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Properties

Label 27.27.1513872271...3209.2
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![9868941829, -25035315705, -96914645181, 211536642519, 237369275820, -531464834277, -177767378010, 513818842734, 63310449366, -263183382240, -12502317699, 81769006062, 1450706658, -16451805348, -98688375, 2213413740, 3645378, -201382767, -56221, 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 - 56221*x^18 - 201382767*x^17 + 3645378*x^16 + 2213413740*x^15 - 98688375*x^14 - 16451805348*x^13 + 1450706658*x^12 + 81769006062*x^11 - 12502317699*x^10 - 263183382240*x^9 + 63310449366*x^8 + 513818842734*x^7 - 177767378010*x^6 - 531464834277*x^5 + 237369275820*x^4 + 211536642519*x^3 - 96914645181*x^2 - 25035315705*x + 9868941829)
 
gp: K = bnfinit(x^27 - 171*x^25 + 12312*x^23 - 493905*x^21 + 12316959*x^19 - 56221*x^18 - 201382767*x^17 + 3645378*x^16 + 2213413740*x^15 - 98688375*x^14 - 16451805348*x^13 + 1450706658*x^12 + 81769006062*x^11 - 12502317699*x^10 - 263183382240*x^9 + 63310449366*x^8 + 513818842734*x^7 - 177767378010*x^6 - 531464834277*x^5 + 237369275820*x^4 + 211536642519*x^3 - 96914645181*x^2 - 25035315705*x + 9868941829, 1)
 

Normalized defining polynomial

\( x^{27} - 171 x^{25} + 12312 x^{23} - 493905 x^{21} + 12316959 x^{19} - 56221 x^{18} - 201382767 x^{17} + 3645378 x^{16} + 2213413740 x^{15} - 98688375 x^{14} - 16451805348 x^{13} + 1450706658 x^{12} + 81769006062 x^{11} - 12502317699 x^{10} - 263183382240 x^{9} + 63310449366 x^{8} + 513818842734 x^{7} - 177767378010 x^{6} - 531464834277 x^{5} + 237369275820 x^{4} + 211536642519 x^{3} - 96914645181 x^{2} - 25035315705 x + 9868941829 \)

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$$\chi_{513}(256,·)$, $\chi_{513}(1,·)$, $\chi_{513}(43,·)$, $\chi_{513}(4,·)$, $\chi_{513}(214,·)$, $\chi_{513}(385,·)$, $\chi_{513}(64,·)$, $\chi_{513}(139,·)$, $\chi_{513}(334,·)$, $\chi_{513}(16,·)$, $\chi_{513}(340,·)$, $\chi_{513}(85,·)$, $\chi_{513}(406,·)$, $\chi_{513}(343,·)$, $\chi_{513}(346,·)$, $\chi_{513}(481,·)$, $\chi_{513}(163,·)$, $\chi_{513}(358,·)$, $\chi_{513}(169,·)$, $\chi_{513}(427,·)$, $\chi_{513}(172,·)$, $\chi_{513}(175,·)$, $\chi_{513}(235,·)$, $\chi_{513}(310,·)$, $\chi_{513}(505,·)$, $\chi_{513}(187,·)$, $\chi_{513}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{38} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{38} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{38} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{38} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{38} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{38} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{38} a^{15} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{38} a^{16} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{722} a^{17} - \frac{11}{38} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{1444} a^{18} - \frac{1}{76} a^{14} - \frac{1}{76} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{1444} a^{19} - \frac{1}{76} a^{15} - \frac{1}{76} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1444} a^{20} - \frac{1}{76} a^{16} - \frac{1}{76} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{1444} a^{21} - \frac{1}{1444} a^{17} - \frac{1}{76} a^{13} - \frac{2}{19} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{1444} a^{22} - \frac{1}{76} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{97523428} a^{23} - \frac{56}{1283203} a^{22} - \frac{3307}{24380857} a^{21} + \frac{20073}{97523428} a^{20} + \frac{2535}{24380857} a^{19} - \frac{2057}{97523428} a^{18} - \frac{32753}{48761714} a^{17} - \frac{42523}{5132812} a^{16} + \frac{4486}{1283203} a^{15} + \frac{64593}{5132812} a^{14} + \frac{3970}{1283203} a^{13} + \frac{45135}{5132812} a^{12} + \frac{11237}{5132812} a^{11} + \frac{17929}{5132812} a^{10} + \frac{5961}{2566406} a^{9} + \frac{431597}{1283203} a^{8} + \frac{72099}{270148} a^{7} + \frac{25191}{67537} a^{6} - \frac{91083}{270148} a^{5} - \frac{132013}{270148} a^{4} + \frac{30681}{270148} a^{3} + \frac{538}{67537} a^{2} - \frac{81321}{270148} a + \frac{58635}{270148}$, $\frac{1}{97523428} a^{24} - \frac{6712}{24380857} a^{22} - \frac{9987}{48761714} a^{21} + \frac{6523}{97523428} a^{20} - \frac{590}{24380857} a^{19} + \frac{27249}{97523428} a^{18} - \frac{32915}{48761714} a^{17} - \frac{28321}{5132812} a^{16} + \frac{24955}{2566406} a^{15} - \frac{19439}{5132812} a^{14} + \frac{681}{135074} a^{13} - \frac{9242}{1283203} a^{12} - \frac{10283}{1283203} a^{11} - \frac{66601}{5132812} a^{10} + \frac{14427}{1283203} a^{9} - \frac{364057}{5132812} a^{8} + \frac{16495}{67537} a^{7} - \frac{99449}{270148} a^{6} + \frac{41865}{135074} a^{5} + \frac{5914}{67537} a^{4} - \frac{18035}{135074} a^{3} - \frac{107441}{270148} a^{2} + \frac{20896}{67537} a + \frac{1345}{270148}$, $\frac{1}{120002285583716} a^{25} - \frac{29267}{6315909767564} a^{24} + \frac{10177}{6315909767564} a^{23} + \frac{45502551}{1578977441891} a^{22} + \frac{178228312}{1578977441891} a^{21} - \frac{438768215}{1578977441891} a^{20} - \frac{130380912}{1578977441891} a^{19} + \frac{403071893}{1578977441891} a^{18} + \frac{1852544429}{3157954883782} a^{17} - \frac{37388136399}{3157954883782} a^{16} + \frac{204937153}{83104075889} a^{15} + \frac{1038791250}{83104075889} a^{14} - \frac{209146059}{332416303556} a^{13} + \frac{842173583}{332416303556} a^{12} + \frac{3756343975}{332416303556} a^{11} - \frac{1045237341}{166208151778} a^{10} + \frac{2726804825}{332416303556} a^{9} - \frac{128477884159}{332416303556} a^{8} + \frac{10434518816}{83104075889} a^{7} - \frac{2568787789}{17495594924} a^{6} - \frac{793145160}{4373898731} a^{5} - \frac{3388046961}{17495594924} a^{4} - \frac{1358913833}{8747797462} a^{3} - \frac{4612120109}{17495594924} a^{2} + \frac{7155908693}{17495594924} a - \frac{1196685474}{4373898731}$, $\frac{1}{665662426933356484876281403693860699841963600691123419071058047984704644372} a^{26} - \frac{820365658764347535946488998350603758795207300362642376250385}{665662426933356484876281403693860699841963600691123419071058047984704644372} a^{25} + \frac{119696783288620158747756798505597010671173080484332125710481281723}{35034864575439814993488494931255826307471768457427548372160949893931823388} a^{24} + \frac{5922350232374305540720582751620105558616541985685783048707423856}{8758716143859953748372123732813956576867942114356887093040237473482955847} a^{23} + \frac{338104784650490254255326139258513653962372320799633166941739563498513}{1843940240812621841762552364802938226709040445127765703797944731259569652} a^{22} + \frac{2736982469619836705441810803148386559480612757047543915937374936428907}{17517432287719907496744247465627913153735884228713774186080474946965911694} a^{21} - \frac{797491995528586288264323034107570877350874927525441050784337141684921}{35034864575439814993488494931255826307471768457427548372160949893931823388} a^{20} - \frac{1569365878806590424419465514939798433928333937552408446071108102152001}{8758716143859953748372123732813956576867942114356887093040237473482955847} a^{19} - \frac{474298970518353845282574598540938309251228604562857914127502735057466}{8758716143859953748372123732813956576867942114356887093040237473482955847} a^{18} - \frac{2052435775084857257713815543962086339373888830694028841131928515307411}{17517432287719907496744247465627913153735884228713774186080474946965911694} a^{17} + \frac{12251170849581450516110134930337408073285597826500769929863105497844863}{35034864575439814993488494931255826307471768457427548372160949893931823388} a^{16} - \frac{9171774944680795174239791491857132494314899980382443430233105796932069}{921970120406310920881276182401469113354520222563882851898972365629784826} a^{15} - \frac{752334435450795414654321998357283835767199160677360318866399262329015}{1843940240812621841762552364802938226709040445127765703797944731259569652} a^{14} + \frac{18380403122604670886173078728196423593035389103396718430646994087558727}{1843940240812621841762552364802938226709040445127765703797944731259569652} a^{13} - \frac{1369511701868313671152666496039874978214025996738482953101914161546826}{460985060203155460440638091200734556677260111281941425949486182814892413} a^{12} + \frac{5746123376108011728792428738586562031585095090172796932956651923742206}{460985060203155460440638091200734556677260111281941425949486182814892413} a^{11} - \frac{58422674210903870130312302545528623151597032095497159870951024830991}{4680051372620867618686681128941467580479798084080623613700367338222258} a^{10} - \frac{4061697527042000034903572119698714358248646402471912506333334788525315}{1843940240812621841762552364802938226709040445127765703797944731259569652} a^{9} + \frac{33052668555615704641093622153406053816466211519326904208936198889273689}{921970120406310920881276182401469113354520222563882851898972365629784826} a^{8} + \frac{148300067374775968460516910352661071436060329963427756993446395676594275}{1843940240812621841762552364802938226709040445127765703797944731259569652} a^{7} - \frac{64800249658788414322076143668837825238688161284357092178896782227491}{542175901444463934655263853220505212204951615738831433048498891872852} a^{6} - \frac{13706912764334495046421512814500379882160649926640176748287102117278559}{97049486358559044303292229726470432984686339217250826515681301645240508} a^{5} + \frac{8609873331487423221324713699561957966149285655002695329703973935239188}{24262371589639761075823057431617608246171584804312706628920325411310127} a^{4} + \frac{47560472268932128874732216084423478439862883656736122338957308563040013}{97049486358559044303292229726470432984686339217250826515681301645240508} a^{3} + \frac{42177075974737837333095161598248943862479129305989052622141762503730643}{97049486358559044303292229726470432984686339217250826515681301645240508} a^{2} - \frac{12102024365991194320102937353745164900896681362972077080525699161852079}{24262371589639761075823057431617608246171584804312706628920325411310127} a + \frac{28320988803147854647222676119984903058380001289508552314050907539}{67450233111851943906749458224399135815593588974139997342065400068}$

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:  \( 9861840342079951000000 \) (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

\(\Q(\zeta_{9})^+\), 3.3.29241.2, 3.3.361.1, 3.3.29241.1, 9.9.25002110044521.1, 9.9.532962204162830310969.8, 9.9.532962204162830310969.6, 9.9.532962204162830310969.5

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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$19.9.8.3$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}$