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Properties

Label 28.0.26011077544...4773.1
Degree $28$
Signature $[0, 14]$
Discriminant $13^{21}\cdot 29^{26}$
Root discriminant $156.10$
Ramified primes $13, 29$
Class number $28831552$ (GRH)
Class group $[2, 2, 2, 2, 1801972]$ (GRH)
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15073109483, 12497398748, -16100534512, -220732122, 11576546984, -11788049168, 12498470472, -3142097964, 36632420, -1574285300, 656858657, 661911232, -244119948, 112739846, -68765767, -44127190, 12889086, 6735427, 2289507, -28565, -5884, -36341, -7876, -1155, 310, -64, 51, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 51*x^26 - 64*x^25 + 310*x^24 - 1155*x^23 - 7876*x^22 - 36341*x^21 - 5884*x^20 - 28565*x^19 + 2289507*x^18 + 6735427*x^17 + 12889086*x^16 - 44127190*x^15 - 68765767*x^14 + 112739846*x^13 - 244119948*x^12 + 661911232*x^11 + 656858657*x^10 - 1574285300*x^9 + 36632420*x^8 - 3142097964*x^7 + 12498470472*x^6 - 11788049168*x^5 + 11576546984*x^4 - 220732122*x^3 - 16100534512*x^2 + 12497398748*x + 15073109483)
 
gp: K = bnfinit(x^28 - x^27 + 51*x^26 - 64*x^25 + 310*x^24 - 1155*x^23 - 7876*x^22 - 36341*x^21 - 5884*x^20 - 28565*x^19 + 2289507*x^18 + 6735427*x^17 + 12889086*x^16 - 44127190*x^15 - 68765767*x^14 + 112739846*x^13 - 244119948*x^12 + 661911232*x^11 + 656858657*x^10 - 1574285300*x^9 + 36632420*x^8 - 3142097964*x^7 + 12498470472*x^6 - 11788049168*x^5 + 11576546984*x^4 - 220732122*x^3 - 16100534512*x^2 + 12497398748*x + 15073109483, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 51 x^{26} - 64 x^{25} + 310 x^{24} - 1155 x^{23} - 7876 x^{22} - 36341 x^{21} - 5884 x^{20} - 28565 x^{19} + 2289507 x^{18} + 6735427 x^{17} + 12889086 x^{16} - 44127190 x^{15} - 68765767 x^{14} + 112739846 x^{13} - 244119948 x^{12} + 661911232 x^{11} + 656858657 x^{10} - 1574285300 x^{9} + 36632420 x^{8} - 3142097964 x^{7} + 12498470472 x^{6} - 11788049168 x^{5} + 11576546984 x^{4} - 220732122 x^{3} - 16100534512 x^{2} + 12497398748 x + 15073109483 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26011077544761738747855517343331362930821153062567249262964773=13^{21}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.10$
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:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(377=13\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{377}(1,·)$, $\chi_{377}(194,·)$, $\chi_{377}(5,·)$, $\chi_{377}(326,·)$, $\chi_{377}(265,·)$, $\chi_{377}(138,·)$, $\chi_{377}(339,·)$, $\chi_{377}(86,·)$, $\chi_{377}(151,·)$, $\chi_{377}(216,·)$, $\chi_{377}(25,·)$, $\chi_{377}(285,·)$, $\chi_{377}(96,·)$, $\chi_{377}(34,·)$, $\chi_{377}(294,·)$, $\chi_{377}(103,·)$, $\chi_{377}(168,·)$, $\chi_{377}(233,·)$, $\chi_{377}(170,·)$, $\chi_{377}(109,·)$, $\chi_{377}(125,·)$, $\chi_{377}(53,·)$, $\chi_{377}(248,·)$, $\chi_{377}(57,·)$, $\chi_{377}(122,·)$, $\chi_{377}(187,·)$, $\chi_{377}(313,·)$, $\chi_{377}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} + \frac{7}{17} a^{13} + \frac{4}{17} a^{10} - \frac{6}{17} a^{9} - \frac{1}{17} a^{6} - \frac{7}{17} a^{5} - \frac{4}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{17} a^{15} + \frac{2}{17} a^{13} + \frac{4}{17} a^{11} + \frac{8}{17} a^{9} - \frac{1}{17} a^{7} - \frac{2}{17} a^{5} - \frac{4}{17} a^{3} - \frac{8}{17} a$, $\frac{1}{17} a^{16} + \frac{3}{17} a^{13} + \frac{4}{17} a^{12} - \frac{5}{17} a^{9} - \frac{1}{17} a^{8} - \frac{3}{17} a^{5} - \frac{4}{17} a^{4} + \frac{5}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{289} a^{22} + \frac{7}{289} a^{21} - \frac{3}{289} a^{20} - \frac{2}{289} a^{19} + \frac{6}{289} a^{18} - \frac{5}{289} a^{17} + \frac{2}{289} a^{16} + \frac{7}{289} a^{15} + \frac{8}{289} a^{14} + \frac{127}{289} a^{13} - \frac{94}{289} a^{12} - \frac{108}{289} a^{11} - \frac{121}{289} a^{10} - \frac{138}{289} a^{9} - \frac{138}{289} a^{8} - \frac{75}{289} a^{7} + \frac{93}{289} a^{6} + \frac{70}{289} a^{5} + \frac{131}{289} a^{4} + \frac{42}{289} a^{3} - \frac{72}{289} a^{2} + \frac{58}{289} a - \frac{7}{17}$, $\frac{1}{289} a^{23} - \frac{1}{289} a^{21} + \frac{2}{289} a^{20} + \frac{3}{289} a^{19} + \frac{4}{289} a^{18} + \frac{3}{289} a^{17} - \frac{7}{289} a^{16} - \frac{7}{289} a^{15} + \frac{3}{289} a^{14} + \frac{54}{289} a^{13} - \frac{28}{289} a^{12} - \frac{96}{289} a^{11} - \frac{141}{289} a^{10} + \frac{63}{289} a^{9} + \frac{24}{289} a^{8} + \frac{6}{289} a^{7} + \frac{65}{289} a^{6} - \frac{2}{289} a^{5} + \frac{9}{289} a^{4} + \frac{93}{289} a^{3} - \frac{84}{289} a^{2} - \frac{15}{289} a - \frac{2}{17}$, $\frac{1}{289} a^{24} - \frac{8}{289} a^{21} + \frac{2}{289} a^{19} - \frac{8}{289} a^{18} + \frac{5}{289} a^{17} - \frac{5}{289} a^{16} - \frac{7}{289} a^{15} - \frac{6}{289} a^{14} - \frac{122}{289} a^{13} + \frac{99}{289} a^{12} - \frac{28}{289} a^{11} - \frac{41}{289} a^{10} - \frac{131}{289} a^{9} - \frac{132}{289} a^{8} + \frac{7}{289} a^{7} - \frac{130}{289} a^{6} + \frac{28}{289} a^{5} - \frac{65}{289} a^{4} + \frac{26}{289} a^{3} - \frac{87}{289} a^{2} + \frac{24}{289} a - \frac{7}{17}$, $\frac{1}{289} a^{25} + \frac{5}{289} a^{21} - \frac{5}{289} a^{20} - \frac{7}{289} a^{19} + \frac{2}{289} a^{18} + \frac{6}{289} a^{17} - \frac{8}{289} a^{16} - \frac{1}{289} a^{15} - \frac{7}{289} a^{14} - \frac{126}{289} a^{13} + \frac{19}{289} a^{12} + \frac{47}{289} a^{11} - \frac{28}{289} a^{10} - \frac{131}{289} a^{9} + \frac{76}{289} a^{8} - \frac{101}{289} a^{7} + \frac{143}{289} a^{6} + \frac{53}{289} a^{5} - \frac{31}{289} a^{4} - \frac{142}{289} a^{3} - \frac{127}{289} a^{2} + \frac{56}{289} a - \frac{5}{17}$, $\frac{1}{6142338195109} a^{26} - \frac{9842859557}{6142338195109} a^{25} + \frac{6612566363}{6142338195109} a^{24} - \frac{7844893735}{6142338195109} a^{23} + \frac{7737564557}{6142338195109} a^{22} - \frac{47092104612}{6142338195109} a^{21} - \frac{108180978182}{6142338195109} a^{20} + \frac{95756267650}{6142338195109} a^{19} - \frac{44938370620}{6142338195109} a^{18} + \frac{67103296755}{6142338195109} a^{17} + \frac{159697383612}{6142338195109} a^{16} + \frac{11399526139}{6142338195109} a^{15} + \frac{2811487394}{6142338195109} a^{14} - \frac{1295161459897}{6142338195109} a^{13} + \frac{2739678105600}{6142338195109} a^{12} - \frac{486085291523}{6142338195109} a^{11} + \frac{782580589127}{6142338195109} a^{10} - \frac{699828991935}{6142338195109} a^{9} + \frac{551516329649}{6142338195109} a^{8} - \frac{676216973897}{6142338195109} a^{7} - \frac{1051149174438}{6142338195109} a^{6} + \frac{56816212090}{6142338195109} a^{5} - \frac{330444372992}{6142338195109} a^{4} - \frac{2993891440093}{6142338195109} a^{3} - \frac{230699141933}{6142338195109} a^{2} + \frac{777814595506}{6142338195109} a + \frac{49156642746}{361314011477}$, $\frac{1}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{27} - \frac{31494528376106345178576581729388345178502662473709050588540008545710886287867855078528102745852978826623651430759380010452336}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{26} - \frac{649390787099952522160452167640043482889299422378960631569117287673324557011678541211745238180193417748380471317357084289300846116810025}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{25} - \frac{2252996349840042680914372692406917447765537077384788034656255687017801050885676548781459648921626070459332437432262408881464370645905816}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{24} - \frac{1243319070745134580925009973709320298191579172858994705743846221203265573274980921724106917404409075099210875479340013029580823869660579}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{23} + \frac{254538849571083461329526904292576956615242175354238614087741774759134866567539569016326826487810586435831290943099125648572650304646692}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{22} + \frac{34766294462533824924412085018543436088097056374071858299240109014990232322201676657528269315744722354106560980535416460402711397873226478}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{21} + \frac{889986812582423330937757270100866725740458526031259555977684358183811684465779699000395220057806062055215781621083427119306917272871320}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{20} - \frac{2465457306661299416816787270356844081693883326944034479713349709611573581720905189967573920772712585056985934642163612153458809720130731}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{19} - \frac{9549578552640742410397342135557709605613478749672481297212872581243672856882327090914665140785458957378466961514421475768298019036462447}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{18} + \frac{32359233298520395903839133668868458792155396252569730910594150516683617530068325407111860169491765031324488185003132047795591278082064428}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{17} - \frac{16711440392328973104088819701251485670721930535175180778718750129710323414701041938098666123144302244601706280138976404607508149451146698}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{16} - \frac{17723606217530604050488748495387447228864890732039250914628433223293146661877096287205023167016148085700518323672192069470473080549920999}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{15} - \frac{30652759064274579219838076290194436682001319734944874410172322454045756658725194506239850770460748959356009708392018573944449044091119114}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{14} - \frac{522122509699618892358340740039353151901227572460583968852480967137835343235919186361346722646752839220354504878765316145729954725657701585}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{13} + \frac{624094329735527447957725771703643820983658963419437458691321863691912184064399540807249555713403124534216173759076119516005234892780171675}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{12} - \frac{490617805560026283860215177898443566023205066948001472922535554883654258587295956861283255615966334472643848992031298126931503169643908376}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{11} + \frac{26516655548508369968675825025885191675796201141040713564539265450187050903857815318013462467354525296076294166540701099062413393643063004}{82505941651818760816973211618526570995945908417827837479188636363273849527555797996595446115576804511191749225003408606545593609539955543} a^{10} + \frac{607731639223031281084789761640603242199063769987009345247581109412890874216854695155627269613459903596119614514833266582761485499475747555}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{9} + \frac{85033351971588706335923308786382152430528274291200224797001710641361172802860700747888973009286866756493058624478485010847413731889336014}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{8} + \frac{523527661021994334253092935600457718686395650262479048595567554057577982226933373370728476827409639351389883188245049115076661163256129183}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{7} - \frac{91965932880069497472348402667421320084563937966192776895601123891075314541629413119307344585522870133102653304207129863473593851592150987}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{6} + \frac{497072018830865111756685645607648079739912248397756622422366319461918666301701134528544147813520334989132784765247735408515323554391770550}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{5} + \frac{473612741508679655227843994069878652755775633600552667569238452509554444246858112696605438163682649797517443652370370279921559052954971108}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{4} + \frac{25125464164062465062040584288007330915631868244202526473273642566300412942247631248178961399787607566159657980447345133464767660770572582}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{3} - \frac{547760476116581323087762636262167752339085227417473441568138368953553887887216275215074880423073943853943187683212803568716148681172782163}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a^{2} - \frac{289655843158939382315071020770625108876153344415999697383288322844347795478569776193000730702675216478901199670661973077785596280056735687}{1402601008080918933888544597514951706931080443103073237146206818175655441968448565942122583964805676690259736825057946311275091362179244231} a - \frac{26413261715743965376611603759710340857065285065646171046258122729495483039445653187302329534515230577020258083492056060965893866981922057}{82505941651818760816973211618526570995945908417827837479188636363273849527555797996595446115576804511191749225003408606545593609539955543}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1801972}$, which has order $28831552$ (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:  $13$
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:  \( 1523489837639.8035 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{28}$ (as 28T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.0.1847677.1, 7.7.594823321.1, 14.14.22201352938819688612162197.1

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 $28$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ $28$ $28$ $28$ R ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{7}$

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
13Data not computed
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$