Properties

Label 27.1.490...711.1
Degree $27$
Signature $[1, 13]$
Discriminant $-4.907\times 10^{45}$
Root discriminant $49.23$
Ramified prime $3271$
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 - 3*x^26 + 15*x^25 - 53*x^24 + 106*x^23 - 298*x^22 + 714*x^21 - 199*x^20 + 2747*x^19 + 4180*x^18 + 4767*x^17 + 27487*x^16 - 7928*x^15 + 93863*x^14 - 80586*x^13 + 257840*x^12 - 253867*x^11 + 459460*x^10 - 363114*x^9 + 981160*x^8 + 143119*x^7 - 202978*x^6 + 143097*x^5 + 629777*x^4 + 25669*x^3 - 205103*x^2 - 65505*x + 40293)
 
gp: K = bnfinit(x^27 - 3*x^26 + 15*x^25 - 53*x^24 + 106*x^23 - 298*x^22 + 714*x^21 - 199*x^20 + 2747*x^19 + 4180*x^18 + 4767*x^17 + 27487*x^16 - 7928*x^15 + 93863*x^14 - 80586*x^13 + 257840*x^12 - 253867*x^11 + 459460*x^10 - 363114*x^9 + 981160*x^8 + 143119*x^7 - 202978*x^6 + 143097*x^5 + 629777*x^4 + 25669*x^3 - 205103*x^2 - 65505*x + 40293, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40293, -65505, -205103, 25669, 629777, 143097, -202978, 143119, 981160, -363114, 459460, -253867, 257840, -80586, 93863, -7928, 27487, 4767, 4180, 2747, -199, 714, -298, 106, -53, 15, -3, 1]);
 

\( x^{27} - 3 x^{26} + 15 x^{25} - 53 x^{24} + 106 x^{23} - 298 x^{22} + 714 x^{21} - 199 x^{20} + 2747 x^{19} + 4180 x^{18} + 4767 x^{17} + 27487 x^{16} - 7928 x^{15} + 93863 x^{14} - 80586 x^{13} + 257840 x^{12} - 253867 x^{11} + 459460 x^{10} - 363114 x^{9} + 981160 x^{8} + 143119 x^{7} - 202978 x^{6} + 143097 x^{5} + 629777 x^{4} + 25669 x^{3} - 205103 x^{2} - 65505 x + 40293 \)

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:  \(-4907350451606845763597379498834801987276076711\)\(\medspace = -\,3271^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $49.23$
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:  $3271$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{4}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} - \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{3} + \frac{4}{9} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{13} - \frac{2}{9} a^{5}$, $\frac{1}{189} a^{22} + \frac{1}{27} a^{21} - \frac{2}{189} a^{20} - \frac{1}{27} a^{19} + \frac{1}{27} a^{18} - \frac{2}{63} a^{17} + \frac{1}{63} a^{16} + \frac{1}{21} a^{15} - \frac{11}{189} a^{14} - \frac{8}{189} a^{13} - \frac{23}{189} a^{12} + \frac{23}{189} a^{11} - \frac{8}{189} a^{10} - \frac{1}{63} a^{9} - \frac{16}{63} a^{8} - \frac{2}{21} a^{7} + \frac{4}{27} a^{6} - \frac{44}{189} a^{5} + \frac{79}{189} a^{4} - \frac{1}{27} a^{3} - \frac{17}{189} a^{2} + \frac{2}{7} a - \frac{1}{21}$, $\frac{1}{2079} a^{23} + \frac{4}{2079} a^{22} + \frac{61}{2079} a^{21} + \frac{62}{2079} a^{20} + \frac{13}{297} a^{19} - \frac{10}{231} a^{18} - \frac{2}{99} a^{17} - \frac{2}{99} a^{16} + \frac{67}{2079} a^{15} - \frac{164}{2079} a^{14} + \frac{64}{2079} a^{13} - \frac{118}{2079} a^{12} - \frac{4}{27} a^{11} - \frac{8}{99} a^{10} - \frac{2}{231} a^{9} - \frac{40}{99} a^{8} - \frac{23}{2079} a^{7} + \frac{124}{2079} a^{6} + \frac{127}{2079} a^{5} + \frac{37}{189} a^{4} - \frac{584}{2079} a^{3} - \frac{4}{99} a^{2} - \frac{1}{693} a - \frac{8}{21}$, $\frac{1}{193347} a^{24} + \frac{1}{193347} a^{23} + \frac{29}{27621} a^{22} + \frac{11}{651} a^{21} - \frac{2020}{193347} a^{20} - \frac{89}{17577} a^{19} - \frac{10706}{193347} a^{18} + \frac{170}{3069} a^{17} - \frac{5120}{193347} a^{16} + \frac{328}{193347} a^{15} - \frac{8992}{193347} a^{14} + \frac{4799}{64449} a^{13} + \frac{4820}{193347} a^{12} - \frac{109}{891} a^{11} - \frac{18071}{193347} a^{10} - \frac{2726}{64449} a^{9} - \frac{1873}{17577} a^{8} + \frac{59098}{193347} a^{7} - \frac{5920}{27621} a^{6} + \frac{21928}{64449} a^{5} - \frac{91972}{193347} a^{4} - \frac{95275}{193347} a^{3} + \frac{42907}{193347} a^{2} + \frac{23378}{64449} a - \frac{802}{1953}$, $\frac{1}{1008691299} a^{25} + \frac{145}{112076811} a^{24} - \frac{124136}{1008691299} a^{23} - \frac{846347}{1008691299} a^{22} - \frac{55208749}{1008691299} a^{21} + \frac{4695385}{336230433} a^{20} - \frac{1099199}{21461517} a^{19} - \frac{17751754}{1008691299} a^{18} - \frac{7124984}{1008691299} a^{17} + \frac{3418972}{336230433} a^{16} + \frac{4218328}{1008691299} a^{15} + \frac{1568753}{91699209} a^{14} + \frac{46695806}{1008691299} a^{13} + \frac{2114044}{16010973} a^{12} + \frac{57269867}{1008691299} a^{11} - \frac{23004979}{144098757} a^{10} + \frac{3747532}{27261927} a^{9} + \frac{14305892}{336230433} a^{8} + \frac{7411189}{32538429} a^{7} - \frac{223140836}{1008691299} a^{6} + \frac{199017374}{1008691299} a^{5} + \frac{30441338}{336230433} a^{4} + \frac{233127791}{1008691299} a^{3} - \frac{200521219}{1008691299} a^{2} - \frac{124807565}{336230433} a - \frac{50198}{275373}$, $\frac{1}{1054952583997643476982439464202581492383801576710248736082266321} a^{26} + \frac{105613948423037278105581474752540846831827291249120946}{1054952583997643476982439464202581492383801576710248736082266321} a^{25} + \frac{830493897082652002498826420782086907439677481915221370764}{1054952583997643476982439464202581492383801576710248736082266321} a^{24} + \frac{6881328904995054740802754722387298636609864586857480798364}{50235837333221117951544736390599118684942932224297558861060301} a^{23} + \frac{2035093792399297394503987532563976802482773401029490059359288}{1054952583997643476982439464202581492383801576710248736082266321} a^{22} - \frac{32284859514101612734002145273169816327217913050488404031723735}{1054952583997643476982439464202581492383801576710248736082266321} a^{21} - \frac{485785480504845931008443861086323322703287312343021750467319}{13700682909060304895875837197436123277711708788444788780289173} a^{20} - \frac{2845156107994192662352386631557248832208493653148529773849372}{117216953777515941886937718244731276931533508523360970675807369} a^{19} - \frac{5966213435988863064055760392493084418389098880062428297205396}{150707511999663353854634209171797356054828796672892676583180903} a^{18} - \frac{15143260743772669641325146148411696542051758451306817121973385}{1054952583997643476982439464202581492383801576710248736082266321} a^{17} - \frac{5108580922630745014694534194399288771934622652852161720023422}{95904780363422134271130860382052862943981961519113521462024211} a^{16} + \frac{420633749146037270181905741504009187441668032912840660210345}{10656086707046903807903428931339206993775773502123724606891579} a^{15} - \frac{34987106114869368459083475389245632044915095037410512883301705}{1054952583997643476982439464202581492383801576710248736082266321} a^{14} - \frac{129597118597178847580671008565990719215836767160292781439272708}{1054952583997643476982439464202581492383801576710248736082266321} a^{13} + \frac{266976356875653727219134000234562567391225556062693039321570}{150707511999663353854634209171797356054828796672892676583180903} a^{12} - \frac{6912956824845855258066614345781707809272361941861273508515610}{351650861332547825660813154734193830794600525570082912027422107} a^{11} - \frac{16275215136050693864636260598060955517555846448184494634607369}{150707511999663353854634209171797356054828796672892676583180903} a^{10} + \frac{2385008239044427133831669219410581601337132462482002545259340}{95904780363422134271130860382052862943981961519113521462024211} a^{9} - \frac{40217597787434146361694063865932035880477458719087452500170967}{1054952583997643476982439464202581492383801576710248736082266321} a^{8} - \frac{46658995552902623200875816241482287574042175889474757092356352}{351650861332547825660813154734193830794600525570082912027422107} a^{7} - \frac{23992448023348135114136508494317535441664537092783733442157213}{1054952583997643476982439464202581492383801576710248736082266321} a^{6} - \frac{261755904339040413996409506283409024487522318372505591411155347}{1054952583997643476982439464202581492383801576710248736082266321} a^{5} - \frac{84407066484298428145759687380495447755323611301580704603683073}{1054952583997643476982439464202581492383801576710248736082266321} a^{4} - \frac{25385586992019492607594098012096830948100489255236983426427381}{351650861332547825660813154734193830794600525570082912027422107} a^{3} - \frac{300096369582430663011281902995327711695649478384644726906166382}{1054952583997643476982439464202581492383801576710248736082266321} a^{2} - \frac{99217659820509455901675135845307051195129236622009671784760351}{351650861332547825660813154734193830794600525570082912027422107} a - \frac{73154212636475155272514995297257786037955963063530454191307}{288002343433700102916308890036194783615561446003343908294367}$

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:  \( 2789984396899.383 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 2789984396899.383 \cdot 2}{2\sqrt{4907350451606845763597379498834801987276076711}}\approx 1.89472646812597$ (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

3.1.3271.1, 9.1.114478037712481.1

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 $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3271Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3271.2t1.a.a$1$ $ 3271 $ \(\Q(\sqrt{-3271}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3271.3t2.a.a$2$ $ 3271 $ 3.1.3271.1 $S_3$ (as 3T2) $1$ $0$
* 2.3271.9t3.a.b$2$ $ 3271 $ 9.1.114478037712481.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3271.9t3.a.c$2$ $ 3271 $ 9.1.114478037712481.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3271.9t3.a.a$2$ $ 3271 $ 9.1.114478037712481.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3271.27t8.a.b$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.g$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.f$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.e$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.d$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.c$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.a$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.i$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3271.27t8.a.h$2$ $ 3271 $ 27.1.4907350451606845763597379498834801987276076711.1 $D_{27}$ (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 *.