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Properties

Label 27.27.1410763180...5561.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 109^{26}$
Root discriminant $396.39$
Ramified primes $3, 109$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![4826968628849, 16368292431012, 1114518996171, -45243443721079, -28185634575723, 46953795423150, 41739348314777, -21922844820627, -27072768198312, 3708053868470, 9167918905077, 460247647320, -1695101848663, -261230425254, 175020065793, 38866417215, -10299084090, -2891823192, 347122817, 121903638, -6507954, -3018210, 61476, 43164, -218, -327, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 327*x^25 - 218*x^24 + 43164*x^23 + 61476*x^22 - 3018210*x^21 - 6507954*x^20 + 121903638*x^19 + 347122817*x^18 - 2891823192*x^17 - 10299084090*x^16 + 38866417215*x^15 + 175020065793*x^14 - 261230425254*x^13 - 1695101848663*x^12 + 460247647320*x^11 + 9167918905077*x^10 + 3708053868470*x^9 - 27072768198312*x^8 - 21922844820627*x^7 + 41739348314777*x^6 + 46953795423150*x^5 - 28185634575723*x^4 - 45243443721079*x^3 + 1114518996171*x^2 + 16368292431012*x + 4826968628849)
 
gp: K = bnfinit(x^27 - 327*x^25 - 218*x^24 + 43164*x^23 + 61476*x^22 - 3018210*x^21 - 6507954*x^20 + 121903638*x^19 + 347122817*x^18 - 2891823192*x^17 - 10299084090*x^16 + 38866417215*x^15 + 175020065793*x^14 - 261230425254*x^13 - 1695101848663*x^12 + 460247647320*x^11 + 9167918905077*x^10 + 3708053868470*x^9 - 27072768198312*x^8 - 21922844820627*x^7 + 41739348314777*x^6 + 46953795423150*x^5 - 28185634575723*x^4 - 45243443721079*x^3 + 1114518996171*x^2 + 16368292431012*x + 4826968628849, 1)
 

Normalized defining polynomial

\( x^{27} - 327 x^{25} - 218 x^{24} + 43164 x^{23} + 61476 x^{22} - 3018210 x^{21} - 6507954 x^{20} + 121903638 x^{19} + 347122817 x^{18} - 2891823192 x^{17} - 10299084090 x^{16} + 38866417215 x^{15} + 175020065793 x^{14} - 261230425254 x^{13} - 1695101848663 x^{12} + 460247647320 x^{11} + 9167918905077 x^{10} + 3708053868470 x^{9} - 27072768198312 x^{8} - 21922844820627 x^{7} + 41739348314777 x^{6} + 46953795423150 x^{5} - 28185634575723 x^{4} - 45243443721079 x^{3} + 1114518996171 x^{2} + 16368292431012 x + 4826968628849 \)

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:  \(14107631800772057462614083047171090763038947897293819627757634951745561=3^{36}\cdot 109^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $396.39$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(981=3^{2}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{981}(1,·)$, $\chi_{981}(343,·)$, $\chi_{981}(772,·)$, $\chi_{981}(517,·)$, $\chi_{981}(838,·)$, $\chi_{981}(136,·)$, $\chi_{981}(457,·)$, $\chi_{981}(907,·)$, $\chi_{981}(910,·)$, $\chi_{981}(661,·)$, $\chi_{981}(727,·)$, $\chi_{981}(25,·)$, $\chi_{981}(154,·)$, $\chi_{981}(751,·)$, $\chi_{981}(157,·)$, $\chi_{981}(187,·)$, $\chi_{981}(172,·)$, $\chi_{981}(877,·)$, $\chi_{981}(541,·)$, $\chi_{981}(112,·)$, $\chi_{981}(625,·)$, $\chi_{981}(349,·)$, $\chi_{981}(376,·)$, $\chi_{981}(634,·)$, $\chi_{981}(571,·)$, $\chi_{981}(124,·)$, $\chi_{981}(829,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{173} a^{23} + \frac{47}{173} a^{22} - \frac{49}{173} a^{21} - \frac{64}{173} a^{20} + \frac{69}{173} a^{19} - \frac{44}{173} a^{18} + \frac{36}{173} a^{17} + \frac{28}{173} a^{16} + \frac{72}{173} a^{15} + \frac{77}{173} a^{14} + \frac{65}{173} a^{13} + \frac{67}{173} a^{12} + \frac{12}{173} a^{11} + \frac{66}{173} a^{10} - \frac{42}{173} a^{9} + \frac{9}{173} a^{8} + \frac{76}{173} a^{7} - \frac{72}{173} a^{6} + \frac{12}{173} a^{5} - \frac{59}{173} a^{4} - \frac{14}{173} a^{3} - \frac{80}{173} a^{2} - \frac{46}{173} a + \frac{16}{173}$, $\frac{1}{43423} a^{24} - \frac{88}{43423} a^{23} + \frac{6581}{43423} a^{22} - \frac{17669}{43423} a^{21} + \frac{3173}{43423} a^{20} + \frac{18667}{43423} a^{19} + \frac{2343}{43423} a^{18} - \frac{14174}{43423} a^{17} + \frac{1655}{43423} a^{16} - \frac{18466}{43423} a^{15} - \frac{8946}{43423} a^{14} + \frac{13090}{43423} a^{13} + \frac{3423}{43423} a^{12} - \frac{10723}{43423} a^{11} + \frac{12846}{43423} a^{10} - \frac{30}{43423} a^{9} + \frac{3532}{43423} a^{8} + \frac{48}{43423} a^{7} + \frac{14922}{43423} a^{6} - \frac{8253}{43423} a^{5} + \frac{13833}{43423} a^{4} - \frac{10992}{43423} a^{3} + \frac{19750}{43423} a^{2} + \frac{20931}{43423} a - \frac{3025}{43423}$, $\frac{1}{43423} a^{25} + \frac{92}{43423} a^{23} + \frac{12522}{43423} a^{22} - \frac{6543}{43423} a^{21} + \frac{456}{43423} a^{20} - \frac{5286}{43423} a^{19} + \frac{6521}{43423} a^{18} + \frac{15367}{43423} a^{17} - \frac{11378}{43423} a^{16} + \frac{19634}{43423} a^{15} + \frac{17245}{43423} a^{14} + \frac{21074}{43423} a^{13} - \frac{16221}{43423} a^{12} - \frac{3835}{43423} a^{11} - \frac{2596}{43423} a^{10} - \frac{8395}{43423} a^{9} + \frac{18198}{43423} a^{8} - \frac{91}{251} a^{7} - \frac{1321}{43423} a^{6} - \frac{2603}{43423} a^{5} + \frac{3269}{43423} a^{4} - \frac{9810}{43423} a^{3} + \frac{8457}{43423} a^{2} + \frac{830}{43423} a + \frac{14418}{43423}$, $\frac{1}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{26} + \frac{17956813985054103337871644122792818869606431817739191385451628979293008396774432559220936271149271194656977623035733194658987574758502038064683569028383770318901}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{25} + \frac{15771026874289350995157855435034611155956976689582449872870819507307495377564783479847864067074187051162966619640522311795930973126185260478285010415818345293528}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{24} - \frac{4064699662828642890766087642964813942449169474429380924936865371682003065083531492100078347354281114015397885024498883715884181876917713541447272965049154244746213}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{23} + \frac{833401712646865123360320150052209650889122109950323505039614607508092116385036983920478721154673748296404013045819497085475498043875271257362608550862940976165265904}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{22} - \frac{102147825469165691966215230178491529363048533673018788250528527405389582806072650512279476309014615963566816523831547734194742850577165398676714275709126504230801593}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{21} + \frac{895846019101916305650128239248511329282503460214792849053386848168608093065598235251265336804123399347880938294539016770068772978869971454514640766477416954985670652}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{20} + \frac{733984877889206006087784511591390056559023313696519599378220212584325012845363530151137522463945811197401823331751346191819648385289368978967893738914481944185840974}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{19} + \frac{820711888418920368358300418473782641208946475676542293194151409409472115807153079153150917763763985231185847832888167001899341306706623534995323175928022244811185831}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{18} - \frac{921470799414268868275619426778124959688699812127972144163067228829525754033227648320080421463751617981155753199192413798276884537746540940538171592735112608623206636}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{17} + \frac{826567734939402125713665040644210325237498387547580974352837633960983951528739922689729497699298925493426237579832318302785991063175831080283822501285136896961552968}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{16} + \frac{80632542673563761021622748835175394033049283948370424762237180039899427348926932104256668603935095560516842075230718003471474323586235734900255983859106874597041752}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{15} - \frac{487924994594084105133189272069291377363674483548853669024885115255735971453165541674684292107707515867905605617536131482937978888195782431289518639306471567614038330}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{14} + \frac{66467332019018900799403311978100228088084054476881033372876140784121028487602288686046519257188919928978431309465195848901263140443114387533403872110044686188817164}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{13} + \frac{266408190346476648489284265904725136168568000684919437140208594380872029197200915014261666287480764915307815227710513310402864120293042212344530558775711323983369430}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{12} + \frac{799501888277754943357376946419291296459229147215544580040298826420155547786709584216119882847937716358417395079494316480559521041842833918168504715478547363968669242}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{11} + \frac{471215957067863328436401317058632059030184396822083948674206788969053349015583299765341038423343547102794278325658202273286566389208197732700951640687446479569663318}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{10} - \frac{955038542844910106166865606444151231961724804315265981119941191784919383926226341518601850188598650481708605432467700682565371732814990585631018176571919630064884427}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{9} + \frac{122278133102209991805461780351390181412169761410680674080946614787239971190851194578661674545469019467361851922695438354635580278102095636856990800911079211424255368}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{8} + \frac{673208143857000253861655700422345739744857643065246917145060102490941584868667976057331616563594984241956065235282463439920871718724739192267163677436299901799858245}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{7} + \frac{761011610207695157006281288832040508380893590759499416830813021780219545065452030054708938722399744032071205781726064168291055367890971000344062485154267175367412423}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{6} - \frac{366692271755553251728168510754537558441226865922987721201292190672920333507301695978890308170592082254466569786922133561287517456303091805395013604463933602435827125}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{5} + \frac{858051269153364361286664891193104069290633297983782163321643464262829487886710215016225400829483369304036617207489693910121776432236693983280754708379364104925051381}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{4} - \frac{36480861647003672295879557895716406270236583298531917901397080155502397220385731778791314154154888505436720885908895727570077278716631459073216194132593751569179887}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{3} - \frac{1023458664330211088007908985690178199963915615600264928673483809159171069419218044162104733564389696859974333074867522552569997304865395925125763617204749815950624143}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a^{2} - \frac{812343762820013420449858330825849370931090993897520382661645001424541435988901657564164214823680685301676551895397197119327042686318333959225132486067875351136758406}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657} a + \frac{482057927059778519525080172338084042807870868228633255611514708060093165770941964268359515058268854210367146739007322957141661486038167364280831195961833346732478073}{2210518888053736260363265017954737994970810624378502496432763531623264144148931441567930248137804715281909687487239105869794403210827892849639336699640791381461931657}$

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:  \( 75678932334826550000000000 \) (assuming GRH)
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

3.3.11881.1, 9.9.19925626416901921.1

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 $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $27$ $27$ $27$

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
109Data not computed