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Properties

Label 27.27.1112252257...6081.1
Degree $27$
Signature $[27, 0]$
Discriminant $919^{26}$
Root discriminant $713.78$
Ramified prime $919$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-384744919097023, 397514834392024, 1009917274857026, -1546434511258730, -245718011944824, 1367425404919687, -401896444043157, -473016326020054, 272496425836143, 61821212164769, -71555147996156, 2167507434225, 9607454297019, -1504988947931, -703058979520, 175097584226, 28281288192, -10054716230, -609480375, 330851832, 6342342, -6491047, -23792, 74120, 57, -442, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - x^26 - 442*x^25 + 57*x^24 + 74120*x^23 - 23792*x^22 - 6491047*x^21 + 6342342*x^20 + 330851832*x^19 - 609480375*x^18 - 10054716230*x^17 + 28281288192*x^16 + 175097584226*x^15 - 703058979520*x^14 - 1504988947931*x^13 + 9607454297019*x^12 + 2167507434225*x^11 - 71555147996156*x^10 + 61821212164769*x^9 + 272496425836143*x^8 - 473016326020054*x^7 - 401896444043157*x^6 + 1367425404919687*x^5 - 245718011944824*x^4 - 1546434511258730*x^3 + 1009917274857026*x^2 + 397514834392024*x - 384744919097023)
 
gp: K = bnfinit(x^27 - x^26 - 442*x^25 + 57*x^24 + 74120*x^23 - 23792*x^22 - 6491047*x^21 + 6342342*x^20 + 330851832*x^19 - 609480375*x^18 - 10054716230*x^17 + 28281288192*x^16 + 175097584226*x^15 - 703058979520*x^14 - 1504988947931*x^13 + 9607454297019*x^12 + 2167507434225*x^11 - 71555147996156*x^10 + 61821212164769*x^9 + 272496425836143*x^8 - 473016326020054*x^7 - 401896444043157*x^6 + 1367425404919687*x^5 - 245718011944824*x^4 - 1546434511258730*x^3 + 1009917274857026*x^2 + 397514834392024*x - 384744919097023, 1)
 

Normalized defining polynomial

\( x^{27} - x^{26} - 442 x^{25} + 57 x^{24} + 74120 x^{23} - 23792 x^{22} - 6491047 x^{21} + 6342342 x^{20} + 330851832 x^{19} - 609480375 x^{18} - 10054716230 x^{17} + 28281288192 x^{16} + 175097584226 x^{15} - 703058979520 x^{14} - 1504988947931 x^{13} + 9607454297019 x^{12} + 2167507434225 x^{11} - 71555147996156 x^{10} + 61821212164769 x^{9} + 272496425836143 x^{8} - 473016326020054 x^{7} - 401896444043157 x^{6} + 1367425404919687 x^{5} - 245718011944824 x^{4} - 1546434511258730 x^{3} + 1009917274857026 x^{2} + 397514834392024 x - 384744919097023 \)

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:  \(111225225787238495736527863011990180372275621447089344667888529197779462256081=919^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $713.78$
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:  $919$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(919\)
Dirichlet character group:    $\lbrace$$\chi_{919}(704,·)$, $\chi_{919}(129,·)$, $\chi_{919}(515,·)$, $\chi_{919}(1,·)$, $\chi_{919}(267,·)$, $\chi_{919}(610,·)$, $\chi_{919}(526,·)$, $\chi_{919}(655,·)$, $\chi_{919}(824,·)$, $\chi_{919}(275,·)$, $\chi_{919}(771,·)$, $\chi_{919}(474,·)$, $\chi_{919}(207,·)$, $\chi_{919}(866,·)$, $\chi_{919}(611,·)$, $\chi_{919}(553,·)$, $\chi_{919}(492,·)$, $\chi_{919}(367,·)$, $\chi_{919}(99,·)$, $\chi_{919}(754,·)$, $\chi_{919}(52,·)$, $\chi_{919}(440,·)$, $\chi_{919}(57,·)$, $\chi_{919}(767,·)$, $\chi_{919}(701,·)$, $\chi_{919}(574,·)$, $\chi_{919}(575,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{229} a^{25} - \frac{67}{229} a^{24} - \frac{87}{229} a^{23} + \frac{53}{229} a^{22} + \frac{114}{229} a^{21} - \frac{5}{229} a^{20} - \frac{83}{229} a^{19} - \frac{106}{229} a^{18} + \frac{101}{229} a^{17} + \frac{18}{229} a^{16} + \frac{68}{229} a^{15} - \frac{46}{229} a^{14} + \frac{73}{229} a^{13} - \frac{48}{229} a^{12} - \frac{16}{229} a^{11} + \frac{83}{229} a^{10} + \frac{65}{229} a^{9} + \frac{110}{229} a^{8} - \frac{98}{229} a^{7} - \frac{9}{229} a^{6} - \frac{11}{229} a^{5} - \frac{70}{229} a^{4} - \frac{55}{229} a^{3} + \frac{2}{229} a^{2} + \frac{5}{229} a + \frac{44}{229}$, $\frac{1}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{26} + \frac{467340166259317503962102662873964018915892987748320296167107987373419647021387356462291726336307075534905398208570673342876520130841472431941446934259993368369085403271823305139}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{25} - \frac{52043281503272860232852519833019498812241761342180218567824779437842748805090484599735601923709497343530607985656103778455203072419179384299371733162858871841969502631702570192436}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{24} - \frac{14500787947017706780448083757360525105772467591354975735443381823202278257939659404333719682372633431411513097195818644350373982188570015680449368375199196214573757121992545202102}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{23} - \frac{18145639194915112387143491052247202908308148741514975835368623264817703025377869139286989327513375439285895280375101539939697929762070597776799441440645826024955040575126429875380}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{22} + \frac{108463030653428173204943094908819527939575569951421495930242637447867170667912684961132072871530927883241247746033536524415648788467185816327127616814400536228617092896809382979888}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{21} - \frac{37510024297006762224839384652658240848966916328684585253356184788681302169905415062607836284357009827869301039881952694401557557393730601709040202761861436497579653116850539073394}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{20} - \frac{34964228148482227090350698622601662243987317286531359730568632115386925653041394153239140067086804887542326298996125034932253011666257773841326923643921808989863146882304481364429}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{19} - \frac{40144820622055742486491008757091297682239644236378808411089664724755830847097528136619360799067409921758007112361494531499290773761552096184654531146409610289198976607125957316106}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{18} + \frac{60081949347180720285980666987053235348591039370858294389182750136623049029924796484720290089211582098845615439202448814682444247761652712694825485312183320471101081773700279302162}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{17} + \frac{75594890655073897173694595365342897085219143081859732924494798671652939973859174540359342697723689786649932482781558595151371930447452951881885615462974622095047605777849752762116}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{16} - \frac{20755013609276925389905677893544938480614113965777701114915872114916406227865313088637758978708344722302460936999578460958845971261663961456116341887814781591515071746024034168894}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{15} + \frac{143359202687632947684821193714375460739172428967876260881474173736558677916530007681290662977029736938924453751800413963469627568766831537349082979955322187086041617452558746925}{364069740650659837352873608129195624939020837004240006348983221355371990403687725335777488450164108938235963288550842153038537942590830727754677583411798711849155105658033378289} a^{14} - \frac{100597968520091317037779863714571294591304469014612510042073819436779985697597740978376019853208826622277163235216804346855096737975884235751698418201331814194418380019835410519502}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{13} + \frac{75243857647790236091247456421795404838951671027382666263587990860484166389617702588925765279638657061040822480661921249612066120389769232823226968788611076245989220870713108238478}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{12} + \frac{4123532350817552618230643771083446636432945503811983524027084604465254759451521607646832553771155623026863305984592758670403488417416604349917679069780446053862632239902492163757}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{11} + \frac{44057055046575626885982197676199009968761882270237385497825367643747247486560903510909408649932229893934085339449692417526107372223883578886252126442406535002783708479136903421082}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{10} + \frac{13358125917463668872170256200452183685580431971042877888637798459259092437603803853957928864096441155979720525266262304601880300173230592069877932043018368605990872856771341780850}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{9} + \frac{59958965817589157613599491806280841200361784349486864151027016177983352284754913476797195931276814822056781541094338187021634018235234949248957324580401191262215277291917993962789}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{8} + \frac{71820619735265530628861403111115046826324178822180961294010339812475177074464663882400255681247504787986280684069409727552402815531722469453819119592658321736817968939003766341049}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{7} + \frac{67899803444981018550521750925052940482346004226801198862631442257365743008787904397131583485021288685060686932618109092871406814843880715010561267012608979569271978309921296420909}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{6} + \frac{98848680808790058569438954317211979461215957904364920331309278436636434211078799238321399046526474851222698771838988817073568370692589958635702052316943528866677921668071272807757}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{5} - \frac{41474213992358834183241689307260047564927210996935658062665724439188448563698875316267259637202070630353442637706702141373082668881990503976565533427881732894964931645194764555670}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{4} + \frac{52199760653671474671384764222041374887005294824795307811754217516582752804547190827787523752356118166007997892962850100162756540363520977570008213970775085385117145648572729469958}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{3} + \frac{32185387147145345759441512183879455496969265757114424192542160656298368054045254523290687468714226466088177542785371240514209865946458157330071860527403336969048414717759793507900}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a^{2} + \frac{25965780573661450125011819565345702719507777081832482460456207614801866898842453275745406784325869489850057107182967782803238828406621162693258819671461982933709053741582980254693}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359} a + \frac{10032667912420476798182159634035247929651599303663978728498726560003635820072449592467887402964440933028705154709213024564206555142858435116590942731137274210150209448624246193226}{229728006350566357369663246729522439336522148149675444006208412675239725944726954686875595212053552740026892835075581398567317441774814189213201555132844987176816871670219061700359}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

3.3.844561.1, 9.9.508773041409246017163841.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $27$ $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ $27$

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
919Data not computed