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Properties

Label 28.0.12966284311...6073.1
Degree $28$
Signature $[0, 14]$
Discriminant $3^{14}\cdot 113^{27}$
Root discriminant $165.32$
Ramified primes $3, 113$
Class number Not computed
Class group Not computed
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![805522987, -3736246457, 4490511812, -4978220620, 14821845636, -3903580904, 14632734546, -3024360615, 6832621830, -2466399578, 1606739712, -906030787, 432226710, -175996687, 105198385, -32869029, 17538561, -5335265, 2001244, -608973, 201884, -49859, 18024, -2583, 1338, -64, 59, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 59*x^26 - 64*x^25 + 1338*x^24 - 2583*x^23 + 18024*x^22 - 49859*x^21 + 201884*x^20 - 608973*x^19 + 2001244*x^18 - 5335265*x^17 + 17538561*x^16 - 32869029*x^15 + 105198385*x^14 - 175996687*x^13 + 432226710*x^12 - 906030787*x^11 + 1606739712*x^10 - 2466399578*x^9 + 6832621830*x^8 - 3024360615*x^7 + 14632734546*x^6 - 3903580904*x^5 + 14821845636*x^4 - 4978220620*x^3 + 4490511812*x^2 - 3736246457*x + 805522987)
 
gp: K = bnfinit(x^28 - x^27 + 59*x^26 - 64*x^25 + 1338*x^24 - 2583*x^23 + 18024*x^22 - 49859*x^21 + 201884*x^20 - 608973*x^19 + 2001244*x^18 - 5335265*x^17 + 17538561*x^16 - 32869029*x^15 + 105198385*x^14 - 175996687*x^13 + 432226710*x^12 - 906030787*x^11 + 1606739712*x^10 - 2466399578*x^9 + 6832621830*x^8 - 3024360615*x^7 + 14632734546*x^6 - 3903580904*x^5 + 14821845636*x^4 - 4978220620*x^3 + 4490511812*x^2 - 3736246457*x + 805522987, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 59 x^{26} - 64 x^{25} + 1338 x^{24} - 2583 x^{23} + 18024 x^{22} - 49859 x^{21} + 201884 x^{20} - 608973 x^{19} + 2001244 x^{18} - 5335265 x^{17} + 17538561 x^{16} - 32869029 x^{15} + 105198385 x^{14} - 175996687 x^{13} + 432226710 x^{12} - 906030787 x^{11} + 1606739712 x^{10} - 2466399578 x^{9} + 6832621830 x^{8} - 3024360615 x^{7} + 14632734546 x^{6} - 3903580904 x^{5} + 14821845636 x^{4} - 4978220620 x^{3} + 4490511812 x^{2} - 3736246457 x + 805522987 \)

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:  \(129662843111551841878714037360432823103337384758484270764976073=3^{14}\cdot 113^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $165.32$
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:  $3, 113$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(339=3\cdot 113\)
Dirichlet character group:    $\lbrace$$\chi_{339}(128,·)$, $\chi_{339}(1,·)$, $\chi_{339}(2,·)$, $\chi_{339}(4,·)$, $\chi_{339}(7,·)$, $\chi_{339}(8,·)$, $\chi_{339}(224,·)$, $\chi_{339}(194,·)$, $\chi_{339}(256,·)$, $\chi_{339}(109,·)$, $\chi_{339}(16,·)$, $\chi_{339}(212,·)$, $\chi_{339}(14,·)$, $\chi_{339}(196,·)$, $\chi_{339}(218,·)$, $\chi_{339}(28,·)$, $\chi_{339}(32,·)$, $\chi_{339}(97,·)$, $\chi_{339}(98,·)$, $\chi_{339}(64,·)$, $\chi_{339}(170,·)$, $\chi_{339}(173,·)$, $\chi_{339}(112,·)$, $\chi_{339}(49,·)$, $\chi_{339}(53,·)$, $\chi_{339}(56,·)$, $\chi_{339}(106,·)$, $\chi_{339}(85,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{131} a^{24} - \frac{28}{131} a^{23} + \frac{18}{131} a^{22} - \frac{9}{131} a^{21} + \frac{40}{131} a^{20} + \frac{55}{131} a^{19} - \frac{29}{131} a^{18} - \frac{6}{131} a^{17} - \frac{55}{131} a^{16} - \frac{21}{131} a^{15} + \frac{1}{131} a^{14} + \frac{29}{131} a^{13} - \frac{65}{131} a^{12} - \frac{55}{131} a^{11} + \frac{4}{131} a^{10} - \frac{5}{131} a^{9} - \frac{42}{131} a^{8} + \frac{54}{131} a^{7} + \frac{45}{131} a^{6} + \frac{6}{131} a^{5} + \frac{65}{131} a^{4} - \frac{57}{131} a^{3} + \frac{53}{131} a^{2} + \frac{34}{131} a - \frac{39}{131}$, $\frac{1}{131} a^{25} + \frac{20}{131} a^{23} - \frac{29}{131} a^{22} + \frac{50}{131} a^{21} - \frac{4}{131} a^{20} - \frac{61}{131} a^{19} - \frac{32}{131} a^{18} + \frac{39}{131} a^{17} + \frac{11}{131} a^{16} - \frac{63}{131} a^{15} + \frac{57}{131} a^{14} - \frac{39}{131} a^{13} - \frac{41}{131} a^{12} + \frac{36}{131} a^{11} - \frac{24}{131} a^{10} - \frac{51}{131} a^{9} + \frac{57}{131} a^{8} - \frac{15}{131} a^{7} - \frac{44}{131} a^{6} - \frac{29}{131} a^{5} + \frac{60}{131} a^{4} + \frac{29}{131} a^{3} - \frac{54}{131} a^{2} - \frac{4}{131} a - \frac{44}{131}$, $\frac{1}{131} a^{26} + \frac{7}{131} a^{23} - \frac{48}{131} a^{22} + \frac{45}{131} a^{21} + \frac{56}{131} a^{20} + \frac{47}{131} a^{19} - \frac{36}{131} a^{18} - \frac{11}{131} a^{16} - \frac{47}{131} a^{15} - \frac{59}{131} a^{14} + \frac{34}{131} a^{13} + \frac{26}{131} a^{12} + \frac{28}{131} a^{11} + \frac{26}{131} a^{9} + \frac{39}{131} a^{8} + \frac{55}{131} a^{7} - \frac{12}{131} a^{6} - \frac{60}{131} a^{5} + \frac{39}{131} a^{4} + \frac{38}{131} a^{3} - \frac{16}{131} a^{2} + \frac{62}{131} a - \frac{6}{131}$, $\frac{1}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{27} - \frac{467774040103953416204910276671336123479556577750063387281414260433403449897766635376254843623307722105023289322712835976983067472362169497054461928}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{26} + \frac{479094093414147552916988460200812481779890014253224328845676170387056095064844404660001039806877171066160865525673821226962589529888273560415769092}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{25} + \frac{198161064590513401483534883835616734858608422932395997037097435130978870153631337900536774240496303842993157650545101708963629106789620705091808237}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{24} + \frac{17519092224623812266350582295108121166794595545888032263618485277229458808035368137404032280668852539987477779601319473309869085588035072647917763884}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{23} + \frac{42057691300679490774085578494119299438126986117029321204712625944928529374380268865803043933532737812452954103455360875264542071912756914338269306632}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{22} - \frac{29544775123003404342215046278540148931823783860585171903941126355114749696667727861279836923761110473253862435449641986960297558375971451778120129602}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{21} - \frac{57049859874310059964307021686655747340369599187849291578087790714648288633424389027990669443028338329887110028405645935771264680171072786856286930162}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{20} - \frac{10318023319600464949640829108783425988280550083117458966797726958606868572939890210322019572340181230400282799703673484645233622253747824380291440120}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{19} - \frac{6168426355827618789980379004786258674666445274227123337208917893273512730950746103207230360184553313259400978957693901298537037193512940445376649005}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{18} + \frac{8769641859086587677578472426027960501376615014327633601960311578218646080404714447425433345454916429470140493451469247199959363445912039438671995}{533458380575959045017788045873013555700119645668658773337267624469416903675990981525899712212077274940006774929520044868250845795783397559371215059} a^{17} + \frac{13179070854596307142806350351084868091387999892649232306903023441172795113922687146360155138152313023989851802965897528382316296861183083170120555813}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{16} - \frac{14249159537157986954060369582338507459601891920088180081470739311070261362781763190497104112011638516611496759740303086063859466253406489097742034248}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{15} + \frac{51176159739474738941863181910231280460700503458792490270089792521891896109980473020312428973192771731516689341644261872173519727243439940174652561531}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{14} - \frac{66431726369687450361676209809883094921207718929310957053779095708297005573873739326890499748407019427874676388428880628145583226835482857709257560041}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{13} + \frac{13745615194486537833008630111229027402253253948786148277702812003600016983404082360167077309679998471301821694437121255212808963314163295558097930365}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{12} + \frac{54428945425732262174111138789622399894955777504406660526081256727368374705547617096089281288668341160209884529693100205511287246686242628563505834886}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{11} - \frac{5267797492573234040128873321920188892242208458152885006899253673556880420847232314560776663280633747724383042869025126906771933233314779220373642271}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{10} + \frac{2917576987422604539053920554670159094762780373719391071273992650149411600197309259370168025844551683307588431540545457233737431990771272016456977704}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{9} + \frac{46900186740412714460670837250147659938740669713980650505281306667016953427511145581305671716231677164498636857293127091723991348320477594302961749047}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{8} - \frac{65676455445125283520685926072083139718179827623031180259190574910839231275926430656220451192877226424322805398203056999469028004241198721544192727343}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{7} + \frac{55226011117988666686725917615225311170543620202009034339272885953067848187300066889265949721451336791142051867035469741707997728896903766098142161640}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{6} - \frac{75365937298690673753540402877384342037534106591269218473116286204584658900962299969130460663326673083066345681418494356011727141398052988140304900904}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{5} + \frac{39126027650032968667565021059115172284580836256736483334909170786738885534248193785634601846348553607746458236835058946618812276240657223862555273865}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{4} + \frac{62074178651444312938684836841870241958501915564438169872703359510177899332651966797201536379553419768760824878948168871945067900903169901234534241869}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{3} - \frac{3078583959137022759490712544858652277065173365162534114491321128461316867151393660565213739536193191565631387067018616531365229908335719695696299806}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a^{2} + \frac{33265525583483521197684729057269581055946587748358004283281092347445669563949171445887443668600762579407389884596230583635337330294925442392640991044}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597} a - \frac{23211463905850685356329808954294963053053884335967277605071051073904892438940987422176548281817592505793432982029751780819507520301587897309247853831}{204314559760592314241812821569364191833145824291096310188173500171786674107904545924419589777225596302022594798006177184540073939785041265239175367597}$

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:  $13$
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:  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

$C_{28}$ (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

\(\Q(\sqrt{113}) \), 4.0.12986073.1, 7.7.2081951752609.1, 14.14.489801110321660601428677553.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R $28$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ $28$ $28$ $28$ $28$ ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ $28$

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
113Data not computed