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Properties

Label 27.27.3463254175...9849.5
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 19^{24}$
Root discriminant $627.71$
Ramified primes $3, 19$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
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![249008793852866527, -920607736797292131, -170783754800326866, 1775016705261914199, 59924124491342760, -856864467175196223, -8095013308479636, 189345054725982270, 547782855461028, -23533430642034312, -20822155324542, 1824136571981646, 464928508044, -93545465229828, -6050255310, 3282297025608, 42457932, -80025043581, -124146, 1354686795, 0, -15617943, 0, 116964, 0, -513, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 124146*x^18 - 80025043581*x^17 + 42457932*x^16 + 3282297025608*x^15 - 6050255310*x^14 - 93545465229828*x^13 + 464928508044*x^12 + 1824136571981646*x^11 - 20822155324542*x^10 - 23533430642034312*x^9 + 547782855461028*x^8 + 189345054725982270*x^7 - 8095013308479636*x^6 - 856864467175196223*x^5 + 59924124491342760*x^4 + 1775016705261914199*x^3 - 170783754800326866*x^2 - 920607736797292131*x + 249008793852866527)
 
gp: K = bnfinit(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 124146*x^18 - 80025043581*x^17 + 42457932*x^16 + 3282297025608*x^15 - 6050255310*x^14 - 93545465229828*x^13 + 464928508044*x^12 + 1824136571981646*x^11 - 20822155324542*x^10 - 23533430642034312*x^9 + 547782855461028*x^8 + 189345054725982270*x^7 - 8095013308479636*x^6 - 856864467175196223*x^5 + 59924124491342760*x^4 + 1775016705261914199*x^3 - 170783754800326866*x^2 - 920607736797292131*x + 249008793852866527, 1)
 

Normalized defining polynomial

\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 124146 x^{18} - 80025043581 x^{17} + 42457932 x^{16} + 3282297025608 x^{15} - 6050255310 x^{14} - 93545465229828 x^{13} + 464928508044 x^{12} + 1824136571981646 x^{11} - 20822155324542 x^{10} - 23533430642034312 x^{9} + 547782855461028 x^{8} + 189345054725982270 x^{7} - 8095013308479636 x^{6} - 856864467175196223 x^{5} + 59924124491342760 x^{4} + 1775016705261914199 x^{3} - 170783754800326866 x^{2} - 920607736797292131 x + 249008793852866527 \)

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:  \(3463254175600113063839837232871102966170246194081956872095753304432278349849=3^{94}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $627.71$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1539=3^{4}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1539}(64,·)$, $\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(1366,·)$, $\chi_{1539}(577,·)$, $\chi_{1539}(139,·)$, $\chi_{1539}(652,·)$, $\chi_{1539}(1165,·)$, $\chi_{1539}(340,·)$, $\chi_{1539}(853,·)$, $\chi_{1539}(214,·)$, $\chi_{1539}(727,·)$, $\chi_{1539}(1240,·)$, $\chi_{1539}(1090,·)$, $\chi_{1539}(358,·)$, $\chi_{1539}(871,·)$, $\chi_{1539}(1384,·)$, $\chi_{1539}(427,·)$, $\chi_{1539}(940,·)$, $\chi_{1539}(1453,·)$, $\chi_{1539}(175,·)$, $\chi_{1539}(688,·)$, $\chi_{1539}(1201,·)$, $\chi_{1539}(505,·)$, $\chi_{1539}(1018,·)$, $\chi_{1539}(1531,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{323} a^{10} + \frac{6}{17} a^{8} + \frac{1}{17} a^{6} - \frac{8}{17} a^{4} + \frac{5}{17} a^{2} + \frac{4}{17}$, $\frac{1}{323} a^{11} - \frac{5}{323} a^{9} + \frac{1}{17} a^{7} - \frac{8}{17} a^{5} + \frac{5}{17} a^{3} + \frac{4}{17} a$, $\frac{1}{323} a^{12} - \frac{3}{17} a^{8} - \frac{3}{17} a^{6} - \frac{1}{17} a^{4} - \frac{5}{17} a^{2} + \frac{3}{17}$, $\frac{1}{323} a^{13} - \frac{6}{323} a^{9} - \frac{3}{17} a^{7} - \frac{1}{17} a^{5} - \frac{5}{17} a^{3} + \frac{3}{17} a$, $\frac{1}{104329} a^{14} - \frac{3}{5491} a^{13} + \frac{3}{5491} a^{12} - \frac{7}{5491} a^{11} + \frac{1}{5491} a^{10} + \frac{104}{5491} a^{9} - \frac{97}{289} a^{8} + \frac{2}{289} a^{7} + \frac{3}{289} a^{6} - \frac{511}{5491} a^{5} - \frac{29}{289} a^{4} - \frac{3}{289} a^{3} + \frac{32}{289} a^{2} + \frac{14}{289} a - \frac{94}{289}$, $\frac{1}{104329} a^{15} + \frac{2}{5491} a^{13} - \frac{6}{5491} a^{12} - \frac{7}{5491} a^{11} + \frac{8}{5491} a^{10} - \frac{46}{5491} a^{9} + \frac{134}{289} a^{8} - \frac{2}{289} a^{7} - \frac{1461}{5491} a^{6} + \frac{53}{289} a^{5} + \frac{27}{289} a^{4} + \frac{99}{289} a^{3} - \frac{100}{289} a^{2} - \frac{112}{289} a - \frac{122}{289}$, $\frac{1}{104329} a^{16} + \frac{6}{5491} a^{13} - \frac{2}{5491} a^{12} + \frac{2}{5491} a^{11} + \frac{1}{5491} a^{10} - \frac{12}{5491} a^{9} + \frac{80}{289} a^{8} - \frac{2259}{5491} a^{7} - \frac{44}{289} a^{6} - \frac{141}{289} a^{5} + \frac{113}{289} a^{4} + \frac{31}{289} a^{3} - \frac{53}{289} a^{2} - \frac{25}{289} a - \frac{66}{289}$, $\frac{1}{104329} a^{17} - \frac{7}{5491} a^{10} + \frac{2}{323} a^{9} - \frac{1138}{5491} a^{8} - \frac{3}{17} a^{7} + \frac{61}{289} a^{6} + \frac{5}{17} a^{5} + \frac{22}{289} a^{4} + \frac{1}{17} a^{3} + \frac{135}{289} a^{2} - \frac{98}{289} a + \frac{74}{289}$, $\frac{1}{4921540511596649} a^{18} - \frac{18}{259028447978771} a^{16} + \frac{135}{13633076209409} a^{14} - \frac{546}{717530326811} a^{12} + \frac{24453}{717530326811} a^{10} + \frac{23487485883}{896292207539} a^{9} - \frac{643302}{717530326811} a^{8} - \frac{22694276623}{47173274081} a^{7} + \frac{9506574}{717530326811} a^{6} + \frac{1047124596}{2482803899} a^{5} - \frac{70373340}{717530326811} a^{4} - \frac{588329902}{2482803899} a^{3} + \frac{200564019}{717530326811} a^{2} - \frac{619005797}{2482803899} a - \frac{302996942232}{717530326811}$, $\frac{1}{4921540511596649} a^{19} - \frac{18}{259028447978771} a^{17} + \frac{135}{13633076209409} a^{15} - \frac{546}{717530326811} a^{13} + \frac{24453}{717530326811} a^{11} + \frac{21901067363}{15236967528163} a^{10} - \frac{643302}{717530326811} a^{9} - \frac{244282880348}{801945659377} a^{8} + \frac{9506574}{717530326811} a^{7} - \frac{2061313060}{42207666283} a^{6} - \frac{70373340}{717530326811} a^{5} - \frac{19932823930}{42207666283} a^{4} + \frac{200564019}{717530326811} a^{3} + \frac{16787744340}{42207666283} a^{2} - \frac{302996942232}{717530326811} a + \frac{2}{17}$, $\frac{1}{4921540511596649} a^{20} - \frac{189}{13633076209409} a^{16} + \frac{1884}{717530326811} a^{14} - \frac{162279}{717530326811} a^{12} + \frac{21901067363}{15236967528163} a^{11} + \frac{7719624}{717530326811} a^{10} + \frac{20830529438}{801945659377} a^{9} - \frac{210502710}{717530326811} a^{8} + \frac{17754976997}{42207666283} a^{7} + \frac{3180874968}{717530326811} a^{6} - \frac{9854367538}{42207666283} a^{5} - \frac{1403948133}{42207666283} a^{4} + \frac{15058663035}{42207666283} a^{3} - \frac{234404047734}{717530326811} a^{2} - \frac{6282461905}{42207666283} a - \frac{300587182560}{717530326811}$, $\frac{1}{93509269720336331} a^{21} - \frac{189}{259028447978771} a^{17} + \frac{1884}{13633076209409} a^{15} - \frac{8541}{717530326811} a^{13} - \frac{213965303042}{289502383035097} a^{12} + \frac{406296}{717530326811} a^{11} + \frac{8416509943}{15236967528163} a^{10} - \frac{11079090}{717530326811} a^{9} - \frac{314940745469}{801945659377} a^{8} + \frac{167414472}{717530326811} a^{7} - \frac{1433368149}{42207666283} a^{6} - \frac{73892007}{42207666283} a^{5} + \frac{2229973996}{42207666283} a^{4} + \frac{5505838566754}{13633076209409} a^{3} - \frac{18102304851}{42207666283} a^{2} + \frac{286297654312}{717530326811} a + \frac{8}{17}$, $\frac{1}{93509269720336331} a^{22} - \frac{1518}{13633076209409} a^{16} + \frac{16974}{717530326811} a^{14} - \frac{213965303042}{289502383035097} a^{13} - \frac{1554390}{717530326811} a^{12} + \frac{8416509943}{15236967528163} a^{11} + \frac{76731633}{717530326811} a^{10} + \frac{20705412870}{801945659377} a^{9} - \frac{2142683010}{717530326811} a^{8} + \frac{16703179176}{42207666283} a^{7} + \frac{32881943115}{717530326811} a^{6} - \frac{146214431}{332343829} a^{5} + \frac{41431526582}{801945659377} a^{4} - \frac{15153825412}{42207666283} a^{3} + \frac{288992719730}{717530326811} a^{2} + \frac{7276865018}{42207666283} a - \frac{286044109636}{717530326811}$, $\frac{1}{132623542679464975902983} a^{23} - \frac{7307}{6980186456813946100157} a^{22} - \frac{23}{6980186456813946100157} a^{21} + \frac{11460}{367378234569155057903} a^{20} + \frac{230}{367378234569155057903} a^{19} + \frac{86}{1463658305056394653} a^{18} - \frac{69}{1017668239803753623} a^{17} + \frac{9953358}{1137393915074783461} a^{16} + \frac{276}{59862837635514919} a^{15} - \frac{3262441377629281}{6980186456813946100157} a^{14} + \frac{190513089538994442}{367378234569155057903} a^{13} + \frac{45552628296298868}{367378234569155057903} a^{12} - \frac{18015849341813642}{19335696556271318837} a^{11} - \frac{10900512143512378}{19335696556271318837} a^{10} - \frac{466927034003799256}{19335696556271318837} a^{9} + \frac{660916643477726}{8013135746486249} a^{8} - \frac{396122637930744426}{1017668239803753623} a^{7} + \frac{21267066702872347}{53561486305460717} a^{6} + \frac{1359888344001207893}{19335696556271318837} a^{5} + \frac{483750408333814846}{1017668239803753623} a^{4} + \frac{6628717431711308}{59862837635514919} a^{3} - \frac{25662566396204443}{53561486305460717} a^{2} - \frac{7818811594503938}{53561486305460717} a + \frac{13840120823380097}{53561486305460717}$, $\frac{1}{132623542679464975902983} a^{24} - \frac{24}{6980186456813946100157} a^{22} + \frac{10461}{6980186456813946100157} a^{21} + \frac{252}{367378234569155057903} a^{20} + \frac{60}{5174341331959930393} a^{19} - \frac{80}{1017668239803753623} a^{18} - \frac{2053809}{19335696556271318837} a^{17} + \frac{18}{3150675665027101} a^{16} - \frac{3248972368273417}{6980186456813946100157} a^{15} - \frac{864}{3150675665027101} a^{14} + \frac{48736043617173396}{367378234569155057903} a^{13} - \frac{335476550520986201}{367378234569155057903} a^{12} + \frac{6895725699072821}{19335696556271318837} a^{11} - \frac{21439857720584760}{19335696556271318837} a^{10} + \frac{665542376974565}{59862837635514919} a^{9} - \frac{69198561983464750}{1017668239803753623} a^{8} - \frac{184871749323154}{53561486305460717} a^{7} - \frac{9460407153144328301}{19335696556271318837} a^{6} + \frac{16954710008558046}{53561486305460717} a^{5} - \frac{62823318771283680}{1017668239803753623} a^{4} - \frac{468705783899551182}{1017668239803753623} a^{3} + \frac{24081995131594739}{53561486305460717} a^{2} + \frac{10474273006187475}{53561486305460717} a - \frac{14081330434424903}{53561486305460717}$, $\frac{1}{2519847310909834542156677} a^{25} - \frac{33381}{6980186456813946100157} a^{22} - \frac{300}{6980186456813946100157} a^{21} - \frac{19395}{367378234569155057903} a^{20} + \frac{4000}{367378234569155057903} a^{19} + \frac{31842}{367378234569155057903} a^{18} - \frac{1350}{1017668239803753623} a^{17} + \frac{130569014791592202}{132623542679464975902983} a^{16} + \frac{5760}{59862837635514919} a^{15} - \frac{29320576974813321}{6980186456813946100157} a^{14} + \frac{71641213727585988}{367378234569155057903} a^{13} - \frac{471546174124900292}{367378234569155057903} a^{12} + \frac{16527923431311818}{19335696556271318837} a^{11} - \frac{19968263386996534}{19335696556271318837} a^{10} + \frac{110496777479105000}{19335696556271318837} a^{9} + \frac{111130298165984986}{1017668239803753623} a^{8} + \frac{112521930968529663715}{367378234569155057903} a^{7} - \frac{5026286352257485}{53561486305460717} a^{6} + \frac{3515713422080962742}{19335696556271318837} a^{5} + \frac{377307808334068257}{1017668239803753623} a^{4} - \frac{218167473375897745}{1017668239803753623} a^{3} + \frac{1373697770540533}{3150675665027101} a^{2} + \frac{9309294992951102}{53561486305460717} a + \frac{6494847862009434}{53561486305460717}$, $\frac{1}{2519847310909834542156677} a^{26} - \frac{325}{6980186456813946100157} a^{22} - \frac{26602}{6980186456813946100157} a^{21} + \frac{4550}{367378234569155057903} a^{20} - \frac{28073}{367378234569155057903} a^{19} + \frac{10551}{367378234569155057903} a^{18} + \frac{7680116193538533}{7801384863497939758999} a^{17} - \frac{491514}{1137393915074783461} a^{16} - \frac{29242373197983654}{6980186456813946100157} a^{15} + \frac{4371780}{59862837635514919} a^{14} - \frac{256883036744247258}{367378234569155057903} a^{13} + \frac{325451868939448777}{367378234569155057903} a^{12} - \frac{945488097134398}{19335696556271318837} a^{11} + \frac{1804225447715853}{19335696556271318837} a^{10} + \frac{172993860239677697}{19335696556271318837} a^{9} - \frac{711645988550529090}{2892742004481535889} a^{8} - \frac{473804344395819598}{1017668239803753623} a^{7} + \frac{261428409989759749}{19335696556271318837} a^{6} - \frac{14006338704601158}{53561486305460717} a^{5} + \frac{2567442884285488}{14333355490193713} a^{4} + \frac{16103571645164588}{59862837635514919} a^{3} - \frac{195347946342993}{53561486305460717} a^{2} + \frac{543372572120461}{53561486305460717} a - \frac{24604977589704228}{53561486305460717}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 158707282147788600000000000000 \) (assuming GRH)
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

\(\Q(\zeta_{9})^+\), 9.9.1476349596018920529.1

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{27}$ R $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ $27$

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
3Data not computed
$19$19.9.8.5$x^{9} + 19456$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.5$x^{9} + 19456$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.5$x^{9} + 19456$$9$$1$$8$$C_9$$[\ ]_{9}$