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Properties

Label 27.27.1513872271...3209.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 19^{24}$
Root discriminant $200.89$
Ramified primes $3, 19$
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![3329578088, 43969262832, -237404462490, 156639881529, 478960952040, -515019140298, -352985018796, 511737088221, 130647735324, -263091419295, -27287694702, 81769006062, 3352712412, -16451805348, -239483790, 2213413740, 9162180, -201382767, -144362, 12316959, 0, -493905, 0, 12312, 0, -171, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 171*x^25 + 12312*x^23 - 493905*x^21 + 12316959*x^19 - 144362*x^18 - 201382767*x^17 + 9162180*x^16 + 2213413740*x^15 - 239483790*x^14 - 16451805348*x^13 + 3352712412*x^12 + 81769006062*x^11 - 27287694702*x^10 - 263091419295*x^9 + 130647735324*x^8 + 511737088221*x^7 - 352985018796*x^6 - 515019140298*x^5 + 478960952040*x^4 + 156639881529*x^3 - 237404462490*x^2 + 43969262832*x + 3329578088)
 
gp: K = bnfinit(x^27 - 171*x^25 + 12312*x^23 - 493905*x^21 + 12316959*x^19 - 144362*x^18 - 201382767*x^17 + 9162180*x^16 + 2213413740*x^15 - 239483790*x^14 - 16451805348*x^13 + 3352712412*x^12 + 81769006062*x^11 - 27287694702*x^10 - 263091419295*x^9 + 130647735324*x^8 + 511737088221*x^7 - 352985018796*x^6 - 515019140298*x^5 + 478960952040*x^4 + 156639881529*x^3 - 237404462490*x^2 + 43969262832*x + 3329578088, 1)
 

Normalized defining polynomial

\( x^{27} - 171 x^{25} + 12312 x^{23} - 493905 x^{21} + 12316959 x^{19} - 144362 x^{18} - 201382767 x^{17} + 9162180 x^{16} + 2213413740 x^{15} - 239483790 x^{14} - 16451805348 x^{13} + 3352712412 x^{12} + 81769006062 x^{11} - 27287694702 x^{10} - 263091419295 x^{9} + 130647735324 x^{8} + 511737088221 x^{7} - 352985018796 x^{6} - 515019140298 x^{5} + 478960952040 x^{4} + 156639881529 x^{3} - 237404462490 x^{2} + 43969262832 x + 3329578088 \)

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:  \(151387227139400874793584512894425905310908185375198495733583209=3^{66}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.89$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(513=3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{513}(64,·)$, $\chi_{513}(1,·)$, $\chi_{513}(130,·)$, $\chi_{513}(505,·)$, $\chi_{513}(196,·)$, $\chi_{513}(454,·)$, $\chi_{513}(328,·)$, $\chi_{513}(142,·)$, $\chi_{513}(403,·)$, $\chi_{513}(334,·)$, $\chi_{513}(406,·)$, $\chi_{513}(343,·)$, $\chi_{513}(472,·)$, $\chi_{513}(25,·)$, $\chi_{513}(283,·)$, $\chi_{513}(157,·)$, $\chi_{513}(163,·)$, $\chi_{513}(484,·)$, $\chi_{513}(232,·)$, $\chi_{513}(235,·)$, $\chi_{513}(172,·)$, $\chi_{513}(301,·)$, $\chi_{513}(367,·)$, $\chi_{513}(112,·)$, $\chi_{513}(499,·)$, $\chi_{513}(313,·)$, $\chi_{513}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{38} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{76} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{76} a^{11} - \frac{1}{76} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{76} a^{12} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{76} a^{13} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{76} a^{14} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{152} a^{15} - \frac{1}{152} a^{14} - \frac{1}{152} a^{12} - \frac{1}{152} a^{11} - \frac{1}{152} a^{10} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{152} a^{16} - \frac{1}{152} a^{14} - \frac{1}{152} a^{13} - \frac{1}{152} a^{10} - \frac{1}{76} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{3}{8} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{2888} a^{17} - \frac{1}{152} a^{12} - \frac{1}{152} a^{10} - \frac{31}{152} a^{8} - \frac{1}{4} a^{6} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2888} a^{18} - \frac{1}{152} a^{13} - \frac{1}{152} a^{11} + \frac{1}{152} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{5776} a^{19} - \frac{1}{304} a^{15} + \frac{1}{304} a^{11} - \frac{1}{152} a^{10} - \frac{1}{76} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{5776} a^{20} - \frac{1}{304} a^{16} + \frac{1}{304} a^{12} - \frac{1}{152} a^{11} - \frac{1}{152} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{5}{16} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{5776} a^{21} - \frac{1}{5776} a^{17} + \frac{1}{304} a^{13} - \frac{1}{152} a^{9} - \frac{4}{19} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{11552} a^{22} - \frac{1}{11552} a^{20} - \frac{1}{11552} a^{19} - \frac{1}{11552} a^{18} - \frac{1}{5776} a^{17} - \frac{1}{608} a^{16} - \frac{1}{608} a^{15} + \frac{1}{608} a^{14} - \frac{1}{304} a^{13} + \frac{3}{608} a^{12} - \frac{1}{608} a^{11} - \frac{1}{304} a^{10} - \frac{1}{304} a^{9} + \frac{31}{304} a^{8} + \frac{1}{16} a^{7} - \frac{3}{32} a^{6} - \frac{1}{8} a^{5} - \frac{15}{32} a^{4} - \frac{1}{32} a^{3} + \frac{3}{16} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{222098752} a^{23} - \frac{2695}{222098752} a^{22} - \frac{17783}{222098752} a^{21} + \frac{4409}{55524688} a^{20} + \frac{883}{111049376} a^{19} + \frac{24765}{222098752} a^{18} - \frac{5899}{222098752} a^{17} + \frac{4563}{2922352} a^{16} + \frac{2293}{1461176} a^{15} + \frac{9763}{11689408} a^{14} + \frac{14551}{11689408} a^{13} - \frac{2397}{2922352} a^{12} - \frac{54887}{11689408} a^{11} - \frac{463}{153808} a^{10} - \frac{36483}{2922352} a^{9} - \frac{101737}{2922352} a^{8} - \frac{17305}{615232} a^{7} - \frac{127871}{615232} a^{6} - \frac{114061}{615232} a^{5} - \frac{72437}{307616} a^{4} - \frac{71531}{615232} a^{3} - \frac{20733}{307616} a^{2} - \frac{8723}{38452} a - \frac{22095}{76904}$, $\frac{1}{222098752} a^{24} + \frac{2923}{111049376} a^{22} + \frac{3643}{222098752} a^{21} - \frac{1889}{27762344} a^{20} - \frac{3139}{222098752} a^{19} + \frac{1165}{111049376} a^{18} + \frac{22043}{222098752} a^{17} - \frac{15151}{5844704} a^{16} - \frac{21655}{11689408} a^{15} + \frac{2721}{5844704} a^{14} - \frac{54135}{11689408} a^{13} - \frac{74029}{11689408} a^{12} - \frac{11583}{11689408} a^{11} + \frac{9325}{2922352} a^{10} + \frac{23311}{2922352} a^{9} + \frac{2635093}{11689408} a^{8} - \frac{61285}{307616} a^{7} + \frac{1875}{76904} a^{6} + \frac{76731}{615232} a^{5} - \frac{276839}{615232} a^{4} + \frac{115399}{615232} a^{3} + \frac{98571}{307616} a^{2} + \frac{6957}{76904} a - \frac{3103}{76904}$, $\frac{1}{1970682226496} a^{25} - \frac{159}{103720117184} a^{24} - \frac{81}{51860058592} a^{23} - \frac{4434607}{103720117184} a^{22} + \frac{2763483}{103720117184} a^{21} - \frac{4813159}{103720117184} a^{20} - \frac{7056303}{103720117184} a^{19} + \frac{8337033}{103720117184} a^{18} + \frac{339811}{5458953536} a^{17} - \frac{153128339}{103720117184} a^{16} + \frac{17891661}{5458953536} a^{15} - \frac{28382177}{5458953536} a^{14} + \frac{6286091}{2729476768} a^{13} - \frac{1780297}{682369192} a^{12} + \frac{34452495}{5458953536} a^{11} + \frac{4254723}{1364738384} a^{10} + \frac{15194399}{5458953536} a^{9} - \frac{4478731}{287313344} a^{8} + \frac{565846805}{2729476768} a^{7} + \frac{20352267}{287313344} a^{6} - \frac{9942021}{143656672} a^{5} + \frac{17329747}{71828336} a^{4} + \frac{66834179}{287313344} a^{3} - \frac{38988927}{143656672} a^{2} - \frac{808773}{17957084} a + \frac{860375}{35914168}$, $\frac{1}{17222199080131668420773327521284988410979683194875080979179555968} a^{26} + \frac{144493003424987056072365441088891470067720797147109}{906431530533245706356490922172894126893667536572372683114713472} a^{25} - \frac{682201531091524669877932325183983450798621551590588121}{453215765266622853178245461086447063446833768286186341557356736} a^{24} + \frac{331672462928407100523033001303665228197822181377682225}{453215765266622853178245461086447063446833768286186341557356736} a^{23} - \frac{1276565835247862024348347770855321570390018970986771319739}{226607882633311426589122730543223531723416884143093170778678368} a^{22} - \frac{23207046792080897233946259911388460085530421825509979432431}{453215765266622853178245461086447063446833768286186341557356736} a^{21} + \frac{10136723775459282171791548602456066859329985558954133096529}{906431530533245706356490922172894126893667536572372683114713472} a^{20} + \frac{49827892193678723698775013013692072776958920397231093987575}{906431530533245706356490922172894126893667536572372683114713472} a^{19} + \frac{7870102839893377690189906857410304729411990701827267987095}{453215765266622853178245461086447063446833768286186341557356736} a^{18} - \frac{43105818270590848891035160575100941279823044067035363408235}{453215765266622853178245461086447063446833768286186341557356736} a^{17} - \frac{120286341536773223276984134537458728668470733471218022354339}{47706922659644510860867943272257585625982501924861720163932288} a^{16} - \frac{59491100124237807603317016753043507334396823760917843410433}{47706922659644510860867943272257585625982501924861720163932288} a^{15} + \frac{103549458672627560251822318646886478855369365111389213797913}{47706922659644510860867943272257585625982501924861720163932288} a^{14} + \frac{109662847468136942644697844569349587850832165626849708182515}{47706922659644510860867943272257585625982501924861720163932288} a^{13} - \frac{139260270963706172397962968975255988770589453277542353169545}{47706922659644510860867943272257585625982501924861720163932288} a^{12} - \frac{313088782802925762482217902737177198616370180717250687053715}{47706922659644510860867943272257585625982501924861720163932288} a^{11} + \frac{257931762078662343765452682245355761133155062925657217931723}{47706922659644510860867943272257585625982501924861720163932288} a^{10} + \frac{43906163733518912810800458309596336167329032404068967385415}{47706922659644510860867943272257585625982501924861720163932288} a^{9} - \frac{2515123712427825683011549730913280545094294470512727376930257}{23853461329822255430433971636128792812991250962430860081966144} a^{8} + \frac{77959219880134754279342791824818302059388113171581290375257}{1255445333148539759496524822954146990157434261180571583261376} a^{7} - \frac{310592957650328523715331927981513732423777565696083580456521}{2510890666297079518993049645908293980314868522361143166522752} a^{6} + \frac{150707343543330074293277102877524579436796662621866404254651}{2510890666297079518993049645908293980314868522361143166522752} a^{5} - \frac{1112451638006126694097568172281153777780502173111850979492243}{2510890666297079518993049645908293980314868522361143166522752} a^{4} - \frac{850993164073923797727430590576170019388964221035351925131809}{2510890666297079518993049645908293980314868522361143166522752} a^{3} - \frac{353504658088512641843035334502735990885176135555251679554021}{1255445333148539759496524822954146990157434261180571583261376} a^{2} + \frac{58924288244459486200207260632457135454675816967495442841333}{313861333287134939874131205738536747539358565295142895815344} a + \frac{1483702025752730828707743504077250781100992738588599087}{5172486911240049108820699183218852445481279607362397136}$

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:  \( 27543066172679904000000 \) (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

\(\Q(\zeta_{9})^+\), 3.3.29241.1, 3.3.29241.2, 3.3.361.1, 9.9.25002110044521.1, 9.9.532962204162830310969.10, 9.9.532962204162830310969.7, 9.9.532962204162830310969.9

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$19$19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.6$x^{9} + 1216$$9$$1$$8$$C_9$$[\ ]_{9}$