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Properties

Label 27.27.1530571656...8809.2
Degree $27$
Signature $[27, 0]$
Discriminant $7^{18}\cdot 109^{26}$
Root discriminant $335.25$
Ramified primes $7, 109$
Class number $3$ (GRH)
Class group $[3]$ (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![-5532747569, -284679967632, -3131913897999, -7804346234595, -3967891227901, 7880177941710, 8688783015200, -2048147111227, -5450598257003, -470512221204, 1730848478672, 393103177385, -319085698582, -100258381849, 35784378577, 13795117056, -2444680389, -1133435022, 97784729, 56748543, -2065966, -1706248, 17071, 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 + 17071*x^22 - 1706248*x^21 - 2065966*x^20 + 56748543*x^19 + 97784729*x^18 - 1133435022*x^17 - 2444680389*x^16 + 13795117056*x^15 + 35784378577*x^14 - 100258381849*x^13 - 319085698582*x^12 + 393103177385*x^11 + 1730848478672*x^10 - 470512221204*x^9 - 5450598257003*x^8 - 2048147111227*x^7 + 8688783015200*x^6 + 7880177941710*x^5 - 3967891227901*x^4 - 7804346234595*x^3 - 3131913897999*x^2 - 284679967632*x - 5532747569)
 
gp: K = bnfinit(x^27 - x^26 - 270*x^25 + 47*x^24 + 29577*x^23 + 17071*x^22 - 1706248*x^21 - 2065966*x^20 + 56748543*x^19 + 97784729*x^18 - 1133435022*x^17 - 2444680389*x^16 + 13795117056*x^15 + 35784378577*x^14 - 100258381849*x^13 - 319085698582*x^12 + 393103177385*x^11 + 1730848478672*x^10 - 470512221204*x^9 - 5450598257003*x^8 - 2048147111227*x^7 + 8688783015200*x^6 + 7880177941710*x^5 - 3967891227901*x^4 - 7804346234595*x^3 - 3131913897999*x^2 - 284679967632*x - 5532747569, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 270 x^{25} + 47 x^{24} + 29577 x^{23} + 17071 x^{22} - 1706248 x^{21} - 2065966 x^{20} + 56748543 x^{19} + 97784729 x^{18} - 1133435022 x^{17} - 2444680389 x^{16} + 13795117056 x^{15} + 35784378577 x^{14} - 100258381849 x^{13} - 319085698582 x^{12} + 393103177385 x^{11} + 1730848478672 x^{10} - 470512221204 x^{9} - 5450598257003 x^{8} - 2048147111227 x^{7} + 8688783015200 x^{6} + 7880177941710 x^{5} - 3967891227901 x^{4} - 7804346234595 x^{3} - 3131913897999 x^{2} - 284679967632 x - 5532747569 \)

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$$\chi_{763}(1,·)$, $\chi_{763}(130,·)$, $\chi_{763}(323,·)$, $\chi_{763}(198,·)$, $\chi_{763}(135,·)$, $\chi_{763}(393,·)$, $\chi_{763}(548,·)$, $\chi_{763}(463,·)$, $\chi_{763}(144,·)$, $\chi_{763}(25,·)$, $\chi_{763}(408,·)$, $\chi_{763}(729,·)$, $\chi_{763}(732,·)$, $\chi_{763}(669,·)$, $\chi_{763}(158,·)$, $\chi_{763}(291,·)$, $\chi_{763}(676,·)$, $\chi_{763}(625,·)$, $\chi_{763}(365,·)$, $\chi_{763}(751,·)$, $\chi_{763}(561,·)$, $\chi_{763}(114,·)$, $\chi_{763}(372,·)$, $\chi_{763}(443,·)$, $\chi_{763}(281,·)$, $\chi_{763}(445,·)$, $\chi_{763}(702,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{1091} a^{25} + \frac{220}{1091} a^{24} - \frac{527}{1091} a^{23} + \frac{273}{1091} a^{22} + \frac{117}{1091} a^{21} - \frac{112}{1091} a^{20} - \frac{261}{1091} a^{19} + \frac{116}{1091} a^{18} - \frac{536}{1091} a^{17} + \frac{148}{1091} a^{16} + \frac{381}{1091} a^{15} + \frac{500}{1091} a^{14} - \frac{354}{1091} a^{13} - \frac{79}{1091} a^{12} - \frac{485}{1091} a^{11} - \frac{174}{1091} a^{10} - \frac{110}{1091} a^{9} - \frac{192}{1091} a^{8} - \frac{435}{1091} a^{7} + \frac{418}{1091} a^{6} + \frac{116}{1091} a^{5} - \frac{240}{1091} a^{4} + \frac{275}{1091} a^{3} - \frac{306}{1091} a^{2} + \frac{269}{1091} a - \frac{362}{1091}$, $\frac{1}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{26} - \frac{1145680455230795550107675286317749035815890813598796309942706616893541847981863346336216647759688996283026932406896398496616671564596737606657292214062354}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{25} + \frac{1114640705327458369900048622294358795027994242125632108992456378684017973535786854852069862452317250994108272101518216229215618775453958045154674852166410323}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{24} - \frac{1078514445868416171193937864123896060044857188040503169004732198073016339438121973406669391995711502960456288785753247077976341663485158945168147758747410653}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{23} + \frac{44352665195137433493581009837117104851572177778925181699318227880129499999323758566685940980214944100275366754100609905866435180838976396017263608265876466}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{22} - \frac{352091122134379165340737684445704714815664197720285924660429770114719354393492855362152675502256306486319851714004293297418224232449513992162015544972698423}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{21} + \frac{979726033355207878721175094329449475407865980644490594062048223843597862418105773268811479045142924697983755043488650981088469271184856141590466219036788399}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{20} - \frac{1013237417998667324314639178411458041538240304651490977412193230931861023733536367339164956067697265843008933778629637964661957269593266829406168915972781303}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{19} + \frac{129969829076672782217551011130425950591879723065686844038416809948216123044559049570925802637878375302178838105715900255113330545647584035344681652411649669}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{18} + \frac{673948453545383173224184006671684230031703560819169877653073856734275046477532040390693467964923367649476683640299307502881211463076058691835472361002367877}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{17} - \frac{714759998888703235295547570048809563285018818126045010135781428056412943411190090857659167754132554355829178118681642330556293521487244813339301678697230662}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{16} - \frac{372889876457981582800373318523974002348097010562398474054236078760921938915594431436098751863336966771090153479748569658757936167250305405219800320386030015}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{15} - \frac{703880096705418670136403149946000213065287878718860305497644497411447343432035902379206864539165346069790095870823341780602049358880898028974050506218150166}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{14} - \frac{608516602027763066053549565208564084823191128368423702446174139821783510138597134852333095133697213709815390165936402490173492217583753466167609443611510110}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{13} + \frac{147469839853465230547004054545982154371860370965469666305748282933761618300235548756010226198461659055184593053127719499135007922557239461795915570156849645}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{12} + \frac{1173790731593239380547473649912089747514870426779727885112275054469283179208372569533377133714880887983763278217722129155097928078871206156934298614505719033}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{11} + \frac{929290114454953872094867961087263748949894710980689476372948609445668967830209673672963850306836206077276602878094358905351141196893449882486844279741548707}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{10} + \frac{886068900336566190161846138963396081150574574892710403792058196808376541813932460214329129384542456701251720807511219719080936650598683745750762966418033164}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{9} + \frac{216187277066889271470499857681622496467632503032378683340031119856572144549001124634853187185935033174712639665635290093115167444504189008657685961894165698}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{8} + \frac{389368985078753964734852966049155371832968376199807231141509122063072758792838133138693848385826478243472594332593178788128156227882979227251455860344817454}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{7} + \frac{1135134693110307721903878992151329070089384631785019149835012531574761895557241274046714994522043147593436759860427325231701825997151670247402865317193244256}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{6} - \frac{638917211903351086932874764080063613793041068521852594813345600338825990048507959536793418013006768406501985493283618420775596193258643843025309684826168949}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{5} - \frac{315388375443584138406443259368452077395483507028560908047125917964927030389500197395556525390154238385341633878675112380507755565011270762272886331874787825}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{4} + \frac{325562501123113805024148883361825792291056048187734241707852601300693832230714918464267969748071182267221517243915243769431263865330174279341433817795311701}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{3} - \frac{1080741725203934658010752465989209191317212616061419182988709354427786891557632515610560377266634493549961972172343974857601369373910439008672122330783825707}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a^{2} - \frac{983473102968773478490716326364380183282432953948019909566902351448115634839737061437087056926081640583336368414481348684720418620955517909168643935413344316}{2527914053853855348669040973255871609082203791063606366085207429298132136104947835094356837377709181043066025180056516982781657239001357608618106886673391893} a - \frac{14996299534288720170837848570314400777342698290678212864557855817008213048283941923745188075184859535888002684111080152026569143898620549974248274610552}{38294185294621594968703754915787369292142514217860214898356496891493071608697496479395828660683640813825549894415592640582637620453567594392287987012761}$

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:  \( 7760416104753259000000000 \) (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

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