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Properties

Label 27.27.5659106580...6209.3
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 7^{18}\cdot 13^{18}$
Root discriminant $296.70$
Ramified primes $3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![230754987, 969338907, -6710328360, -2061705231, 42662749410, -31916666358, -72744969402, 81498609540, 53008275636, -79554132019, -18541949166, 40044205305, 3103240140, -11538409068, -233589906, 1997088057, 7745112, -211280076, -92880, 13841163, 0, -560310, 0, 13581, 0, -180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 180*x^25 + 13581*x^23 - 560310*x^21 + 13841163*x^19 - 92880*x^18 - 211280076*x^17 + 7745112*x^16 + 1997088057*x^15 - 233589906*x^14 - 11538409068*x^13 + 3103240140*x^12 + 40044205305*x^11 - 18541949166*x^10 - 79554132019*x^9 + 53008275636*x^8 + 81498609540*x^7 - 72744969402*x^6 - 31916666358*x^5 + 42662749410*x^4 - 2061705231*x^3 - 6710328360*x^2 + 969338907*x + 230754987)
 
gp: K = bnfinit(x^27 - 180*x^25 + 13581*x^23 - 560310*x^21 + 13841163*x^19 - 92880*x^18 - 211280076*x^17 + 7745112*x^16 + 1997088057*x^15 - 233589906*x^14 - 11538409068*x^13 + 3103240140*x^12 + 40044205305*x^11 - 18541949166*x^10 - 79554132019*x^9 + 53008275636*x^8 + 81498609540*x^7 - 72744969402*x^6 - 31916666358*x^5 + 42662749410*x^4 - 2061705231*x^3 - 6710328360*x^2 + 969338907*x + 230754987, 1)
 

Normalized defining polynomial

\( x^{27} - 180 x^{25} + 13581 x^{23} - 560310 x^{21} + 13841163 x^{19} - 92880 x^{18} - 211280076 x^{17} + 7745112 x^{16} + 1997088057 x^{15} - 233589906 x^{14} - 11538409068 x^{13} + 3103240140 x^{12} + 40044205305 x^{11} - 18541949166 x^{10} - 79554132019 x^{9} + 53008275636 x^{8} + 81498609540 x^{7} - 72744969402 x^{6} - 31916666358 x^{5} + 42662749410 x^{4} - 2061705231 x^{3} - 6710328360 x^{2} + 969338907 x + 230754987 \)

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:  \(5659106580522258442740055894009895600932750567734671570320232266209=3^{66}\cdot 7^{18}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.70$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2457=3^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2457}(1,·)$, $\chi_{2457}(898,·)$, $\chi_{2457}(1348,·)$, $\chi_{2457}(841,·)$, $\chi_{2457}(1738,·)$, $\chi_{2457}(1303,·)$, $\chi_{2457}(79,·)$, $\chi_{2457}(529,·)$, $\chi_{2457}(1810,·)$, $\chi_{2457}(22,·)$, $\chi_{2457}(919,·)$, $\chi_{2457}(2200,·)$, $\chi_{2457}(100,·)$, $\chi_{2457}(991,·)$, $\chi_{2457}(2146,·)$, $\chi_{2457}(484,·)$, $\chi_{2457}(1381,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(1660,·)$, $\chi_{2457}(172,·)$, $\chi_{2457}(1327,·)$, $\chi_{2457}(562,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1717,·)$, $\chi_{2457}(2167,·)$, $\chi_{2457}(508,·)$, $\chi_{2457}(2122,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{22} + \frac{1}{3} a^{16} - \frac{1}{9} a^{4}$, $\frac{1}{4441707} a^{23} + \frac{229850}{4441707} a^{22} + \frac{2945}{1480569} a^{21} - \frac{5594}{493523} a^{20} + \frac{11184}{493523} a^{19} - \frac{153994}{493523} a^{18} + \frac{689710}{1480569} a^{17} + \frac{57872}{1480569} a^{16} - \frac{21903}{493523} a^{15} - \frac{28145}{493523} a^{14} + \frac{39330}{493523} a^{13} - \frac{58076}{493523} a^{12} + \frac{173939}{493523} a^{11} - \frac{30116}{493523} a^{10} - \frac{58022}{493523} a^{9} - \frac{118377}{493523} a^{8} + \frac{177548}{493523} a^{7} - \frac{89319}{493523} a^{6} - \frac{1744759}{4441707} a^{5} - \frac{1760795}{4441707} a^{4} - \frac{173294}{1480569} a^{3} - \frac{78251}{493523} a^{2} - \frac{217577}{493523} a - \frac{198228}{493523}$, $\frac{1}{43879623453} a^{24} + \frac{230}{14626541151} a^{23} - \frac{49797948}{1625171239} a^{22} + \frac{527987263}{4875513717} a^{21} - \frac{753850151}{4875513717} a^{20} + \frac{795323070}{1625171239} a^{19} + \frac{179279665}{395311923} a^{18} - \frac{1171684882}{4875513717} a^{17} - \frac{1835762110}{4875513717} a^{16} + \frac{169487705}{1625171239} a^{15} + \frac{518493313}{1625171239} a^{14} + \frac{76921638}{1625171239} a^{13} + \frac{1579308958}{4875513717} a^{12} - \frac{219621981}{1625171239} a^{11} + \frac{103606132}{1625171239} a^{10} + \frac{187440899}{1625171239} a^{9} + \frac{16276968}{1625171239} a^{8} - \frac{643049080}{1625171239} a^{7} - \frac{9262981612}{43879623453} a^{6} - \frac{4565236685}{14626541151} a^{5} - \frac{555559661}{4875513717} a^{4} - \frac{1547031241}{4875513717} a^{3} - \frac{434676862}{4875513717} a^{2} - \frac{759332240}{1625171239} a + \frac{163059587}{1625171239}$, $\frac{1}{31549449262707} a^{25} + \frac{262}{31549449262707} a^{24} + \frac{289195}{10516483087569} a^{23} - \frac{551413479665}{10516483087569} a^{22} + \frac{159830335675}{3505494362523} a^{21} + \frac{307062368663}{3505494362523} a^{20} + \frac{2427047354005}{10516483087569} a^{19} - \frac{142815361529}{10516483087569} a^{18} + \frac{32339824831}{1168498120841} a^{17} + \frac{40209690094}{1168498120841} a^{16} + \frac{513814247806}{1168498120841} a^{15} + \frac{1786393993}{1168498120841} a^{14} + \frac{1641469425313}{3505494362523} a^{13} - \frac{1036215059591}{3505494362523} a^{12} - \frac{395873574516}{1168498120841} a^{11} + \frac{192433322493}{1168498120841} a^{10} + \frac{440083553587}{1168498120841} a^{9} + \frac{9824384655}{31581030293} a^{8} + \frac{394657350902}{852687817911} a^{7} + \frac{3545066259242}{31549449262707} a^{6} + \frac{3991843045292}{10516483087569} a^{5} + \frac{1683551514353}{10516483087569} a^{4} + \frac{1265478701705}{3505494362523} a^{3} - \frac{1326007610561}{3505494362523} a^{2} - \frac{546234587250}{1168498120841} a + \frac{379408093298}{1168498120841}$, $\frac{1}{357036489667700755103600561857904739658108158822167450033113272981981550435234861668289663162737764109} a^{26} - \frac{186731020815250989475435302269390567099732004552996654453685340680427537004083202794522}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{25} - \frac{411448112709911384855652957130638674172887705396141085005526050573607812989359735798285088}{39670721074188972789288951317544971073123128758018605559234808109109061159470540185365518129193084901} a^{24} + \frac{59901032349507541328520797597306198985749529721798034947656394377945729714734752674070060279}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{23} + \frac{1883656329595248327458395984333204641844867542335996440378371527221925193941444135533776593637931265}{39670721074188972789288951317544971073123128758018605559234808109109061159470540185365518129193084901} a^{22} + \frac{191354102424436915028602237794858715991505278093680571495986187048122770811299442783888312027923207}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a^{21} - \frac{19458263643459643178262079880150725726585781403522443107105515036034901583729276130826605884974494226}{119012163222566918367866853952634913219369386274055816677704424327327183478411620556096554387579254703} a^{20} + \frac{1359766453836428621521974685895335372697485700676696159107413935292561791838694409346950120912713544}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a^{19} + \frac{1119192287022971004670292729780524394222819351317477692147579308996978122101806161840177340577901232}{39670721074188972789288951317544971073123128758018605559234808109109061159470540185365518129193084901} a^{18} - \frac{457732148083764608598681958966642852267345781728499247684931153193845563948987805921548198819465044}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{17} + \frac{4259795289795838815298396509235303665555911197894418869194105756070648328328590747348718982341223361}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{16} + \frac{5185529003802966232014364158046305008918998958414761592695854499383801174786063105056174212275502415}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{15} - \frac{5596348361023021292992102286236362415286152858852700460051735692395741234865967973911868847902238990}{39670721074188972789288951317544971073123128758018605559234808109109061159470540185365518129193084901} a^{14} - \frac{306016936310225922593498682006028849618797603725809213591289198008461248269996360951270522297173168}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a^{13} - \frac{4910348563633192366684376021572819656327900068622239017933753238591943214879433809848987641021694377}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{12} + \frac{5669971728859770528342782758613837081620645877328881890014410125902738238254968981626743844586367044}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{11} + \frac{982655883395980759857208710822287077589606253254168081807880531869585475226907145447350389245972817}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a^{10} + \frac{2732736670640675731436931284351018412075216696499894695899400651449676677502374755799113164089868316}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{9} + \frac{78635082246087385908561813922277482571442555491312138241363669764394150290287814133826921910723204525}{357036489667700755103600561857904739658108158822167450033113272981981550435234861668289663162737764109} a^{8} - \frac{5352801915246816277222790665915942787377585127955888152880630133779823517596841811206356612477331920}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{7} + \frac{3854264358617960335265656256415850525412880606218727447981885819679369265459927823755220611639482817}{39670721074188972789288951317544971073123128758018605559234808109109061159470540185365518129193084901} a^{6} + \frac{6557648520604958410985948776075318130275727644893814469307190927443321724035828637425047979332075531}{13223573691396324263096317105848323691041042919339535186411602703036353719823513395121839376397694967} a^{5} - \frac{19586396300343062897918609294125282922595405981564676096819751012531724661387305503317614805293570824}{39670721074188972789288951317544971073123128758018605559234808109109061159470540185365518129193084901} a^{4} + \frac{1276068129652528045359504138081259974991535289937263465373055226418296409996816825211580328608523317}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a^{3} + \frac{411838476897099511313028499183846975152334344685954212733351933565058599635675606173361658808833131}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a^{2} - \frac{1627343153947796799410250616740655541847563117088126341556786052491008074118629216301891931169288406}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989} a + \frac{368854460835647721190529937877307017949367772137173198779624554396253908317100568155412971118025739}{4407857897132108087698772368616107897013680973113178395470534234345451239941171131707279792132564989}$

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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:  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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3876170092409284400000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.8281.2, \(\Q(\zeta_{9})^+\), 3.3.670761.2, 3.3.670761.4, 9.9.301789003173921081.10, 9.9.17820338848416865911969.1, 9.9.151470380950257681.1, 9.9.3691950281939241.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$3$3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.2$x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 51$$9$$1$$22$$C_9$$[2, 3]$
7Data not computed
13Data not computed