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Properties

Label 27.27.1530571656...8809.1
Degree $27$
Signature $[27, 0]$
Discriminant $7^{18}\cdot 109^{26}$
Root discriminant $335.25$
Ramified primes $7, 109$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-89453361143, 1824983191615, -7952131380114, -2408923070548, 30919931451508, -14229439088616, -27783814512537, 16962632962667, 10946482266142, -7833979925097, -2242620213654, 1920886337379, 254179800245, -281167865555, -16092388946, 26245873136, 516991856, -1617344119, -4080349, 66361580, -192801, -1780259, 3337, 29577, 47, -270, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 270*x^25 + 47*x^24 + 29577*x^23 + 3337*x^22 - 1780259*x^21 - 192801*x^20 + 66361580*x^19 - 4080349*x^18 - 1617344119*x^17 + 516991856*x^16 + 26245873136*x^15 - 16092388946*x^14 - 281167865555*x^13 + 254179800245*x^12 + 1920886337379*x^11 - 2242620213654*x^10 - 7833979925097*x^9 + 10946482266142*x^8 + 16962632962667*x^7 - 27783814512537*x^6 - 14229439088616*x^5 + 30919931451508*x^4 - 2408923070548*x^3 - 7952131380114*x^2 + 1824983191615*x - 89453361143)
 
gp: K = bnfinit(x^27 - x^26 - 270*x^25 + 47*x^24 + 29577*x^23 + 3337*x^22 - 1780259*x^21 - 192801*x^20 + 66361580*x^19 - 4080349*x^18 - 1617344119*x^17 + 516991856*x^16 + 26245873136*x^15 - 16092388946*x^14 - 281167865555*x^13 + 254179800245*x^12 + 1920886337379*x^11 - 2242620213654*x^10 - 7833979925097*x^9 + 10946482266142*x^8 + 16962632962667*x^7 - 27783814512537*x^6 - 14229439088616*x^5 + 30919931451508*x^4 - 2408923070548*x^3 - 7952131380114*x^2 + 1824983191615*x - 89453361143, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 270 x^{25} + 47 x^{24} + 29577 x^{23} + 3337 x^{22} - 1780259 x^{21} - 192801 x^{20} + 66361580 x^{19} - 4080349 x^{18} - 1617344119 x^{17} + 516991856 x^{16} + 26245873136 x^{15} - 16092388946 x^{14} - 281167865555 x^{13} + 254179800245 x^{12} + 1920886337379 x^{11} - 2242620213654 x^{10} - 7833979925097 x^{9} + 10946482266142 x^{8} + 16962632962667 x^{7} - 27783814512537 x^{6} - 14229439088616 x^{5} + 30919931451508 x^{4} - 2408923070548 x^{3} - 7952131380114 x^{2} + 1824983191615 x - 89453361143 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(153057165655742785694240270408062158957719092936539775009144398818809=7^{18}\cdot 109^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $335.25$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(763=7\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{763}(1,·)$, $\chi_{763}(323,·)$, $\chi_{763}(9,·)$, $\chi_{763}(729,·)$, $\chi_{763}(375,·)$, $\chi_{763}(463,·)$, $\chi_{763}(424,·)$, $\chi_{763}(81,·)$, $\chi_{763}(471,·)$, $\chi_{763}(281,·)$, $\chi_{763}(221,·)$, $\chi_{763}(352,·)$, $\chi_{763}(457,·)$, $\chi_{763}(485,·)$, $\chi_{763}(550,·)$, $\chi_{763}(296,·)$, $\chi_{763}(233,·)$, $\chi_{763}(298,·)$, $\chi_{763}(365,·)$, $\chi_{763}(240,·)$, $\chi_{763}(561,·)$, $\chi_{763}(372,·)$, $\chi_{763}(393,·)$, $\chi_{763}(116,·)$, $\chi_{763}(634,·)$, $\chi_{763}(571,·)$, $\chi_{763}(618,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{101} a^{24} + \frac{33}{101} a^{23} + \frac{34}{101} a^{22} + \frac{49}{101} a^{21} - \frac{26}{101} a^{20} + \frac{5}{101} a^{19} + \frac{17}{101} a^{18} + \frac{43}{101} a^{17} + \frac{35}{101} a^{16} + \frac{34}{101} a^{15} + \frac{6}{101} a^{14} + \frac{36}{101} a^{13} + \frac{20}{101} a^{12} - \frac{41}{101} a^{11} + \frac{50}{101} a^{10} + \frac{32}{101} a^{9} + \frac{3}{101} a^{8} - \frac{31}{101} a^{7} - \frac{3}{101} a^{6} - \frac{36}{101} a^{5} - \frac{41}{101} a^{4} - \frac{12}{101} a^{3} - \frac{13}{101} a^{2} - \frac{25}{101} a$, $\frac{1}{101} a^{25} - \frac{45}{101} a^{23} + \frac{38}{101} a^{22} - \frac{27}{101} a^{21} - \frac{46}{101} a^{20} - \frac{47}{101} a^{19} - \frac{13}{101} a^{18} + \frac{30}{101} a^{17} - \frac{10}{101} a^{16} - \frac{5}{101} a^{15} + \frac{40}{101} a^{14} + \frac{44}{101} a^{13} + \frac{6}{101} a^{12} - \frac{11}{101} a^{11} - \frac{2}{101} a^{10} - \frac{43}{101} a^{9} - \frac{29}{101} a^{8} + \frac{10}{101} a^{7} - \frac{38}{101} a^{6} + \frac{36}{101} a^{5} + \frac{28}{101} a^{4} - \frac{21}{101} a^{3} + \frac{17}{101} a$, $\frac{1}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{26} - \frac{8081084639930685175719677962401252739980696419894521618990615465166089333371219071788507845570142864898394874003546525198990120227689601278369553605881443}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{25} - \frac{9258601279955857361636531344024274866635886099631894908335408821913239977396921397386960972990467928387602525316962663621978034688617113070912175419939309}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{24} + \frac{187995320159062694735143344796223392067832931664230126585824388249810397397197608896006911251922759291127020836425310045058993624184828563829243959864340333}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{23} - \frac{158591417373891379959610622429637919659190999300937608780909539899702532853935340786260826110613896140610872636403726052640651132639506968170858694627085292}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{22} - \frac{723768432833620054104527561583162114300033035430204287724232697613421318603008355478213332817878596222671469281644666314923704866525764479234862743071125221}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{21} + \frac{3854988855299107709931044418819274073604060264322234709288371081678807325355544754795189256851947100482810374257878051646688021731725515245143877511054602830}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{20} + \frac{900044867607994297415803950191904956355801613130958432984398765608863783550966369004666841926778122635731365405150281271898976417420434445506973876395695470}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{19} - \frac{2466808026468342922989677929553371071399209270828390836304360852016773892821797746300227372434240330647840061905465057445475116796087846618006225127754009979}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{18} - \frac{655923003011176643958475549566815135965655062193675857982153376795556579139969631659619334243702046212783190569023827294271148615006094633208123593799979854}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{17} - \frac{875543558602641757395247141840112068408623222613414588940500350122754034367343738927948169268176275534552874325069106866345972520744048359586985625816837176}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{16} + \frac{552746399893216852193686382757330627251784657025263802518706095475956634601333810615477139638897849012754457736078100965274380183878301697739531431459371543}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{15} - \frac{1896825183214704477728862192347797847373791856727367392685757853799878522542024197051558041919499614119626102563672002178928641244788242106125591147375294408}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{14} + \frac{716275549441714917558641737876506627476416806658942633341858946812233612264530354524697604812809172771403051866530890890540871605124109222800554573811335837}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{13} + \frac{1984836122784178334869179430969832938572937529888312257429274265210872429002011917850244858450674154553201956719121086188162210233023975265242373643500865759}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{12} + \frac{2020810276710583939941425348797179753390285013893251600745283241472839064612247740827446027862066213323098250391651828976428852807014255976358649764594355446}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{11} + \frac{1501685888187628064635834975489645714480407285522986781906518316092538513016840244260567636525122127736671568958013544980115079866749086015689433456797929550}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{10} + \frac{488092222274251589527817470669836981705090124101097472612693027806484762950175795643654276844091982523251856087380928599865777235542524889729326604487942613}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{9} - \frac{319874746954244042628287054696250032047729558812860198840218343639788120565963292184605654572295042749707882999055418035532659716483358450631306510501932105}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{8} + \frac{2284442294313038803744703079648238436880107633584441396259750991258658224537147483586480849198543966802715272302259947294764353141525913879271849415087888039}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{7} + \frac{1298249596790356861237998580912575775697428627252084013227416948304366557502367474792353496270188700458490462791211141920615852513971631611294619775317009809}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{6} + \frac{2182877981339070305240921269871366700474415749268278210383455647162810030659927783323000932703301078919258263051906410821833349434390970271077231496984463214}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{5} - \frac{2731262705404139082397570936278705356837240315217797718124452959749933659942699696773251938069486698201092932011400161657729869538342609205392208562983723573}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{4} + \frac{38764910135496510832042073829914786811205066683713495544484637066051579001234278832701411804120709435679947525151840338459456362548878861197630854638475249}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{3} - \frac{205181962212422391791066005991664969719033736470400963944753083521773357676539025037868598109856446409793601048388569577576571334331221414869168610436919480}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a^{2} + \frac{2916306678876951157987712320151340298208572969882689343964528024146381678778038630216666355946746596371035328507195434238083234112816721415424526193517453580}{8049713476849356495659254880538770538800131410728895463323247869749508806851645582221741361252833746299273821178342893578616104887588301276143414862156470577} a + \frac{8631464872512201774909446582979604822975203019802233041105096573277968712415763068514290903286828977050999394810377279214461726745551658897697897047301388}{79700133434152044511477771094443272661387439710187083795279681878708007988630154279423181794582512339596770506716266273055604998887012883922212028338182877}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $26$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.11881.1, 9.9.19925626416901921.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ $27$ $27$ R $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ $27$ $27$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
109Data not computed