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Properties

Label 27.27.5047240778...6609.1
Degree $27$
Signature $[27, 0]$
Discriminant $7^{18}\cdot 127^{24}$
Root discriminant $271.30$
Ramified primes $7, 127$
Class number $1$ (GRH)
Class group Trivial (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![-885366943, -475157373, 12919246032, 10338086657, -59290655739, -60609893115, 95741259302, 111110219033, -68473588099, -92576569635, 24409158618, 42042763063, -4069183891, -11332953439, 74350078, 1886796975, 84388495, -197099335, -14262294, 12931133, 1074049, -525887, -41798, 12829, 805, -173, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 173*x^25 + 805*x^24 + 12829*x^23 - 41798*x^22 - 525887*x^21 + 1074049*x^20 + 12931133*x^19 - 14262294*x^18 - 197099335*x^17 + 84388495*x^16 + 1886796975*x^15 + 74350078*x^14 - 11332953439*x^13 - 4069183891*x^12 + 42042763063*x^11 + 24409158618*x^10 - 92576569635*x^9 - 68473588099*x^8 + 111110219033*x^7 + 95741259302*x^6 - 60609893115*x^5 - 59290655739*x^4 + 10338086657*x^3 + 12919246032*x^2 - 475157373*x - 885366943)
 
gp: K = bnfinit(x^27 - 6*x^26 - 173*x^25 + 805*x^24 + 12829*x^23 - 41798*x^22 - 525887*x^21 + 1074049*x^20 + 12931133*x^19 - 14262294*x^18 - 197099335*x^17 + 84388495*x^16 + 1886796975*x^15 + 74350078*x^14 - 11332953439*x^13 - 4069183891*x^12 + 42042763063*x^11 + 24409158618*x^10 - 92576569635*x^9 - 68473588099*x^8 + 111110219033*x^7 + 95741259302*x^6 - 60609893115*x^5 - 59290655739*x^4 + 10338086657*x^3 + 12919246032*x^2 - 475157373*x - 885366943, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 173 x^{25} + 805 x^{24} + 12829 x^{23} - 41798 x^{22} - 525887 x^{21} + 1074049 x^{20} + 12931133 x^{19} - 14262294 x^{18} - 197099335 x^{17} + 84388495 x^{16} + 1886796975 x^{15} + 74350078 x^{14} - 11332953439 x^{13} - 4069183891 x^{12} + 42042763063 x^{11} + 24409158618 x^{10} - 92576569635 x^{9} - 68473588099 x^{8} + 111110219033 x^{7} + 95741259302 x^{6} - 60609893115 x^{5} - 59290655739 x^{4} + 10338086657 x^{3} + 12919246032 x^{2} - 475157373 x - 885366943 \)

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:  \(504724077808035841778557530783708473162637353188340488433492146609=7^{18}\cdot 127^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $271.30$
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:  $7, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(889=7\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{889}(128,·)$, $\chi_{889}(1,·)$, $\chi_{889}(611,·)$, $\chi_{889}(449,·)$, $\chi_{889}(781,·)$, $\chi_{889}(527,·)$, $\chi_{889}(400,·)$, $\chi_{889}(403,·)$, $\chi_{889}(149,·)$, $\chi_{889}(22,·)$, $\chi_{889}(576,·)$, $\chi_{889}(799,·)$, $\chi_{889}(480,·)$, $\chi_{889}(865,·)$, $\chi_{889}(226,·)$, $\chi_{889}(291,·)$, $\chi_{889}(484,·)$, $\chi_{889}(37,·)$, $\chi_{889}(361,·)$, $\chi_{889}(107,·)$, $\chi_{889}(814,·)$, $\chi_{889}(687,·)$, $\chi_{889}(99,·)$, $\chi_{889}(179,·)$, $\chi_{889}(382,·)$, $\chi_{889}(869,·)$, $\chi_{889}(830,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{8} a - \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{4} a^{10} + \frac{3}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} - \frac{5}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{2} a^{3} + \frac{3}{16} a + \frac{5}{16}$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3} + \frac{3}{16} a^{2} + \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{12} - \frac{1}{4} a^{11} + \frac{3}{16} a^{10} + \frac{3}{16} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{5}{16}$, $\frac{1}{128} a^{21} - \frac{3}{128} a^{20} - \frac{3}{128} a^{19} - \frac{1}{64} a^{18} + \frac{3}{64} a^{17} - \frac{7}{128} a^{16} + \frac{7}{128} a^{15} - \frac{1}{128} a^{14} + \frac{5}{128} a^{13} - \frac{19}{128} a^{11} + \frac{7}{32} a^{10} - \frac{7}{128} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{7} + \frac{23}{128} a^{6} - \frac{11}{32} a^{5} - \frac{25}{64} a^{4} + \frac{23}{128} a^{3} + \frac{47}{128} a^{2} - \frac{11}{64} a - \frac{7}{128}$, $\frac{1}{128} a^{22} - \frac{1}{32} a^{20} - \frac{3}{128} a^{19} - \frac{5}{128} a^{17} - \frac{3}{64} a^{16} - \frac{1}{32} a^{15} + \frac{1}{64} a^{14} + \frac{7}{128} a^{13} - \frac{3}{128} a^{12} - \frac{5}{128} a^{11} - \frac{3}{128} a^{10} - \frac{13}{128} a^{9} - \frac{3}{16} a^{8} - \frac{25}{128} a^{7} + \frac{1}{128} a^{6} - \frac{19}{64} a^{5} - \frac{47}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{128} a^{2} - \frac{57}{128} a - \frac{5}{128}$, $\frac{1}{128} a^{23} + \frac{1}{128} a^{20} - \frac{1}{32} a^{19} + \frac{3}{128} a^{18} - \frac{3}{64} a^{17} + \frac{3}{64} a^{15} + \frac{3}{128} a^{14} - \frac{7}{128} a^{13} + \frac{3}{128} a^{12} - \frac{7}{128} a^{11} + \frac{11}{128} a^{10} + \frac{5}{32} a^{9} - \frac{17}{128} a^{8} + \frac{9}{128} a^{7} - \frac{9}{64} a^{6} - \frac{63}{128} a^{5} - \frac{13}{32} a^{4} + \frac{59}{128} a^{3} + \frac{11}{128} a^{2} + \frac{35}{128} a + \frac{11}{32}$, $\frac{1}{33536} a^{24} + \frac{11}{16768} a^{23} - \frac{39}{16768} a^{22} + \frac{1}{16768} a^{21} - \frac{601}{33536} a^{20} - \frac{423}{16768} a^{19} - \frac{191}{16768} a^{18} - \frac{91}{2096} a^{17} - \frac{957}{33536} a^{16} + \frac{331}{16768} a^{15} - \frac{725}{16768} a^{14} - \frac{403}{8384} a^{13} + \frac{1029}{33536} a^{12} + \frac{1491}{8384} a^{11} - \frac{1381}{8384} a^{10} - \frac{2705}{16768} a^{9} - \frac{77}{33536} a^{8} + \frac{1697}{16768} a^{7} + \frac{1567}{16768} a^{6} - \frac{4791}{16768} a^{5} - \frac{9669}{33536} a^{4} + \frac{1283}{8384} a^{3} + \frac{7857}{16768} a^{2} + \frac{3911}{16768} a + \frac{10959}{33536}$, $\frac{1}{167680} a^{25} + \frac{1}{83840} a^{24} - \frac{259}{83840} a^{23} + \frac{257}{83840} a^{22} - \frac{641}{167680} a^{21} + \frac{279}{16768} a^{20} + \frac{1457}{83840} a^{19} - \frac{13}{20960} a^{18} + \frac{10347}{167680} a^{17} - \frac{1627}{83840} a^{16} - \frac{1057}{83840} a^{15} - \frac{1537}{41920} a^{14} - \frac{263}{33536} a^{13} + \frac{331}{10480} a^{12} - \frac{7359}{41920} a^{11} + \frac{1599}{16768} a^{10} - \frac{31261}{167680} a^{9} - \frac{345}{16768} a^{8} - \frac{11937}{83840} a^{7} + \frac{19937}{83840} a^{6} - \frac{73741}{167680} a^{5} - \frac{775}{4192} a^{4} + \frac{11033}{83840} a^{3} - \frac{33233}{83840} a^{2} + \frac{46303}{167680} a - \frac{18377}{41920}$, $\frac{1}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{26} + \frac{173561038614861903993407086215488081157128477500660601785560983263850491137976979871774512974282272297749}{398405656560511828647751600966435334181062842350958983887019847653845977612261482989426133973488640964538684800} a^{25} - \frac{333352918209718757311072212334171613301853170387266471946896035847910118621354701391897338287536126177981}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{24} + \frac{1414674561400395217864324887875085391085576925079015178940365955192313425792219716983670577936972765428605203}{398405656560511828647751600966435334181062842350958983887019847653845977612261482989426133973488640964538684800} a^{23} - \frac{1465552071370591742065940146603525277398812631917486607318975919810756635619391736321045201117558822688148197}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{22} + \frac{1365534564609100044425539942996153467752859418067007701843829733656569523860268860283230628749668819391190457}{398405656560511828647751600966435334181062842350958983887019847653845977612261482989426133973488640964538684800} a^{21} + \frac{8043923564198388444981560468214913855164925850990199582061023861900993000455456280596035935688750589607863569}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{20} + \frac{92657926278893380207249555021415149797262098342355776321799172673692043325017216608949133835711633549972103}{15936226262420473145910064038657413367242513694038359355480793906153839104490459319577045358939545638581547392} a^{19} + \frac{4056630490675541798546816410923616787774543456866081700874678044884512977592501081793564177666677828352910883}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{18} + \frac{2052689535386526605786783506830869496175985771353486644617883808210955367803352670540479368813725529673137911}{99601414140127957161937900241608833545265710587739745971754961913461494403065370747356533493372160241134671200} a^{17} - \frac{19026654663416020529680107895558240413990317651524196617354534631902649813896592730118124241223069258741752833}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{16} + \frac{4121899642894214372098218500393556790213072697560209565657587087541797497340208631954604983863328349712830647}{199202828280255914323875800483217667090531421175479491943509923826922988806130741494713066986744320482269342400} a^{15} - \frac{31454041711698311909158586576532035901098587450616260406918604750749147512031205868877797039766632945729291073}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{14} + \frac{10567035634451662957023755717190080610271558672218441664844078198436807778275397496779108183928899429090621093}{398405656560511828647751600966435334181062842350958983887019847653845977612261482989426133973488640964538684800} a^{13} - \frac{4712842466000070918962415137136238818285933855244382955029330791403931331938946037152568601469374352583812759}{159362262624204731459100640386574133672425136940383593554807939061538391044904593195770453589395456385815473920} a^{12} - \frac{24871202203972958846708115975049541788840929421503822150836823818228509844623142779956057264727527147960124637}{99601414140127957161937900241608833545265710587739745971754961913461494403065370747356533493372160241134671200} a^{11} - \frac{39684303875180339436765165960550243485053551964602763009839895806783202031338901431687428031553362194311833421}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{10} + \frac{41894621918163914680581251555218265316949297693063930446181178706134164813318341820923144527539514664126355321}{199202828280255914323875800483217667090531421175479491943509923826922988806130741494713066986744320482269342400} a^{9} + \frac{104247223448609777075822445029378829803045090208295861527291692728911304139613990904889069291575917967316048651}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{8} + \frac{4304693852657696457744401738132679279741065504277600958542247577097664845911581571392183149865278572440027197}{19920282828025591432387580048321766709053142117547949194350992382692298880613074149471306698674432048226934240} a^{7} - \frac{30628739767477119049659299102559415541959168367938245366779733766769580748923435273757086753908593101832882947}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{6} + \frac{68473490457382748433897033652932679973760231466932608990195141590331548387798202873236452177969578793283947391}{199202828280255914323875800483217667090531421175479491943509923826922988806130741494713066986744320482269342400} a^{5} + \frac{188267814855945093611915802608225162348741890242037042615960542440181088659850321357029100498470990472803889091}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{4} - \frac{1142393546561646638805879312337424320324813705281208030962728065560943369109747743032573888696542275916777555}{15936226262420473145910064038657413367242513694038359355480793906153839104490459319577045358939545638581547392} a^{3} + \frac{369790959094362099156275715443699931806176989513940020488458599681204641046020158507029404074121046917600450207}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600} a^{2} - \frac{21671956254042906573407929439073423551007949482711644352234572640900092874537983695355912972219843900777645489}{79681131312102365729550320193287066836212568470191796777403969530769195522452296597885226794697728192907736960} a - \frac{170953334273212202834922273685716549464783232272210426241813250309999774846323633289934114310376702933577470433}{796811313121023657295503201932870668362125684701917967774039695307691955224522965978852267946977281929077369600}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (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:  \( 4286588194138309400000000 \) (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

3.3.790321.2, 3.3.16129.1, 3.3.790321.1, \(\Q(\zeta_{7})^+\), 9.9.493640252540246161.1, 9.9.67675234241018881.1, 9.9.7961923633221630330769.1, 9.9.7961923633221630330769.2

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
7Data not computed
$127$127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$
127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$
127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$