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Properties

Label 27.27.1197336751...1009.6
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![-111053593, -844847976, 15202545204, -1004242821, -152909392110, -169543892730, 184866910465, 315859629984, -51466910463, -228532520050, -27094243584, 83517640365, 20633038024, -17089817922, -5368801578, 2134780606, 738720423, -174434025, -59973575, 9796200, 2960787, -383528, -86886, 10011, 1380, -153, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 153*x^25 + 1380*x^24 + 10011*x^23 - 86886*x^22 - 383528*x^21 + 2960787*x^20 + 9796200*x^19 - 59973575*x^18 - 174434025*x^17 + 738720423*x^16 + 2134780606*x^15 - 5368801578*x^14 - 17089817922*x^13 + 20633038024*x^12 + 83517640365*x^11 - 27094243584*x^10 - 228532520050*x^9 - 51466910463*x^8 + 315859629984*x^7 + 184866910465*x^6 - 169543892730*x^5 - 152909392110*x^4 - 1004242821*x^3 + 15202545204*x^2 - 844847976*x - 111053593)
 
gp: K = bnfinit(x^27 - 9*x^26 - 153*x^25 + 1380*x^24 + 10011*x^23 - 86886*x^22 - 383528*x^21 + 2960787*x^20 + 9796200*x^19 - 59973575*x^18 - 174434025*x^17 + 738720423*x^16 + 2134780606*x^15 - 5368801578*x^14 - 17089817922*x^13 + 20633038024*x^12 + 83517640365*x^11 - 27094243584*x^10 - 228532520050*x^9 - 51466910463*x^8 + 315859629984*x^7 + 184866910465*x^6 - 169543892730*x^5 - 152909392110*x^4 - 1004242821*x^3 + 15202545204*x^2 - 844847976*x - 111053593, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 153 x^{25} + 1380 x^{24} + 10011 x^{23} - 86886 x^{22} - 383528 x^{21} + 2960787 x^{20} + 9796200 x^{19} - 59973575 x^{18} - 174434025 x^{17} + 738720423 x^{16} + 2134780606 x^{15} - 5368801578 x^{14} - 17089817922 x^{13} + 20633038024 x^{12} + 83517640365 x^{11} - 27094243584 x^{10} - 228532520050 x^{9} - 51466910463 x^{8} + 315859629984 x^{7} + 184866910465 x^{6} - 169543892730 x^{5} - 152909392110 x^{4} - 1004242821 x^{3} + 15202545204 x^{2} - 844847976 x - 111053593 \)

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$$\chi_{1197}(1024,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(898,·)$, $\chi_{1197}(235,·)$, $\chi_{1197}(4,·)$, $\chi_{1197}(1030,·)$, $\chi_{1197}(64,·)$, $\chi_{1197}(940,·)$, $\chi_{1197}(652,·)$, $\chi_{1197}(256,·)$, $\chi_{1197}(16,·)$, $\chi_{1197}(529,·)$, $\chi_{1197}(340,·)$, $\chi_{1197}(85,·)$, $\chi_{1197}(214,·)$, $\chi_{1197}(919,·)$, $\chi_{1197}(856,·)$, $\chi_{1197}(163,·)$, $\chi_{1197}(676,·)$, $\chi_{1197}(358,·)$, $\chi_{1197}(169,·)$, $\chi_{1197}(43,·)$, $\chi_{1197}(172,·)$, $\chi_{1197}(688,·)$, $\chi_{1197}(310,·)$, $\chi_{1197}(823,·)$, $\chi_{1197}(505,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{14} - \frac{1}{7} a^{13} - \frac{2}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{14} + \frac{3}{7} a^{13} + \frac{2}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3}$, $\frac{1}{49} a^{17} - \frac{2}{49} a^{16} - \frac{3}{49} a^{15} + \frac{23}{49} a^{14} + \frac{8}{49} a^{13} + \frac{2}{7} a^{12} + \frac{23}{49} a^{11} - \frac{5}{49} a^{10} + \frac{4}{49} a^{9} - \frac{2}{7} a^{8} - \frac{20}{49} a^{7} + \frac{11}{49} a^{6} + \frac{2}{49} a^{5} - \frac{23}{49} a^{4} + \frac{2}{49} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{18} + \frac{3}{49} a^{15} - \frac{2}{49} a^{14} + \frac{16}{49} a^{13} - \frac{5}{49} a^{12} + \frac{13}{49} a^{11} - \frac{6}{49} a^{10} + \frac{1}{49} a^{9} - \frac{20}{49} a^{8} - \frac{15}{49} a^{7} - \frac{11}{49} a^{6} - \frac{12}{49} a^{5} - \frac{2}{49} a^{4} + \frac{11}{49} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{49} a^{19} + \frac{3}{49} a^{16} - \frac{2}{49} a^{15} + \frac{16}{49} a^{14} - \frac{5}{49} a^{13} + \frac{13}{49} a^{12} - \frac{6}{49} a^{11} + \frac{1}{49} a^{10} - \frac{20}{49} a^{9} - \frac{15}{49} a^{8} - \frac{11}{49} a^{7} - \frac{12}{49} a^{6} - \frac{2}{49} a^{5} + \frac{11}{49} a^{4} + \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a$, $\frac{1}{49} a^{20} - \frac{3}{49} a^{16} - \frac{3}{49} a^{15} + \frac{17}{49} a^{14} - \frac{4}{49} a^{13} - \frac{6}{49} a^{12} + \frac{2}{49} a^{11} + \frac{16}{49} a^{10} + \frac{15}{49} a^{9} + \frac{10}{49} a^{8} + \frac{13}{49} a^{7} + \frac{2}{7} a^{6} + \frac{12}{49} a^{5} - \frac{8}{49} a^{4} + \frac{8}{49} a^{3} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{1029} a^{21} - \frac{3}{343} a^{20} + \frac{10}{1029} a^{18} + \frac{1}{343} a^{17} + \frac{11}{343} a^{16} - \frac{5}{147} a^{15} - \frac{69}{343} a^{14} - \frac{1}{7} a^{13} - \frac{470}{1029} a^{12} + \frac{4}{49} a^{11} - \frac{171}{343} a^{10} - \frac{17}{147} a^{9} + \frac{169}{343} a^{8} - \frac{143}{343} a^{7} - \frac{263}{1029} a^{6} - \frac{5}{49} a^{5} + \frac{16}{343} a^{4} + \frac{50}{1029} a^{3} - \frac{17}{49} a^{2} + \frac{15}{49} a - \frac{1}{147}$, $\frac{1}{1029} a^{22} + \frac{1}{343} a^{20} + \frac{10}{1029} a^{19} + \frac{3}{343} a^{18} - \frac{1}{343} a^{17} - \frac{11}{1029} a^{16} + \frac{15}{343} a^{15} + \frac{65}{343} a^{14} - \frac{155}{1029} a^{13} + \frac{11}{343} a^{12} - \frac{24}{343} a^{11} + \frac{220}{1029} a^{10} + \frac{71}{343} a^{9} + \frac{111}{343} a^{8} - \frac{512}{1029} a^{7} - \frac{61}{343} a^{6} - \frac{12}{343} a^{5} + \frac{104}{1029} a^{4} + \frac{52}{343} a^{3} + \frac{9}{49} a^{2} + \frac{26}{147} a - \frac{24}{49}$, $\frac{1}{7203} a^{23} - \frac{1}{7203} a^{22} - \frac{50}{7203} a^{20} - \frac{64}{7203} a^{19} + \frac{2}{343} a^{18} + \frac{4}{7203} a^{17} - \frac{169}{7203} a^{16} - \frac{69}{2401} a^{15} + \frac{1972}{7203} a^{14} + \frac{2351}{7203} a^{13} - \frac{1182}{2401} a^{12} + \frac{2707}{7203} a^{11} - \frac{337}{7203} a^{10} + \frac{1195}{2401} a^{9} - \frac{290}{1029} a^{8} - \frac{904}{7203} a^{7} - \frac{696}{2401} a^{6} + \frac{284}{1029} a^{5} + \frac{1147}{7203} a^{4} + \frac{696}{2401} a^{3} - \frac{226}{1029} a^{2} + \frac{292}{1029} a - \frac{171}{343}$, $\frac{1}{1236376243809} a^{24} - \frac{9129339}{137375138201} a^{23} + \frac{23387603}{412125414603} a^{22} + \frac{165457676}{1236376243809} a^{21} + \frac{2198986534}{412125414603} a^{20} - \frac{1292451208}{412125414603} a^{19} + \frac{9919583069}{1236376243809} a^{18} + \frac{152696380}{137375138201} a^{17} + \frac{23351502001}{412125414603} a^{16} + \frac{2066963561}{33415574157} a^{15} + \frac{117558828566}{412125414603} a^{14} + \frac{54025290497}{137375138201} a^{13} - \frac{147059393254}{1236376243809} a^{12} + \frac{203996979365}{412125414603} a^{11} + \frac{122435537734}{412125414603} a^{10} - \frac{548836819828}{1236376243809} a^{9} - \frac{45853985371}{412125414603} a^{8} + \frac{121233333254}{412125414603} a^{7} - \frac{62511788693}{137375138201} a^{6} + \frac{1512083371}{11138524719} a^{5} + \frac{46143085841}{137375138201} a^{4} + \frac{28635503422}{1236376243809} a^{3} - \frac{17784501701}{58875059229} a^{2} - \frac{508092359}{19625019743} a + \frac{66142885262}{176625177687}$, $\frac{1}{295493922270351} a^{25} - \frac{2}{32832658030039} a^{24} + \frac{46326395}{1059118000969} a^{23} - \frac{64067246371}{295493922270351} a^{22} - \frac{27224825884}{98497974090117} a^{21} - \frac{73442302}{887369135947} a^{20} + \frac{1111086713936}{295493922270351} a^{19} + \frac{949535899093}{98497974090117} a^{18} - \frac{954710475634}{98497974090117} a^{17} - \frac{8789394072100}{295493922270351} a^{16} + \frac{436500559058}{32832658030039} a^{15} - \frac{30772952963369}{98497974090117} a^{14} + \frac{14592444799886}{295493922270351} a^{13} - \frac{8218069078901}{32832658030039} a^{12} - \frac{482833473691}{2662107407841} a^{11} + \frac{58062216526226}{295493922270351} a^{10} - \frac{323382352113}{670054245511} a^{9} + \frac{4491030761207}{32832658030039} a^{8} + \frac{25166983921832}{98497974090117} a^{7} + \frac{8606183847953}{98497974090117} a^{6} + \frac{32694782595206}{98497974090117} a^{5} - \frac{15643317004505}{42213417467193} a^{4} + \frac{14368860082557}{32832658030039} a^{3} + \frac{6417502143062}{14071139155731} a^{2} + \frac{5131525073273}{42213417467193} a - \frac{582159907801}{14071139155731}$, $\frac{1}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{26} - \frac{5078108421484584101476730205555167438150815264482803418121295727108291386283361882}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{25} + \frac{99798830707321819659894084630463145296879813578854115309679100679736000451719292375}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{24} + \frac{3467361853392017980122225775805746455621788432229805931336838178435183700158199855813300180}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{23} - \frac{979934491355510845703663122761969213365048457719372270344333102604201341127614628088596592419}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{22} + \frac{712929063682290233414422238131052944303571411852008611138683410524343040011583772280559186482}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{21} + \frac{1754587307989944663793632327125083723035790992396456805666326147499339496882594252744851727273}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{20} + \frac{2782877602880898209175147405912493517863994279708604180638094229973696675196798617938951222876}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{19} + \frac{14045148020309941853792663952613932162362633605188919750961633726780709853698577770778494756799}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{18} - \frac{17048192952089911108348031247743056462061336384293570782179936236076580427842551864471562755347}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{17} - \frac{260086176874124618376707677973183610269083799082997075048320357889035652511594807879735550227}{12246387398269805623289918885542761469560310856174166814485408952617694606145960654160624788559} a^{16} - \frac{23865046122150163052607546968318961008570815174153795288850782816950213548002988323601202713054}{453116333735982808061726998765082174373731501678444172135960131246854700427400544203943117176683} a^{15} - \frac{146287075426731122520544464521035601035412681361469276418760780701391539454555735005757092729152}{453116333735982808061726998765082174373731501678444172135960131246854700427400544203943117176683} a^{14} - \frac{3201021739424043743505608333602907907370865657339485289735893309315072736507192234061679284578}{453116333735982808061726998765082174373731501678444172135960131246854700427400544203943117176683} a^{13} - \frac{1122740173425861845681853086635862981440123263184437435079751694162379197702497859189606920655803}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{12} + \frac{606034299706014637934940091083311534862899724183249604542785379161177931083024273196281629586090}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{11} - \frac{1492929153133297136251818705726330409362105912263923344246599426029142549723384334467127011671502}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{10} - \frac{488427001376030594137414848175295633785185024606783364996949666725851421772384177180084417002751}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{9} - \frac{17860325061003310658396271513977543880167704562243734668814591522104344578646906102244221594515}{352423815127986628492454332372841691179568945749901022772413435414220322554644867714177980026309} a^{8} - \frac{242130711718638765127885792245553205501076257243901034552503005454827281302721265889579272587771}{1057271445383959885477362997118525073538706837249703068317240306242660967663934603142533940078927} a^{7} + \frac{380086495794390931710793011592666860224462973856476431220708522493386644195097929295017245537684}{1057271445383959885477362997118525073538706837249703068317240306242660967663934603142533940078927} a^{6} + \frac{1387448720662064344641060203398221153421981725776001367088109482329454231358941215981480057688053}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{5} + \frac{323682277286454343281613908774460365129817854118454717942507444205790114244975629239233871470807}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{4} - \frac{455133610807111792764081516861489047998770400612627917231709765481004220168409499928510560026020}{3171814336151879656432088991355575220616120511749109204951720918727982902991803809427601820236781} a^{3} - \frac{183592762496454173078377317902207833422076949977027968456342069203652892948689521197689438241693}{453116333735982808061726998765082174373731501678444172135960131246854700427400544203943117176683} a^{2} + \frac{223152917984787757137201565439296397357003393170603865444155529417155559339874470367063526017586}{453116333735982808061726998765082174373731501678444172135960131246854700427400544203943117176683} a - \frac{61851943054212758085589297063826629498734329460578060233393426704441281165669613348415380467696}{453116333735982808061726998765082174373731501678444172135960131246854700427400544203943117176683}$

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:  \( 128935368208637150000000 \) (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

3.3.17689.1, 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 9.9.5534900853769.1, 9.9.9025761726072081.2, 9.9.1061871841310654257569.2, 9.9.1061871841310654257569.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ 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
$3$3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
3.9.12.2$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 54$$3$$3$$12$$C_9$$[2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
19Data not computed