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Properties

Label 27.27.3063651216...3609.4
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 37^{24}$
Root discriminant $1135.13$
Ramified primes $3, 37$
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![3792769205762397922849, -5870767524977761710381, -2432977341416493895440, 5327869987480607495559, 438374295750719620800, -1309884548656074508017, -30409748443968838560, 148513623991816139766, 1056709405388878560, -9478067851112544048, -20626459864948080, 377261368227838926, 236503225314720, -9934810805722716, -1580434383600, 179005600103112, 5695259040, -2241127346283, -8551440, 19481903595, 0, -115336881, 0, 443556, 0, -999, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 8551440*x^18 - 2241127346283*x^17 + 5695259040*x^16 + 179005600103112*x^15 - 1580434383600*x^14 - 9934810805722716*x^13 + 236503225314720*x^12 + 377261368227838926*x^11 - 20626459864948080*x^10 - 9478067851112544048*x^9 + 1056709405388878560*x^8 + 148513623991816139766*x^7 - 30409748443968838560*x^6 - 1309884548656074508017*x^5 + 438374295750719620800*x^4 + 5327869987480607495559*x^3 - 2432977341416493895440*x^2 - 5870767524977761710381*x + 3792769205762397922849)
 
gp: K = bnfinit(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 8551440*x^18 - 2241127346283*x^17 + 5695259040*x^16 + 179005600103112*x^15 - 1580434383600*x^14 - 9934810805722716*x^13 + 236503225314720*x^12 + 377261368227838926*x^11 - 20626459864948080*x^10 - 9478067851112544048*x^9 + 1056709405388878560*x^8 + 148513623991816139766*x^7 - 30409748443968838560*x^6 - 1309884548656074508017*x^5 + 438374295750719620800*x^4 + 5327869987480607495559*x^3 - 2432977341416493895440*x^2 - 5870767524977761710381*x + 3792769205762397922849, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 8551440 x^{18} - 2241127346283 x^{17} + 5695259040 x^{16} + 179005600103112 x^{15} - 1580434383600 x^{14} - 9934810805722716 x^{13} + 236503225314720 x^{12} + 377261368227838926 x^{11} - 20626459864948080 x^{10} - 9478067851112544048 x^{9} + 1056709405388878560 x^{8} + 148513623991816139766 x^{7} - 30409748443968838560 x^{6} - 1309884548656074508017 x^{5} + 438374295750719620800 x^{4} + 5327869987480607495559 x^{3} - 2432977341416493895440 x^{2} - 5870767524977761710381 x + 3792769205762397922849 \)

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:  \(30636512167696967158640831949436414926745966008852301931537630975865004722034443609=3^{94}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1135.13$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2997=3^{4}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2997}(1,·)$, $\chi_{2997}(1735,·)$, $\chi_{2997}(2698,·)$, $\chi_{2997}(715,·)$, $\chi_{2997}(2764,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(1489,·)$, $\chi_{2997}(340,·)$, $\chi_{2997}(1237,·)$, $\chi_{2997}(343,·)$, $\chi_{2997}(2713,·)$, $\chi_{2997}(1765,·)$, $\chi_{2997}(736,·)$, $\chi_{2997}(2338,·)$, $\chi_{2997}(1699,·)$, $\chi_{2997}(2341,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(2236,·)$, $\chi_{2997}(490,·)$, $\chi_{2997}(2734,·)$, $\chi_{2997}(238,·)$, $\chi_{2997}(1714,·)$, $\chi_{2997}(766,·)$, $\chi_{2997}(2488,·)$, $\chi_{2997}(1339,·)$, $\chi_{2997}(700,·)$, $\chi_{2997}(1342,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{629} a^{9} + \frac{8}{17} a^{7} - \frac{4}{17} a^{5} + \frac{2}{17} a^{3} + \frac{5}{17} a + \frac{4}{17}$, $\frac{1}{629} a^{10} + \frac{8}{17} a^{8} - \frac{4}{17} a^{6} + \frac{2}{17} a^{4} + \frac{5}{17} a^{2} + \frac{4}{17} a$, $\frac{1}{629} a^{11} + \frac{8}{17} a^{7} - \frac{4}{17} a^{5} + \frac{8}{17} a^{3} + \frac{4}{17} a^{2} - \frac{1}{17} a + \frac{6}{17}$, $\frac{1}{629} a^{12} + \frac{8}{17} a^{8} - \frac{4}{17} a^{6} + \frac{8}{17} a^{4} + \frac{4}{17} a^{3} - \frac{1}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{629} a^{13} + \frac{8}{17} a^{7} + \frac{2}{17} a^{5} + \frac{4}{17} a^{4} + \frac{2}{17} a^{3} + \frac{6}{17} a^{2} - \frac{1}{17} a + \frac{6}{17}$, $\frac{1}{23273} a^{14} - \frac{3}{17} a^{8} - \frac{5}{17} a^{6} - \frac{166}{629} a^{5} - \frac{5}{17} a^{4} + \frac{2}{17} a^{3} - \frac{6}{17} a^{2} + \frac{2}{17} a$, $\frac{1}{23273} a^{15} - \frac{1}{17} a^{7} - \frac{166}{629} a^{6} - \frac{7}{17} a^{5} + \frac{2}{17} a^{4} - \frac{5}{17} a^{3} + \frac{2}{17} a^{2} - \frac{6}{17} a + \frac{2}{17}$, $\frac{1}{23273} a^{16} - \frac{1}{17} a^{8} - \frac{166}{629} a^{7} - \frac{7}{17} a^{6} + \frac{2}{17} a^{5} - \frac{5}{17} a^{4} + \frac{2}{17} a^{3} - \frac{6}{17} a^{2} + \frac{2}{17} a$, $\frac{1}{23273} a^{17} - \frac{166}{629} a^{8} + \frac{2}{17} a^{6} + \frac{2}{17} a^{4} + \frac{2}{17} a^{2} - \frac{2}{17} a - \frac{5}{17}$, $\frac{1}{236823413259969593} a^{18} - \frac{18}{6400632790809989} a^{16} + \frac{135}{172990075427297} a^{14} - \frac{546}{4675407443981} a^{12} + \frac{47619}{4675407443981} a^{10} - \frac{222890154176}{336875410042631} a^{9} - \frac{2439558}{4675407443981} a^{8} + \frac{2006011387584}{9104740811963} a^{7} + \frac{70205058}{4675407443981} a^{6} - \frac{112256338776}{246074075999} a^{5} - \frac{1012046940}{4675407443981} a^{4} + \frac{103624756365}{246074075999} a^{3} + \frac{5616860517}{4675407443981} a^{2} - \frac{42901453656}{246074075999} a - \frac{48039927468}{4675407443981}$, $\frac{1}{236823413259969593} a^{19} - \frac{18}{6400632790809989} a^{17} + \frac{135}{172990075427297} a^{15} - \frac{546}{4675407443981} a^{13} + \frac{47619}{4675407443981} a^{11} - \frac{222890154176}{336875410042631} a^{10} - \frac{2439558}{4675407443981} a^{9} + \frac{2006011387584}{9104740811963} a^{8} + \frac{70205058}{4675407443981} a^{7} - \frac{112256338776}{246074075999} a^{6} - \frac{1012046940}{4675407443981} a^{5} + \frac{103624756365}{246074075999} a^{4} + \frac{5616860517}{4675407443981} a^{3} - \frac{42901453656}{246074075999} a^{2} - \frac{48039927468}{4675407443981} a$, $\frac{1}{236823413259969593} a^{20} - \frac{189}{172990075427297} a^{16} + \frac{1884}{4675407443981} a^{14} - \frac{316017}{4675407443981} a^{12} - \frac{222890154176}{336875410042631} a^{11} + \frac{29274696}{4675407443981} a^{10} + \frac{6006057349}{9104740811963} a^{9} - \frac{1554540570}{4675407443981} a^{8} - \frac{75615914588}{246074075999} a^{7} + \frac{45744521688}{4675407443981} a^{6} - \frac{26277112519}{246074075999} a^{5} - \frac{39318023619}{275023967293} a^{4} - \frac{88779396403}{246074075999} a^{3} - \frac{982618267127}{4675407443981} a^{2} - \frac{56725210306}{246074075999} a - \frac{641859422286}{4675407443981}$, $\frac{1}{8762466290618874941} a^{21} - \frac{189}{6400632790809989} a^{17} + \frac{1884}{172990075427297} a^{15} - \frac{8541}{4675407443981} a^{13} + \frac{8881850657787}{12464390171577347} a^{12} + \frac{791208}{4675407443981} a^{11} + \frac{136280568172}{336875410042631} a^{10} - \frac{42014610}{4675407443981} a^{9} + \frac{1704802399993}{9104740811963} a^{8} + \frac{1236338424}{4675407443981} a^{7} + \frac{5158028978}{246074075999} a^{6} - \frac{1062649287}{275023967293} a^{5} + \frac{104402182844}{246074075999} a^{4} + \frac{13043604064816}{172990075427297} a^{3} + \frac{9420899130}{246074075999} a^{2} - \frac{1139742661717}{4675407443981} a$, $\frac{1}{8762466290618874941} a^{22} - \frac{1518}{172990075427297} a^{16} + \frac{16974}{4675407443981} a^{14} + \frac{8881850657787}{12464390171577347} a^{13} - \frac{3026970}{4675407443981} a^{12} + \frac{136280568172}{336875410042631} a^{11} + \frac{15315003}{246074075999} a^{10} + \frac{7086452800}{9104740811963} a^{9} - \frac{15823490670}{4675407443981} a^{8} + \frac{28008841777}{246074075999} a^{7} + \frac{472878932715}{4675407443981} a^{6} + \frac{28702722145}{246074075999} a^{5} - \frac{3990755674233}{9104740811963} a^{4} + \frac{6702223470}{14474945647} a^{3} + \frac{735703381816}{4675407443981} a^{2} + \frac{70232791549}{246074075999} a + \frac{1511195084787}{4675407443981}$, $\frac{1}{7615290302378628177520718062003} a^{23} - \frac{4779865518}{205818656821044004797857244919} a^{22} - \frac{23}{205818656821044004797857244919} a^{21} + \frac{11202371560}{5562666400568756886428574187} a^{20} + \frac{230}{5562666400568756886428574187} a^{19} + \frac{8462529772}{5562666400568756886428574187} a^{18} - \frac{69}{7912754481605628572444629} a^{17} + \frac{189975416829012}{150342335150506942876447951} a^{16} + \frac{276}{239018020907006268484019} a^{15} - \frac{3643953008301750798869799}{205818656821044004797857244919} a^{14} + \frac{1015996629503099056501150}{5562666400568756886428574187} a^{13} - \frac{107772601508717048490804}{292771915819408257180451273} a^{12} + \frac{96906791498767518685920}{150342335150506942876447951} a^{11} + \frac{29978548204208803654925}{150342335150506942876447951} a^{10} + \frac{59150647563335637809775}{150342335150506942876447951} a^{9} - \frac{127031201913232442814860}{4063306355419106564228323} a^{8} - \frac{1846827360054068854746532}{4063306355419106564228323} a^{7} + \frac{13736502258445215353174}{109819090687002880114279} a^{6} + \frac{3263231173341660500668329}{150342335150506942876447951} a^{5} - \frac{426811546491104378174453}{4063306355419106564228323} a^{4} - \frac{72745457268001204002530}{239018020907006268484019} a^{3} - \frac{11162183954628467045489}{109819090687002880114279} a^{2} + \frac{5673058719997743515235}{109819090687002880114279} a + \frac{46487385018152114775995}{109819090687002880114279}$, $\frac{1}{7615290302378628177520718062003} a^{24} - \frac{24}{205818656821044004797857244919} a^{22} + \frac{11054315506}{205818656821044004797857244919} a^{21} + \frac{252}{5562666400568756886428574187} a^{20} + \frac{161532292}{327215670621691581554622011} a^{19} - \frac{80}{7912754481605628572444629} a^{18} - \frac{57802554378}{4063306355419106564228323} a^{17} + \frac{18}{12579895837210856236001} a^{16} - \frac{3643839490022045674761315}{205818656821044004797857244919} a^{15} - \frac{864}{6459946511000169418487} a^{14} + \frac{1595602956041056930479717}{5562666400568756886428574187} a^{13} - \frac{1446787745627717191487958}{5562666400568756886428574187} a^{12} - \frac{12913897383649574225446}{150342335150506942876447951} a^{11} + \frac{113502910704877464605692}{150342335150506942876447951} a^{10} + \frac{2154696344253050210295}{4063306355419106564228323} a^{9} - \frac{97808685408115474855384}{4063306355419106564228323} a^{8} - \frac{380992465135751247763}{6459946511000169418487} a^{7} + \frac{10254196281883075071616745}{150342335150506942876447951} a^{6} - \frac{22501483600556596429872}{109819090687002880114279} a^{5} + \frac{789238647093754866185952}{4063306355419106564228323} a^{4} + \frac{214194361849868974907397}{4063306355419106564228323} a^{3} - \frac{378234925227693386089}{109819090687002880114279} a^{2} + \frac{45225625297699623608637}{109819090687002880114279} a + \frac{2213123265432059790464}{109819090687002880114279}$, $\frac{1}{281765741188009242568266568294111} a^{25} - \frac{2053300655}{205818656821044004797857244919} a^{22} - \frac{300}{205818656821044004797857244919} a^{21} - \frac{6584586026}{5562666400568756886428574187} a^{20} + \frac{4000}{5562666400568756886428574187} a^{19} - \frac{9523065820}{5562666400568756886428574187} a^{18} - \frac{1350}{7912754481605628572444629} a^{17} + \frac{5199834392486306780248580}{7615290302378628177520718062003} a^{16} + \frac{5760}{239018020907006268484019} a^{15} + \frac{2580013386411536208100895}{205818656821044004797857244919} a^{14} + \frac{52671123121102498808130}{292771915819408257180451273} a^{13} + \frac{3174855031442518312629095}{5562666400568756886428574187} a^{12} - \frac{82942833935686636705993}{150342335150506942876447951} a^{11} - \frac{5797254572469527797630}{7912754481605628572444629} a^{10} + \frac{48620307647278210146791}{150342335150506942876447951} a^{9} + \frac{1138846329537928964959810}{4063306355419106564228323} a^{8} - \frac{840265086241769297186325372}{5562666400568756886428574187} a^{7} + \frac{9898213533432925365354}{109819090687002880114279} a^{6} - \frac{11119423028703687554524126}{150342335150506942876447951} a^{5} + \frac{1879341468844904301117591}{4063306355419106564228323} a^{4} + \frac{20019402493890469908603}{109819090687002880114279} a^{3} + \frac{18451548363562031306258}{109819090687002880114279} a^{2} + \frac{42857275785687495941019}{109819090687002880114279} a + \frac{9910258720696448488618}{109819090687002880114279}$, $\frac{1}{281765741188009242568266568294111} a^{26} - \frac{325}{205818656821044004797857244919} a^{22} + \frac{292010192}{10832560885318105515676697101} a^{21} + \frac{4550}{5562666400568756886428574187} a^{20} - \frac{11503081266}{5562666400568756886428574187} a^{19} - \frac{1625}{7912754481605628572444629} a^{18} + \frac{5199826167215972348183804}{7615290302378628177520718062003} a^{17} + \frac{390}{12579895837210856236001} a^{16} + \frac{2580102471597976055137655}{205818656821044004797857244919} a^{15} - \frac{19500}{6459946511000169418487} a^{14} - \frac{3009887953743882521482595}{5562666400568756886428574187} a^{13} + \frac{3254179622424433710713726}{5562666400568756886428574187} a^{12} - \frac{22949729160082428532071}{150342335150506942876447951} a^{11} - \frac{22751485671916022513527}{150342335150506942876447951} a^{10} + \frac{2570904776071860746343}{4063306355419106564228323} a^{9} - \frac{1391062828027956266910264379}{5562666400568756886428574187} a^{8} + \frac{43210034052216828271406}{109819090687002880114279} a^{7} - \frac{36703034664498678531180184}{150342335150506942876447951} a^{6} + \frac{667824177912294165497}{6459946511000169418487} a^{5} + \frac{273615098038421445178132}{4063306355419106564228323} a^{4} + \frac{765673081712063238851078}{4063306355419106564228323} a^{3} - \frac{22332067348132577872379}{109819090687002880114279} a^{2} + \frac{2165915445559340951154}{6459946511000169418487} a + \frac{48868135124801325343113}{109819090687002880114279}$

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

\(\Q(\zeta_{9})^+\), 9.9.80515213381214514081.3

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 $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ $27$ $27$ $27$ R $27$ $27$ $27$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ $27$

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
$37$37.9.8.9$x^{9} - 9472$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.9$x^{9} - 9472$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.9$x^{9} - 9472$$9$$1$$8$$C_9$$[\ ]_{9}$