac{11805}{58097}$, $\frac{1}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{26} + \frac{57320805861205357401647202063609315957811346936476626962195709918827702138530459377707}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{25} - \frac{8677382960764353220491851417522368602334534376966970570436683756912303607275989904874672}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{24} - \frac{7725937265016692230064903044657144038299408129857295614458192184245296310683121614133871639}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{23} + \frac{3908164781593493398187911744379466952407911546835945106974053697220702189226384002096148090}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{22} + \frac{9561026817307812724255592204109205963672115084048053869368730966145900896307664859191121123}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{21} - \frac{4230711715971983819234399847323927942037344569885663979976740474710811866019811731491882601}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{20} + \frac{2025976245758935953642715865630179518993693351986525336145918408013163355980197224513503807}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{19} - \frac{2857356501074620223756980054425225758287196084647904093586019498466163059307993305637562351}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{18} + \frac{1686660920828930266028708941242730535326221859759244454245730766531336469540887340210150666}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{17} + \frac{39633853244717789203976790487746293026063845705873289173808012474472280206215047599418286}{814923019198487105984805626493473606944452932546173967750406702559222599033725944876608061} a^{16} + \frac{6195906776521555948426295564116388541236058119884368804119594967545897064272012722705355653}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{15} - \frac{12246341219971942204933288989213029800574859128207018597379580833716009758514959928634213561}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{14} + \frac{7965050889729577705399547278869488006714720507737748410440937729905947870910307721025115645}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{13} - \frac{13536046493089423335478928633465169209755033887587921334581553789645955358220225807290420776}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{12} + \frac{10390703010485216130992806273555436630760150891517006292253432396058267235913378750103142}{56570641107587285030840165441385597480196545036038342977045118188914139144930318875111629} a^{11} + \frac{9792168937635013623626172258298269034580603473554337957213167131578042917262579477065877488}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{10} + \frac{13898601260789307807469790508346347879673319021177367817895913384037870836888461084358978098}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{9} - \frac{5574145767749655174655626043582593507666891524048058002543891799985987825384923074775309729}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{8} - \frac{117955820527930159211483295783551647720116511409233072042769291280667067150589517232235975}{735418334398634705400922150738012767242555085468498458701586536455883808884094145376451177} a^{7} - \frac{3948647809623634740272372569571645811186992725646364731174275677162024991742777641394957870}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{6} - \frac{9258370502553220816323270441004838612694377276855597003165653917344706858983077686267119457}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{5} + \frac{12130814961061398252306097462903059185054234129245860002629259098598611480368062646126035476}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{4} + \frac{5336551016660678161695301324151103845570829884974824083237455078610539020433599736126936657}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{3} + \frac{11694468694063677515493714595544175397976059730309846675553082015490131241615916111096798542}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257} a^{2} + \frac{177977843869630163262637135857349208309176826059623797387355793227364301379456229150849754}{2319396285411078686264446783096809496688058346477572062058849845745479704942143073879576789} a + \frac{6709086900800485315754640455414913081404899059835593206528925192852768391201481288061141620}{30152151710344022921437808180258523456944758504208436806765047994691236164247859960434498257}$

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:  \( 1347745809122.2559 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 1347745809122.2559 \cdot 1}{2\sqrt{21988570612019400506053514781537922706637489911}}\approx 0.216196062191980$ (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

3.1.3671.1, 9.1.181609071490081.1

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 $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $27$

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
3671Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3671.2t1.a.a$1$ $ 3671 $ \(\Q(\sqrt{-3671}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3671.3t2.a.a$2$ $ 3671 $ 3.1.3671.1 $S_3$ (as 3T2) $1$ $0$
* 2.3671.9t3.a.a$2$ $ 3671 $ 9.1.181609071490081.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3671.9t3.a.b$2$ $ 3671 $ 9.1.181609071490081.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3671.9t3.a.c$2$ $ 3671 $ 9.1.181609071490081.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3671.27t8.a.c$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.b$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.e$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.i$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.f$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.a$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.h$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.g$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3671.27t8.a.d$2$ $ 3671 $ 27.1.21988570612019400506053514781537922706637489911.1 $D_{27}$ (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 *.