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Properties

Label 27.27.1197336751...009.10
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![2164132187, -16629567129, -159434629011, 160140482187, 846276915456, -857479276836, -1025115854027, 1133759797215, 551539138572, -690719521993, -157505340675, 234863069337, 25496147479, -48608855199, -2363379129, 6402152565, 117546663, -546700113, -2255516, 30177957, -36198, -1048518, 1923, 21600, -15, -234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 234*x^25 - 15*x^24 + 21600*x^23 + 1923*x^22 - 1048518*x^21 - 36198*x^20 + 30177957*x^19 - 2255516*x^18 - 546700113*x^17 + 117546663*x^16 + 6402152565*x^15 - 2363379129*x^14 - 48608855199*x^13 + 25496147479*x^12 + 234863069337*x^11 - 157505340675*x^10 - 690719521993*x^9 + 551539138572*x^8 + 1133759797215*x^7 - 1025115854027*x^6 - 857479276836*x^5 + 846276915456*x^4 + 160140482187*x^3 - 159434629011*x^2 - 16629567129*x + 2164132187)
 
gp: K = bnfinit(x^27 - 234*x^25 - 15*x^24 + 21600*x^23 + 1923*x^22 - 1048518*x^21 - 36198*x^20 + 30177957*x^19 - 2255516*x^18 - 546700113*x^17 + 117546663*x^16 + 6402152565*x^15 - 2363379129*x^14 - 48608855199*x^13 + 25496147479*x^12 + 234863069337*x^11 - 157505340675*x^10 - 690719521993*x^9 + 551539138572*x^8 + 1133759797215*x^7 - 1025115854027*x^6 - 857479276836*x^5 + 846276915456*x^4 + 160140482187*x^3 - 159434629011*x^2 - 16629567129*x + 2164132187, 1)
 

Normalized defining polynomial

\( x^{27} - 234 x^{25} - 15 x^{24} + 21600 x^{23} + 1923 x^{22} - 1048518 x^{21} - 36198 x^{20} + 30177957 x^{19} - 2255516 x^{18} - 546700113 x^{17} + 117546663 x^{16} + 6402152565 x^{15} - 2363379129 x^{14} - 48608855199 x^{13} + 25496147479 x^{12} + 234863069337 x^{11} - 157505340675 x^{10} - 690719521993 x^{9} + 551539138572 x^{8} + 1133759797215 x^{7} - 1025115854027 x^{6} - 857479276836 x^{5} + 846276915456 x^{4} + 160140482187 x^{3} - 159434629011 x^{2} - 16629567129 x + 2164132187 \)

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$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(130,·)$, $\chi_{1197}(709,·)$, $\chi_{1197}(1030,·)$, $\chi_{1197}(1033,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(652,·)$, $\chi_{1197}(226,·)$, $\chi_{1197}(142,·)$, $\chi_{1197}(277,·)$, $\chi_{1197}(121,·)$, $\chi_{1197}(1054,·)$, $\chi_{1197}(415,·)$, $\chi_{1197}(802,·)$, $\chi_{1197}(100,·)$, $\chi_{1197}(358,·)$, $\chi_{1197}(424,·)$, $\chi_{1197}(169,·)$, $\chi_{1197}(43,·)$, $\chi_{1197}(1138,·)$, $\chi_{1197}(562,·)$, $\chi_{1197}(1012,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(58,·)$, $\chi_{1197}(85,·)$, $\chi_{1197}(1087,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{28} a^{18} - \frac{13}{28} a^{17} - \frac{1}{7} a^{16} - \frac{5}{14} a^{15} - \frac{5}{28} a^{14} + \frac{1}{4} a^{13} - \frac{5}{14} a^{12} + \frac{9}{28} a^{11} + \frac{3}{28} a^{10} + \frac{5}{14} a^{9} - \frac{1}{4} a^{8} + \frac{3}{28} a^{7} - \frac{1}{7} a^{6} - \frac{5}{28} a^{5} - \frac{1}{14} a^{4} + \frac{13}{28} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{28} a^{19} - \frac{5}{28} a^{17} - \frac{3}{14} a^{16} + \frac{5}{28} a^{15} - \frac{1}{14} a^{14} - \frac{3}{28} a^{13} - \frac{9}{28} a^{12} + \frac{2}{7} a^{11} - \frac{1}{4} a^{10} + \frac{11}{28} a^{9} - \frac{1}{7} a^{8} + \frac{1}{4} a^{7} - \frac{1}{28} a^{6} - \frac{11}{28} a^{5} - \frac{13}{28} a^{4} + \frac{2}{7} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{28} a^{20} + \frac{13}{28} a^{17} + \frac{13}{28} a^{16} + \frac{1}{7} a^{15} - \frac{1}{14} a^{13} - \frac{1}{2} a^{12} + \frac{5}{14} a^{11} - \frac{1}{14} a^{10} - \frac{5}{14} a^{9} - \frac{1}{2} a^{7} - \frac{3}{28} a^{6} - \frac{5}{14} a^{5} - \frac{1}{14} a^{4} - \frac{5}{28} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{196} a^{21} + \frac{3}{196} a^{20} - \frac{3}{196} a^{19} + \frac{1}{98} a^{18} - \frac{1}{14} a^{17} + \frac{11}{28} a^{16} - \frac{61}{196} a^{15} + \frac{31}{196} a^{14} - \frac{22}{49} a^{13} - \frac{13}{28} a^{12} + \frac{73}{196} a^{11} + \frac{79}{196} a^{9} + \frac{47}{196} a^{8} - \frac{43}{196} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{17}{98} a^{4} - \frac{11}{28} a^{3} + \frac{3}{14} a^{2} + \frac{5}{14} a - \frac{1}{4}$, $\frac{1}{154252} a^{22} - \frac{53}{154252} a^{21} + \frac{135}{77126} a^{20} - \frac{1153}{154252} a^{19} - \frac{59}{22036} a^{18} + \frac{864}{5509} a^{17} + \frac{17081}{77126} a^{16} - \frac{1129}{77126} a^{15} - \frac{55563}{154252} a^{14} + \frac{4149}{22036} a^{13} + \frac{14223}{38563} a^{12} - \frac{1555}{22036} a^{11} - \frac{34739}{154252} a^{10} + \frac{16099}{77126} a^{9} + \frac{27597}{77126} a^{8} + \frac{288}{787} a^{7} + \frac{148}{5509} a^{6} + \frac{6561}{38563} a^{5} - \frac{371}{1574} a^{4} + \frac{1483}{11018} a^{3} - \frac{2465}{5509} a^{2} + \frac{727}{1574} a - \frac{651}{3148}$, $\frac{1}{154252} a^{23} - \frac{89}{77126} a^{21} - \frac{449}{38563} a^{20} - \frac{2497}{154252} a^{19} + \frac{379}{38563} a^{18} - \frac{33367}{154252} a^{17} - \frac{59223}{154252} a^{16} - \frac{5443}{38563} a^{15} - \frac{4122}{38563} a^{14} - \frac{9607}{38563} a^{13} - \frac{32061}{77126} a^{12} - \frac{6529}{38563} a^{11} + \frac{17400}{38563} a^{10} + \frac{64129}{154252} a^{9} - \frac{15990}{38563} a^{8} - \frac{26475}{77126} a^{7} + \frac{11888}{38563} a^{6} - \frac{72257}{154252} a^{5} - \frac{3566}{38563} a^{4} + \frac{1026}{5509} a^{3} + \frac{3083}{22036} a^{2} + \frac{1039}{11018} a + \frac{125}{3148}$, $\frac{1}{80902989945932} a^{24} + \frac{412157}{7354817267812} a^{23} - \frac{8870563}{2889392498069} a^{22} - \frac{42358783843}{80902989945932} a^{21} - \frac{31754259764}{1838704316953} a^{20} + \frac{1422976508077}{80902989945932} a^{19} + \frac{607058924207}{40451494972966} a^{18} + \frac{7414642870681}{40451494972966} a^{17} + \frac{4441840383039}{20225747486483} a^{16} - \frac{127411897247}{1093283647918} a^{15} + \frac{752033043609}{5778784996138} a^{14} + \frac{8961999326442}{20225747486483} a^{13} + \frac{527151434344}{1838704316953} a^{12} - \frac{214844711012}{20225747486483} a^{11} - \frac{20192000479337}{80902989945932} a^{10} + \frac{26860618466895}{80902989945932} a^{9} + \frac{6791390237307}{40451494972966} a^{8} + \frac{25524228793981}{80902989945932} a^{7} + \frac{609461689534}{20225747486483} a^{6} + \frac{724879065139}{7354817267812} a^{5} - \frac{3181318334863}{11557569992276} a^{4} - \frac{3199221163079}{11557569992276} a^{3} + \frac{1991713603959}{5778784996138} a^{2} + \frac{223951130027}{825540713734} a + \frac{19641835490}{412770356867}$, $\frac{1}{80902989945932} a^{25} + \frac{13979167}{5778784996138} a^{23} + \frac{2794634}{2889392498069} a^{22} + \frac{10360355723}{7354817267812} a^{21} - \frac{285861630056}{20225747486483} a^{20} + \frac{1383904094703}{80902989945932} a^{19} + \frac{1060297319833}{80902989945932} a^{18} + \frac{5114535529903}{40451494972966} a^{17} + \frac{3773258402458}{20225747486483} a^{16} + \frac{10235606952523}{40451494972966} a^{15} + \frac{2383741815551}{20225747486483} a^{14} - \frac{340510320417}{1838704316953} a^{13} + \frac{19151338212551}{40451494972966} a^{12} - \frac{5532457743873}{80902989945932} a^{11} + \frac{1946497321077}{20225747486483} a^{10} + \frac{6193011820311}{20225747486483} a^{9} + \frac{1993696644157}{5778784996138} a^{8} - \frac{33009976713111}{80902989945932} a^{7} + \frac{558409714231}{3677408633906} a^{6} - \frac{4455110942614}{20225747486483} a^{5} - \frac{36834088795179}{80902989945932} a^{4} + \frac{1281213241123}{5778784996138} a^{3} - \frac{2363646164303}{11557569992276} a^{2} - \frac{913694092517}{2889392498069} a - \frac{7701560252}{37524577897}$, $\frac{1}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{26} + \frac{502993398753676381760250755559509647174061979708638566850217834969763420195949768657064985634009}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{25} - \frac{391288247534810112473268430158128893100635800995071088779286595538147519788690419524810543626005}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{24} - \frac{37913765263375097615179002750366171797103462116158838134994184823740284642343572033857353230381180410129}{55905186704811990283936901543384505141369460681495502311474561150625291164696500628793770655228300499544613698} a^{23} + \frac{2514162448736213676758315249207283059011277996782923322607747662242510341755485870854912736816166898883}{5082289700437453662176081958489500467397223698317772937406778286420481014972409148072160968657118227231328518} a^{22} - \frac{94385067461247062759898567865168996275620857529427318715467928600375978763746092899908216795206691345559041}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{21} + \frac{871766025899961490388699556498994299582010944544803468000313835159588440155215900812824748336537274172498557}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{20} + \frac{6435528325287695242870716857563849656119875961739376337288221922006217115779183897624612423906523924726813}{5082289700437453662176081958489500467397223698317772937406778286420481014972409148072160968657118227231328518} a^{19} - \frac{551719358540435819861841457696759766014870652675719530523956227006956393666989384666017347097036844254164017}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{18} - \frac{3837217338527195146887585771524441830255961274287902563406220112091590464108290492351968539356445127502814415}{10164579400874907324352163916979000934794447396635545874813556572840962029944818296144321937314236454462657036} a^{17} + \frac{32574930828766542634768770686744006077262846258201928601647246015526429424196391184037019716737369532322535407}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{16} + \frac{7734238262020918868898356917004505332501214402325642086709068240059490504684191965979000109993017008438542395}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{15} - \frac{3480138331779388018169451706683129683226901631011426764389420400382066685240613297365992380870458046401834131}{55905186704811990283936901543384505141369460681495502311474561150625291164696500628793770655228300499544613698} a^{14} + \frac{4041329230175702173893508271926959036161159739048713218477983220580931688916487136222001425464477334841240437}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{13} - \frac{2100169502738135422643823912866169562911947641976845233710516308792355385974570834332670077106750543791604377}{55905186704811990283936901543384505141369460681495502311474561150625291164696500628793770655228300499544613698} a^{12} - \frac{1285245661540280051684113531099468306916552061798514174143171176367979125241318409290949852686315771918653027}{3021901984043891366699291975318081358992943820621378503322949251385150873767378412367230846228556783759168308} a^{11} + \frac{10116893096078007110921468386618216977516344077321361429991195869383020022276080041909027560190894957473225171}{27952593352405995141968450771692252570684730340747751155737280575312645582348250314396885327614150249772306849} a^{10} + \frac{24775289766891971369516412051273007103972711780993450150890792664649934354087008728663501005049116793730319387}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{9} + \frac{10420146138962737557612247866966038318703915586594734615213393494549679379167684416634509884925201084549719001}{55905186704811990283936901543384505141369460681495502311474561150625291164696500628793770655228300499544613698} a^{8} - \frac{888056549945161281295757593107104066230723629510430822803658170494867682200999736687266957248675114645531524}{27952593352405995141968450771692252570684730340747751155737280575312645582348250314396885327614150249772306849} a^{7} + \frac{6389870802764979112851461584043399014329378689340313078527204278532764433278404708015457027869818461494219203}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{6} + \frac{5212057238243437807845867600826571230413829999414444288929024518947276835212112192660627956902284058648887803}{15972910487089140081124829012395572897534131623284429231849874614464368904199000179655363044350942999869889628} a^{5} + \frac{13758539907275296716819555034684638047404780396288717352800096234958370777994441605965029538075018189068710287}{111810373409623980567873803086769010282738921362991004622949122301250582329393001257587541310456600999089227396} a^{4} - \frac{7139405540569394991693509244454994251128392809596707128494449703184266845808607005064687352302372115812724145}{15972910487089140081124829012395572897534131623284429231849874614464368904199000179655363044350942999869889628} a^{3} + \frac{2580873949644019145414439421751615911949140553557569251577288654000098376720561871007847874287490105254511591}{15972910487089140081124829012395572897534131623284429231849874614464368904199000179655363044350942999869889628} a^{2} - \frac{565924560377239835491472458964705251313854713334742994745394784180724701897979867330588119851516237427495451}{1452082771553558189193166273854142990684921056662220839259079510405851718563545470877760276759176636351808148} a + \frac{128719249589517955590191053684777387481744183573466842463987536290282983792403023552860230691064508374793}{583890571980155727486651155592761108990134947480787733288853436703625124440671157320345190976419907876513}$

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:  \( 68804553250153435000000 \) (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

3.3.1432809.4, 3.3.361.1, 3.3.3969.1, 3.3.1432809.1, 9.9.2941473244627851129.10, 9.9.9025761726072081.2, 9.9.1061871841310654257569.6, 9.9.1998099208210609.1

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
3Data not computed
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
19Data not computed