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Properties

Label 27.27.1071061216...3001.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 73^{18}$
Root discriminant $256.16$
Ramified primes $3, 73$
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![-2813528454063, -7673425487946, 28868873558292, 29359018756881, -93507662067480, 741345334326, 97874407378491, -48962930005890, -22126898435067, 21186874621942, 107461159776, -3835851698421, 600310861842, 367193241090, -99785979396, -19135943232, 8144357769, 437158845, -393140443, 5617746, 11704797, -634038, -211032, 16893, 2112, -207, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 207*x^25 + 2112*x^24 + 16893*x^23 - 211032*x^22 - 634038*x^21 + 11704797*x^20 + 5617746*x^19 - 393140443*x^18 + 437158845*x^17 + 8144357769*x^16 - 19135943232*x^15 - 99785979396*x^14 + 367193241090*x^13 + 600310861842*x^12 - 3835851698421*x^11 + 107461159776*x^10 + 21186874621942*x^9 - 22126898435067*x^8 - 48962930005890*x^7 + 97874407378491*x^6 + 741345334326*x^5 - 93507662067480*x^4 + 29359018756881*x^3 + 28868873558292*x^2 - 7673425487946*x - 2813528454063)
 
gp: K = bnfinit(x^27 - 9*x^26 - 207*x^25 + 2112*x^24 + 16893*x^23 - 211032*x^22 - 634038*x^21 + 11704797*x^20 + 5617746*x^19 - 393140443*x^18 + 437158845*x^17 + 8144357769*x^16 - 19135943232*x^15 - 99785979396*x^14 + 367193241090*x^13 + 600310861842*x^12 - 3835851698421*x^11 + 107461159776*x^10 + 21186874621942*x^9 - 22126898435067*x^8 - 48962930005890*x^7 + 97874407378491*x^6 + 741345334326*x^5 - 93507662067480*x^4 + 29359018756881*x^3 + 28868873558292*x^2 - 7673425487946*x - 2813528454063, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 207 x^{25} + 2112 x^{24} + 16893 x^{23} - 211032 x^{22} - 634038 x^{21} + 11704797 x^{20} + 5617746 x^{19} - 393140443 x^{18} + 437158845 x^{17} + 8144357769 x^{16} - 19135943232 x^{15} - 99785979396 x^{14} + 367193241090 x^{13} + 600310861842 x^{12} - 3835851698421 x^{11} + 107461159776 x^{10} + 21186874621942 x^{9} - 22126898435067 x^{8} - 48962930005890 x^{7} + 97874407378491 x^{6} + 741345334326 x^{5} - 93507662067480 x^{4} + 29359018756881 x^{3} + 28868873558292 x^{2} - 7673425487946 x - 2813528454063 \)

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:  \(107106121656197668357044180433735940609513745651269801273974583001=3^{66}\cdot 73^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.16$
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:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1971=3^{3}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{1971}(64,·)$, $\chi_{1971}(1,·)$, $\chi_{1971}(1030,·)$, $\chi_{1971}(1159,·)$, $\chi_{1971}(1096,·)$, $\chi_{1971}(592,·)$, $\chi_{1971}(721,·)$, $\chi_{1971}(658,·)$, $\chi_{1971}(1687,·)$, $\chi_{1971}(1816,·)$, $\chi_{1971}(1753,·)$, $\chi_{1971}(154,·)$, $\chi_{1971}(283,·)$, $\chi_{1971}(220,·)$, $\chi_{1971}(1249,·)$, $\chi_{1971}(1378,·)$, $\chi_{1971}(1315,·)$, $\chi_{1971}(811,·)$, $\chi_{1971}(940,·)$, $\chi_{1971}(877,·)$, $\chi_{1971}(1906,·)$, $\chi_{1971}(373,·)$, $\chi_{1971}(502,·)$, $\chi_{1971}(439,·)$, $\chi_{1971}(1468,·)$, $\chi_{1971}(1597,·)$, $\chi_{1971}(1534,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{27} a^{18} - \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{2}{9} a^{15} - \frac{1}{3} a^{14} + \frac{2}{9} a^{12} + \frac{1}{3} a^{10} + \frac{13}{27} a^{9} - \frac{1}{3} a^{8} + \frac{2}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{9}$, $\frac{1}{27} a^{19} - \frac{1}{3} a^{17} + \frac{2}{9} a^{16} - \frac{1}{3} a^{15} + \frac{2}{9} a^{13} + \frac{1}{3} a^{11} + \frac{13}{27} a^{10} + \frac{2}{9} a^{7} - \frac{1}{3} a^{6} + \frac{1}{9} a$, $\frac{1}{27} a^{20} + \frac{2}{9} a^{17} - \frac{1}{3} a^{16} + \frac{2}{9} a^{14} + \frac{1}{3} a^{12} + \frac{13}{27} a^{11} + \frac{1}{3} a^{9} + \frac{2}{9} a^{8} - \frac{1}{3} a^{7} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{21} - \frac{1}{3} a^{17} - \frac{1}{9} a^{15} + \frac{1}{3} a^{13} + \frac{4}{27} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{9} a^{3} + \frac{1}{3}$, $\frac{1}{27} a^{22} - \frac{1}{9} a^{16} + \frac{1}{3} a^{14} + \frac{4}{27} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{9} a^{4} + \frac{1}{3} a$, $\frac{1}{81} a^{23} - \frac{1}{81} a^{22} + \frac{1}{81} a^{21} + \frac{1}{81} a^{20} - \frac{1}{81} a^{19} + \frac{1}{81} a^{18} - \frac{11}{27} a^{17} + \frac{2}{27} a^{16} - \frac{2}{27} a^{15} - \frac{35}{81} a^{14} - \frac{1}{81} a^{13} + \frac{1}{81} a^{12} + \frac{4}{81} a^{11} + \frac{23}{81} a^{10} - \frac{23}{81} a^{9} + \frac{2}{27} a^{8} + \frac{7}{27} a^{7} + \frac{11}{27} a^{6} - \frac{2}{27} a^{5} + \frac{8}{27} a^{4} - \frac{8}{27} a^{3} + \frac{4}{27} a^{2} - \frac{13}{27} a + \frac{13}{27}$, $\frac{1}{1539} a^{24} - \frac{8}{1539} a^{23} + \frac{14}{1539} a^{22} - \frac{2}{513} a^{21} + \frac{16}{1539} a^{20} - \frac{28}{1539} a^{19} - \frac{25}{1539} a^{18} + \frac{244}{513} a^{17} + \frac{248}{513} a^{16} + \frac{421}{1539} a^{15} + \frac{469}{1539} a^{14} + \frac{140}{1539} a^{13} + \frac{182}{513} a^{12} - \frac{125}{1539} a^{11} + \frac{14}{81} a^{10} + \frac{254}{1539} a^{9} + \frac{185}{513} a^{8} + \frac{178}{513} a^{7} - \frac{22}{513} a^{6} - \frac{77}{513} a^{5} - \frac{112}{513} a^{4} + \frac{11}{171} a^{3} - \frac{179}{513} a^{2} + \frac{5}{513} a - \frac{238}{513}$, $\frac{1}{502124913} a^{25} + \frac{68513}{502124913} a^{24} + \frac{966307}{167374971} a^{23} + \frac{7112609}{502124913} a^{22} + \frac{5716789}{502124913} a^{21} - \frac{1893911}{167374971} a^{20} - \frac{8158004}{502124913} a^{19} - \frac{8344756}{502124913} a^{18} + \frac{13403176}{55791657} a^{17} + \frac{2190838}{502124913} a^{16} + \frac{125063354}{502124913} a^{15} + \frac{8276344}{167374971} a^{14} - \frac{42443809}{502124913} a^{13} - \frac{149994653}{502124913} a^{12} - \frac{68336021}{167374971} a^{11} + \frac{117111067}{502124913} a^{10} - \frac{239476840}{502124913} a^{9} + \frac{10836419}{55791657} a^{8} + \frac{26023378}{55791657} a^{7} + \frac{6507221}{18597219} a^{6} - \frac{12325042}{55791657} a^{5} - \frac{34910629}{167374971} a^{4} + \frac{57230617}{167374971} a^{3} - \frac{14802974}{55791657} a^{2} + \frac{43169032}{167374971} a - \frac{61013440}{167374971}$, $\frac{1}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{26} + \frac{18509794039582188664236734469053369421610200534805599319651653984477513463156573695156408488740807751621313761405318153802226393679}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{25} + \frac{1374445645575761363886668113918167355028631967712228905671284232129888311684900795475501070234136146642341328302267760468906213471159973}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{24} + \frac{105049833136999419865286192282857521053902519587208131596076605948262842140518242656828022256647078355961806473161478857581412859569244741}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{23} + \frac{211384419503621117063517115637108446407984351777508352709101267921155362988064314291263815275235594828898528512100818910864603112576666639}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{22} + \frac{36464317410481919574956863424132338227551939336633766780068243040845120363359758616736791080596179029979493208253916906629368126131079206}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{21} + \frac{267755046998632904878345244925678161827342940060434220252598010594016549223070394169278615147978577733433719646258830586817952995358695565}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{20} + \frac{384970094495047542904820470573061675270322224906787246027528523199395520493006934623654537007318239215774721555135228206798193074403655292}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{19} - \frac{100818463253221793389691434153556352055301763545787260559295221431646612753440799058193512281162773002084167513237502109683306289232340361}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{18} + \frac{5596877025820813199167985772736316204391747770104836297537958385280520086626001537122686177807462507310411422869531055042925332729858064524}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{17} + \frac{4763135293267301303213586153548142427689354548505354176969728608442696501544620355497398282647215136326763451427784309435851191177337254983}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{16} - \frac{2886439715744688200186327464221563495287432059135714061411288975038718185304477627084117522678491572901855741936522819093096156744288550951}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{15} + \frac{1420468455787268464313919882304759883718842765556923611328645906892524897314489822270488466283515631152526294209555176056576044131579064538}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{14} + \frac{1887461369456994410947633540567451298723422164839989179966928311861226253601781181323399081577063123294096114542643129375349354881460747}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{13} + \frac{1868451389881285269164880307891015896442982160654638131910693501757880206987086575513311072556285058548706065881742246399152396776495354549}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{12} + \frac{189764217340150344324049783228295455456145323969062587825871775172339318851373432276704525297355328342572151486657106874761114140007471057}{1216921006913473808235211135532776495788014332580986370208632808318165934349840961474638756323754940423257998399698331444726996886831854941} a^{11} - \frac{9529943176960986708935646519184926418455755547251974294151027324574752087171207159589126454873057917046115575052625378930545272437955675336}{23121499131356002356469011575122753419972272319038741033964023358045152752646978268018136370151343868041901969594268297449812940849805243879} a^{10} + \frac{1887020262164859519917545372516477428521283506522404755961872028875220002352189624386732044726616819920549835504368743509519479078257879231}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{9} + \frac{430168026053957265689016863282950183174189649109510976160310602329060611457644972489410762221278733854305527857195698952879451793599781594}{2569055459039555817385445730569194824441363591004304559329335928671683639182997585335348485572371540893544663288252033049979215649978360431} a^{8} - \frac{64397832452841117143516518601490314657174477723235776684869086218564430315052527512705629867997402260786000114493222340420238196921246531}{2569055459039555817385445730569194824441363591004304559329335928671683639182997585335348485572371540893544663288252033049979215649978360431} a^{7} + \frac{1115391652167848315062706719079865616796310610083720590129189425589715057787638712984584433477208164818943195067336669043689462301363243608}{2569055459039555817385445730569194824441363591004304559329335928671683639182997585335348485572371540893544663288252033049979215649978360431} a^{6} - \frac{1934513641428674822121659086645907101033790740077304649377020019232242744731554211062349227576275305101771558839708466904328558029898487146}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{5} - \frac{86242264861188698997360767913033127049714536664108691611759333055284024363526262656832634069314287892782607044814264567445490971266902437}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{4} - \frac{855683089510378526402271421753387001883342355830513390315339158357824232414824390939094472334636218369575052598062000331616605405612141411}{2569055459039555817385445730569194824441363591004304559329335928671683639182997585335348485572371540893544663288252033049979215649978360431} a^{3} + \frac{808849193687996125432453707211504901970058747468577958470373360085926301470813475803049346647594017320988497975690554614817642922093408422}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a^{2} + \frac{2635809116370764397319154299232361942826714696750913819872927946186920754142303850875840441693795228590321048727592983949330167610428492071}{7707166377118667452156337191707584473324090773012913677988007786015050917548992756006045456717114622680633989864756099149937646949935081293} a + \frac{1086601220842986595738162328303157902448921537229328191451855556292316650858867333121060879654714312837277965712084082998465980866297635640}{2569055459039555817385445730569194824441363591004304559329335928671683639182997585335348485572371540893544663288252033049979215649978360431}$

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:  \( 803150353858998500000000 \) (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

3.3.431649.2, \(\Q(\zeta_{9})^+\), 3.3.5329.1, 3.3.431649.1, 9.9.80425212553252449.3, \(\Q(\zeta_{27})^+\), 9.9.4749028376057003861001.3, 9.9.4749028376057003861001.2

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
$73$73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$