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Properties

Label 27.27.1197336751...1009.4
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 7^{18}\cdot 19^{24}$
Root discriminant $216.89$
Ramified primes $3, 7, 19$
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![-1834540021, -6912437058, 10730880483, 44013206071, -22163323110, -103424572098, 19715601263, 120762893079, -7402115451, -79891828714, 34580448, 32083489470, 949692990, -8127116829, -360894987, 1325068846, 67545369, -140140161, -7334897, 9585783, 475152, -417005, -17940, 11058, 362, -162, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 162*x^25 + 362*x^24 + 11058*x^23 - 17940*x^22 - 417005*x^21 + 475152*x^20 + 9585783*x^19 - 7334897*x^18 - 140140161*x^17 + 67545369*x^16 + 1325068846*x^15 - 360894987*x^14 - 8127116829*x^13 + 949692990*x^12 + 32083489470*x^11 + 34580448*x^10 - 79891828714*x^9 - 7402115451*x^8 + 120762893079*x^7 + 19715601263*x^6 - 103424572098*x^5 - 22163323110*x^4 + 44013206071*x^3 + 10730880483*x^2 - 6912437058*x - 1834540021)
 
gp: K = bnfinit(x^27 - 3*x^26 - 162*x^25 + 362*x^24 + 11058*x^23 - 17940*x^22 - 417005*x^21 + 475152*x^20 + 9585783*x^19 - 7334897*x^18 - 140140161*x^17 + 67545369*x^16 + 1325068846*x^15 - 360894987*x^14 - 8127116829*x^13 + 949692990*x^12 + 32083489470*x^11 + 34580448*x^10 - 79891828714*x^9 - 7402115451*x^8 + 120762893079*x^7 + 19715601263*x^6 - 103424572098*x^5 - 22163323110*x^4 + 44013206071*x^3 + 10730880483*x^2 - 6912437058*x - 1834540021, 1)
 

Normalized defining polynomial

\( x^{27} - 3 x^{26} - 162 x^{25} + 362 x^{24} + 11058 x^{23} - 17940 x^{22} - 417005 x^{21} + 475152 x^{20} + 9585783 x^{19} - 7334897 x^{18} - 140140161 x^{17} + 67545369 x^{16} + 1325068846 x^{15} - 360894987 x^{14} - 8127116829 x^{13} + 949692990 x^{12} + 32083489470 x^{11} + 34580448 x^{10} - 79891828714 x^{9} - 7402115451 x^{8} + 120762893079 x^{7} + 19715601263 x^{6} - 103424572098 x^{5} - 22163323110 x^{4} + 44013206071 x^{3} + 10730880483 x^{2} - 6912437058 x - 1834540021 \)

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:  \(1197336751300349460100016353165918520894496565611371829528951009=3^{36}\cdot 7^{18}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.89$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(1024,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(898,·)$, $\chi_{1197}(4,·)$, $\chi_{1197}(904,·)$, $\chi_{1197}(226,·)$, $\chi_{1197}(256,·)$, $\chi_{1197}(655,·)$, $\chi_{1197}(16,·)$, $\chi_{1197}(403,·)$, $\chi_{1197}(25,·)$, $\chi_{1197}(463,·)$, $\chi_{1197}(1054,·)$, $\chi_{1197}(415,·)$, $\chi_{1197}(400,·)$, $\chi_{1197}(802,·)$, $\chi_{1197}(100,·)$, $\chi_{1197}(424,·)$, $\chi_{1197}(64,·)$, $\chi_{1197}(106,·)$, $\chi_{1197}(814,·)$, $\chi_{1197}(625,·)$, $\chi_{1197}(499,·)$, $\chi_{1197}(862,·)$, $\chi_{1197}(823,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(799,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{11} a^{21} - \frac{2}{11} a^{20} - \frac{4}{11} a^{19} + \frac{2}{11} a^{18} - \frac{5}{11} a^{16} + \frac{2}{11} a^{15} + \frac{5}{11} a^{14} + \frac{1}{11} a^{13} - \frac{3}{11} a^{11} + \frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{1}{11} a^{8} - \frac{1}{11} a^{7} - \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{5}{11} a^{4} - \frac{2}{11} a^{3} - \frac{4}{11} a^{2} + \frac{3}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{22} + \frac{3}{11} a^{20} + \frac{5}{11} a^{19} + \frac{4}{11} a^{18} - \frac{5}{11} a^{17} + \frac{3}{11} a^{16} - \frac{2}{11} a^{15} + \frac{2}{11} a^{13} - \frac{3}{11} a^{12} - \frac{5}{11} a^{11} - \frac{4}{11} a^{10} - \frac{2}{11} a^{9} - \frac{3}{11} a^{8} + \frac{5}{11} a^{7} - \frac{1}{11} a^{6} - \frac{2}{11} a^{5} - \frac{1}{11} a^{4} + \frac{3}{11} a^{3} - \frac{5}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{1969} a^{23} + \frac{69}{1969} a^{22} - \frac{79}{1969} a^{21} + \frac{882}{1969} a^{20} + \frac{270}{1969} a^{19} + \frac{151}{1969} a^{18} - \frac{507}{1969} a^{17} - \frac{881}{1969} a^{16} + \frac{589}{1969} a^{15} + \frac{604}{1969} a^{14} - \frac{167}{1969} a^{13} + \frac{206}{1969} a^{12} - \frac{169}{1969} a^{11} + \frac{366}{1969} a^{10} + \frac{967}{1969} a^{9} + \frac{826}{1969} a^{8} + \frac{85}{1969} a^{7} - \frac{612}{1969} a^{6} + \frac{420}{1969} a^{5} - \frac{338}{1969} a^{4} - \frac{899}{1969} a^{3} + \frac{889}{1969} a^{2} + \frac{620}{1969} a - \frac{596}{1969}$, $\frac{1}{1885866851} a^{24} + \frac{469288}{1885866851} a^{23} - \frac{85090749}{1885866851} a^{22} + \frac{21963022}{1885866851} a^{21} + \frac{932263352}{1885866851} a^{20} + \frac{81829375}{1885866851} a^{19} - \frac{371262865}{1885866851} a^{18} - \frac{9971502}{171442441} a^{17} + \frac{5695061}{1885866851} a^{16} + \frac{202401172}{1885866851} a^{15} + \frac{598734688}{1885866851} a^{14} - \frac{113528213}{1885866851} a^{13} - \frac{176423097}{1885866851} a^{12} + \frac{879120444}{1885866851} a^{11} + \frac{48937004}{1885866851} a^{10} + \frac{59860525}{171442441} a^{9} - \frac{677633615}{1885866851} a^{8} + \frac{667111175}{1885866851} a^{7} + \frac{54167406}{1885866851} a^{6} - \frac{832948595}{1885866851} a^{5} + \frac{920010539}{1885866851} a^{4} - \frac{602561681}{1885866851} a^{3} - \frac{786635825}{1885866851} a^{2} + \frac{64219838}{171442441} a - \frac{677224085}{1885866851}$, $\frac{1}{214500381499591} a^{25} - \frac{5105}{214500381499591} a^{24} - \frac{36312915069}{214500381499591} a^{23} - \frac{3811911989351}{214500381499591} a^{22} + \frac{3289855453859}{214500381499591} a^{21} - \frac{17913561188390}{214500381499591} a^{20} + \frac{85813661989614}{214500381499591} a^{19} + \frac{14042072069430}{214500381499591} a^{18} - \frac{12473230698835}{214500381499591} a^{17} - \frac{5431375720474}{19500034681781} a^{16} + \frac{50984590441684}{214500381499591} a^{15} + \frac{20091075830335}{214500381499591} a^{14} - \frac{64609762275306}{214500381499591} a^{13} + \frac{47359848084121}{214500381499591} a^{12} - \frac{13049669392571}{214500381499591} a^{11} + \frac{88278921246504}{214500381499591} a^{10} + \frac{57150569999472}{214500381499591} a^{9} + \frac{27211007024929}{214500381499591} a^{8} + \frac{64367092602282}{214500381499591} a^{7} - \frac{74103561503076}{214500381499591} a^{6} - \frac{34684488330608}{214500381499591} a^{5} + \frac{49582523746531}{214500381499591} a^{4} + \frac{65859095045763}{214500381499591} a^{3} + \frac{30741514187614}{214500381499591} a^{2} - \frac{6684832378808}{214500381499591} a + \frac{834556032647}{19500034681781}$, $\frac{1}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{26} + \frac{1417051672071787513673220055724897670622690267047077934877701582046406426952925858022608}{2382926686587854901872754163590870624158263384906064226916343245626420508577613049405976381449738109501} a^{25} + \frac{7032964213932274545031383746922947194181424716059473154524789041185894273842323590411955771706945}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{24} - \frac{867316645658494029975996321629607733311365797424426984843087481933507522773710941363121090732596088571}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{23} - \frac{321604677138613449294015830040214115913594664905505793733505251808706889942866068052120630231843782403615}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{22} - \frac{83600962458383040376597193591827284494900158799000201617893936092805064326785202801340823865465663222256}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{21} - \frac{3789776396120740387805309849196583440840255842754096821613440115817367966450574676948133014111515701542521}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{20} - \frac{15573711193321786388321977543413880509109951003479398744557550954710162665321291230412218787838628559483197}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{19} + \frac{790997877576656739785501003837021040010494617921246413510015824700402724356083788648056560831285301811071}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{18} - \frac{12683292387515473025663025716551390225159251340946608467958051014921187599544016459770699589696491150915655}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{17} - \frac{334969706702455353928579600718467829095515611015614995129723234179948121875539961169401227630707028406889}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{16} - \frac{10450283847645211259843051351152957026499685074287981740596116240691436698843301958072322149023363454172979}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{15} - \frac{1098789629118241278817403294840842582416042242787202527457060567074155169090837262954061689635872924885815}{2900021777577419415579141817090089549600606539430680164157189729927353758938955081127073256224331279262717} a^{14} - \frac{241103537133820864887411389098943420907000895058718051088684035096694102801245383217652591888433912107706}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{13} - \frac{6789640500556391931913760137095132212097335975458396872757123001314536949709385147036230890177797570724413}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{12} + \frac{10573195224394599102783036982637258725743858706023780188993650077082565798774980660581535576110542665760980}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{11} - \frac{5322590337770954967482114712307775524478275769155456248231194769113349823232938650137111564343660773630512}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{10} - \frac{4427425997547026512879034091671494613302468695434759254401394464459519411905387208692486180119988862966472}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{9} - \frac{6407213931363039259433113276872761323741955673967404517962315821632996506366601414621760042619789352794634}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{8} + \frac{5508795981181754472293405034348474812389733000542684833064946809332129683071332798845193058512838961084255}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{7} + \frac{10511811944604678690428272059441898519318594340966240493645488767824857766510692650209128567114708110266669}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{6} + \frac{3284845563227292339639508611569191708138579501148959830566573542697989563524913047537049403803704507691699}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{5} - \frac{578963634909851981483863201085306835429763805651092989848812904094772447033789741704265806891459346943601}{2900021777577419415579141817090089549600606539430680164157189729927353758938955081127073256224331279262717} a^{4} + \frac{441045155372770711545035102663213305602046726788299691973235053785880306242390032627590438810363666220074}{2900021777577419415579141817090089549600606539430680164157189729927353758938955081127073256224331279262717} a^{3} + \frac{6388032416438258383162272168621867587017651532346373916599338930453661729415619876744741416630539477473527}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887} a^{2} + \frac{1194684731156980383752133847632151756620790953544485134728440629211747473091541253628909004159656436140444}{2900021777577419415579141817090089549600606539430680164157189729927353758938955081127073256224331279262717} a + \frac{2703258871645717780420611100504373500878580265250799075818213657958601517708856520141720634409362175783537}{31900239553351613571370559987990985045606671933737481805729087029200891348328505892397805818467644071889887}$

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:  \( 30503747567527345000000 \) (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

\(\Q(\zeta_{9})^+\), 3.3.361.1, 3.3.29241.1, 3.3.29241.2, 9.9.25002110044521.1, 9.9.1998099208210609.1, 9.9.1061871841310654257569.5, 9.9.1061871841310654257569.2

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.9.8.3$x^{9} - 77824$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.3$x^{9} - 77824$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.3$x^{9} - 77824$$9$$1$$8$$C_9$$[\ ]_{9}$