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Properties

Label 27.27.1156455411...4041.1
Degree $27$
Signature $[27, 0]$
Discriminant $541^{26}$
Root discriminant $428.52$
Ramified prime $541$
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![9025899413, 19932730946, -114088799811, -246881343980, 423859606824, 747910684042, -895854373113, -809764643713, 1047941306021, 257304955275, -564208979448, 27952037234, 148707111628, -29583496180, -20405620517, 6070505088, 1558672841, -614779252, -67728086, 35778151, 1622160, -1241609, -19668, 24978, 127, -260, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 260*x^25 + 127*x^24 + 24978*x^23 - 19668*x^22 - 1241609*x^21 + 1622160*x^20 + 35778151*x^19 - 67728086*x^18 - 614779252*x^17 + 1558672841*x^16 + 6070505088*x^15 - 20405620517*x^14 - 29583496180*x^13 + 148707111628*x^12 + 27952037234*x^11 - 564208979448*x^10 + 257304955275*x^9 + 1047941306021*x^8 - 809764643713*x^7 - 895854373113*x^6 + 747910684042*x^5 + 423859606824*x^4 - 246881343980*x^3 - 114088799811*x^2 + 19932730946*x + 9025899413)
 
gp: K = bnfinit(x^27 - x^26 - 260*x^25 + 127*x^24 + 24978*x^23 - 19668*x^22 - 1241609*x^21 + 1622160*x^20 + 35778151*x^19 - 67728086*x^18 - 614779252*x^17 + 1558672841*x^16 + 6070505088*x^15 - 20405620517*x^14 - 29583496180*x^13 + 148707111628*x^12 + 27952037234*x^11 - 564208979448*x^10 + 257304955275*x^9 + 1047941306021*x^8 - 809764643713*x^7 - 895854373113*x^6 + 747910684042*x^5 + 423859606824*x^4 - 246881343980*x^3 - 114088799811*x^2 + 19932730946*x + 9025899413, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 260 x^{25} + 127 x^{24} + 24978 x^{23} - 19668 x^{22} - 1241609 x^{21} + 1622160 x^{20} + 35778151 x^{19} - 67728086 x^{18} - 614779252 x^{17} + 1558672841 x^{16} + 6070505088 x^{15} - 20405620517 x^{14} - 29583496180 x^{13} + 148707111628 x^{12} + 27952037234 x^{11} - 564208979448 x^{10} + 257304955275 x^{9} + 1047941306021 x^{8} - 809764643713 x^{7} - 895854373113 x^{6} + 747910684042 x^{5} + 423859606824 x^{4} - 246881343980 x^{3} - 114088799811 x^{2} + 19932730946 x + 9025899413 \)

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:  \(115645541170359168800904668451418896961215191921100766227083814325184041=541^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $428.52$
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:  $541$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(541\)
Dirichlet character group:    $\lbrace$$\chi_{541}(1,·)$, $\chi_{541}(66,·)$, $\chi_{541}(129,·)$, $\chi_{541}(329,·)$, $\chi_{541}(74,·)$, $\chi_{541}(76,·)$, $\chi_{541}(15,·)$, $\chi_{541}(80,·)$, $\chi_{541}(147,·)$, $\chi_{541}(399,·)$, $\chi_{541}(214,·)$, $\chi_{541}(411,·)$, $\chi_{541}(28,·)$, $\chi_{541}(349,·)$, $\chi_{541}(352,·)$, $\chi_{541}(225,·)$, $\chi_{541}(34,·)$, $\chi_{541}(420,·)$, $\chi_{541}(41,·)$, $\chi_{541}(366,·)$, $\chi_{541}(449,·)$, $\chi_{541}(243,·)$, $\chi_{541}(118,·)$, $\chi_{541}(312,·)$, $\chi_{541}(505,·)$, $\chi_{541}(58,·)$, $\chi_{541}(510,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{240697} a^{25} + \frac{51491}{240697} a^{24} + \frac{45043}{240697} a^{23} - \frac{71135}{240697} a^{22} + \frac{8614}{240697} a^{21} - \frac{76422}{240697} a^{20} - \frac{78053}{240697} a^{19} + \frac{13872}{240697} a^{18} + \frac{110471}{240697} a^{17} + \frac{86433}{240697} a^{16} - \frac{88342}{240697} a^{15} - \frac{85672}{240697} a^{14} - \frac{24418}{240697} a^{13} + \frac{93458}{240697} a^{12} + \frac{25034}{240697} a^{11} - \frac{40488}{240697} a^{10} + \frac{8399}{240697} a^{9} + \frac{72139}{240697} a^{8} + \frac{1949}{240697} a^{7} - \frac{36419}{240697} a^{6} + \frac{88273}{240697} a^{5} + \frac{104487}{240697} a^{4} + \frac{8587}{240697} a^{3} + \frac{1746}{240697} a^{2} - \frac{109670}{240697} a + \frac{33512}{240697}$, $\frac{1}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{26} - \frac{1023900490553836841146096049151558560825308504583785757111591404516823286896442452696026687603342656356860409860634944810242044507093}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{25} - \frac{114929601550604245265421393117908722411541580275766478837925689042892144062066397161034363053691467071178251328268651748268531104311290181}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{24} + \frac{121671386439595864628323364138393460787251854681414996445506873374850491842822455895445694713335719403241555229700891857479366259932157310}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{23} - \frac{7516422805268903343936495632121271192773796610861218543837024783119363374064811636367203597758296438519366529363328743013362903911382223}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{22} - \frac{169723702793454199161395217947541889418431126829393701169401224495838769768665039110952981317921887634015168994669248094085674348460046205}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{21} + \frac{220691026549616782590004511038584957034453632256715874625711921321197154584577963282184160975964056647732781399368782383005785210993808394}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{20} - \frac{23479010522682114956043993741720535909692654260879396328193582946814723324948192960683801870424092998643627110564669412762991050408807723}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{19} - \frac{13705319556114477381549935918136928327144254095592007331237138879063471888893442762067181989544351608814673251851167017378058432466060632}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{18} + \frac{204404842692587849051653620742015897016388793159676109806424892468967140851265124298267741431790663728352119062433819528653450418595790051}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{17} - \frac{197481454266120241741454661110621020725016544850425147331313582300957053912131291169620176676486458091972671427460988712647701014893753817}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{16} + \frac{81395299652398070424008893329791139122092432015134093701107352203890357388755155204767255536277430721623703269780618225038270874539345119}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{15} - \frac{123621733986407367183857060013482815900547702309425391889509714709852179861631155380984096948824843563192455308171160138939895358644904085}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{14} - \frac{143420716866753319993105553545392156947925349757299889476504858204548716812662392264725854655799753252370505518469662207546502722694225365}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{13} + \frac{17800037560831876115899730885868968807187674992992572804885111448362894047418922277125898653111377526824897582363865938169329394064230671}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{12} + \frac{34225966218306540797920560969829519329178040144858276353950918407674188266595721583474825477271319345468287452380828271655123675572332678}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{11} + \frac{17067217029586029830572052957409667083515949847510270344926450839088610601454196996590355170214530232801031474349364903298677735916528083}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{10} - \frac{54154375092986176780376966724642435247120944646708733037417401199214282949179597677644356231433639807085953432942658293897758536449363811}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{9} - \frac{176954783387194786045221043886212834810650539292521146262411642430641064835424491630225183913178345369647172357122087082907226504854407894}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{8} - \frac{115723631777330983589051355389270476866196207953836526160493232937733536171750529158466599288635172994095887374667392600831635323653241661}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{7} - \frac{121261160339615326821816095545297174787078557165243057702874068500611384568968597186389263486132687651248495097771838308139649634569981856}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{6} - \frac{39063824925030397435337882529697710991483669122325442287142423440644827165676918322009699434093707514805561693036900390195505908008798497}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{5} - \frac{116861900568481585755401617975783444609878053017468467397022105616908304347093560500236119861445660362968695200581131891086856442023057379}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{4} + \frac{168521839657256616703272686042131198932296393270806268571745374473096070433096285214470617991252254347064606526413195032594588885780971753}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{3} - \frac{108884339500504302148259006068111155425815689045254981646757638359545336437761656090281015999521399762619768002657865072424947235864899664}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a^{2} + \frac{180772734175561889950637490983249455325144092551725097288824534313135357648245097525079591344395521414460672002719324342283172633896265181}{494368925370940198935661812468349086517073675093115908628128430121925120936531783460385800563746263059317543482679654872853121128977739013} a + \frac{96431318027439780441521256422020968723982383973365700940657656556499673613404275185557839940873698402977757272894731045066692390783095}{833674410406307249469918739406996773215975843327345545747265480812689917262279567386822597915255081044380343141112402821000204264717941}$

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

3.3.292681.1, 9.9.7338006985513707753121.1

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 $27$ $27$ $27$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ $27$ $27$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ $27$ $27$ $27$ $27$ $27$ $27$ $27$

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
541Data not computed