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Properties

Label 28.0.89245865645...1613.1
Degree $28$
Signature $[0, 14]$
Discriminant $197^{27}$
Root discriminant $163.13$
Ramified prime $197$
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![493619587, 529082650, -474592366, -977493727, -1271425197, -93877796, 2357319562, 383663419, -1148132126, 111504745, 407111869, -34599131, -35698128, 26658338, 905534, -5830816, 1222711, 913848, -127473, 7993, 45738, 5613, -1294, 544, 110, -20, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 4*x^26 - 20*x^25 + 110*x^24 + 544*x^23 - 1294*x^22 + 5613*x^21 + 45738*x^20 + 7993*x^19 - 127473*x^18 + 913848*x^17 + 1222711*x^16 - 5830816*x^15 + 905534*x^14 + 26658338*x^13 - 35698128*x^12 - 34599131*x^11 + 407111869*x^10 + 111504745*x^9 - 1148132126*x^8 + 383663419*x^7 + 2357319562*x^6 - 93877796*x^5 - 1271425197*x^4 - 977493727*x^3 - 474592366*x^2 + 529082650*x + 493619587)
 
gp: K = bnfinit(x^28 - x^27 + 4*x^26 - 20*x^25 + 110*x^24 + 544*x^23 - 1294*x^22 + 5613*x^21 + 45738*x^20 + 7993*x^19 - 127473*x^18 + 913848*x^17 + 1222711*x^16 - 5830816*x^15 + 905534*x^14 + 26658338*x^13 - 35698128*x^12 - 34599131*x^11 + 407111869*x^10 + 111504745*x^9 - 1148132126*x^8 + 383663419*x^7 + 2357319562*x^6 - 93877796*x^5 - 1271425197*x^4 - 977493727*x^3 - 474592366*x^2 + 529082650*x + 493619587, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 4 x^{26} - 20 x^{25} + 110 x^{24} + 544 x^{23} - 1294 x^{22} + 5613 x^{21} + 45738 x^{20} + 7993 x^{19} - 127473 x^{18} + 913848 x^{17} + 1222711 x^{16} - 5830816 x^{15} + 905534 x^{14} + 26658338 x^{13} - 35698128 x^{12} - 34599131 x^{11} + 407111869 x^{10} + 111504745 x^{9} - 1148132126 x^{8} + 383663419 x^{7} + 2357319562 x^{6} - 93877796 x^{5} - 1271425197 x^{4} - 977493727 x^{3} - 474592366 x^{2} + 529082650 x + 493619587 \)

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:  \(89245865645132062133059821789304344522517366724620268695771613=197^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.13$
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:  $197$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(197\)
Dirichlet character group:    $\lbrace$$\chi_{197}(128,·)$, $\chi_{197}(1,·)$, $\chi_{197}(68,·)$, $\chi_{197}(69,·)$, $\chi_{197}(6,·)$, $\chi_{197}(129,·)$, $\chi_{197}(77,·)$, $\chi_{197}(14,·)$, $\chi_{197}(19,·)$, $\chi_{197}(20,·)$, $\chi_{197}(87,·)$, $\chi_{197}(196,·)$, $\chi_{197}(93,·)$, $\chi_{197}(161,·)$, $\chi_{197}(164,·)$, $\chi_{197}(33,·)$, $\chi_{197}(113,·)$, $\chi_{197}(104,·)$, $\chi_{197}(114,·)$, $\chi_{197}(110,·)$, $\chi_{197}(177,·)$, $\chi_{197}(178,·)$, $\chi_{197}(83,·)$, $\chi_{197}(183,·)$, $\chi_{197}(120,·)$, $\chi_{197}(84,·)$, $\chi_{197}(36,·)$, $\chi_{197}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{464323} a^{26} - \frac{41024}{464323} a^{25} + \frac{54910}{464323} a^{24} + \frac{222477}{464323} a^{23} + \frac{69853}{464323} a^{22} - \frac{119371}{464323} a^{21} + \frac{79500}{464323} a^{20} - \frac{167684}{464323} a^{19} - \frac{54301}{464323} a^{18} + \frac{60272}{464323} a^{17} - \frac{85412}{464323} a^{16} + \frac{46632}{464323} a^{15} - \frac{183925}{464323} a^{14} + \frac{188463}{464323} a^{13} + \frac{201646}{464323} a^{12} + \frac{208160}{464323} a^{11} - \frac{37055}{464323} a^{10} - \frac{33280}{464323} a^{9} + \frac{48223}{464323} a^{8} + \frac{138404}{464323} a^{7} + \frac{153793}{464323} a^{6} - \frac{67586}{464323} a^{5} + \frac{230516}{464323} a^{4} + \frac{16036}{464323} a^{3} - \frac{186530}{464323} a^{2} + \frac{221005}{464323} a + \frac{54491}{464323}$, $\frac{1}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{27} + \frac{951627303677498363438052921828663564904404643718396894607452935187754535970678154992605342746506971350717258524417735684280304103640847}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{26} - \frac{449292243575117199462275962302901365530615598470680871232367958760606662741561566986691854387340786575343104112098544678406152536879644679747}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{25} + \frac{170325517892143317813176311759256307759334651785225335874038194278435566250045080826136451490186747170915440441079523055488991815930143420556}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{24} - \frac{696018061191161386213402904789757847042291310290672017625551879357711685704873717806222600375618210370273503457370837262585120065585368335007}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{23} + \frac{317194393355589643744528339138732968770448540251636301852279343576783516516639401932971657893987065794232908599877167851943428504704404541735}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{22} + \frac{498428159326359586199593736921182814874819602975746646984595736237811049345463058791371176486253031973118709524079347749039246988122517457778}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{21} + \frac{178754374737720890974518882403378583628687917614003627884228984145402511977932185337512569361183590691541024891205944979662147674981375862319}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{20} + \frac{716762884634383781415407542102796289249974699092750767729314524150521993784599720515201297912793706290849043756448421472605346512176079945933}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{19} - \frac{262027443696253774971187182964020991647261282144517989997046217151740605979139976444663411143156510034599535884961329282802113624168317378135}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{18} + \frac{44852511713640777715436806697677283048665475902153175236663296850091318281956347417602848242872724276450052987113841407087074210731586777219}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{17} + \frac{742609128046303521562390951346142726443821891006094141645630314809257651391408802673514591497085055188009547232078438085764175183079906412707}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{16} - \frac{6585920732616519436656422068037019472312039470531433476824407205345404111773096844392180867132148307402530460650014540358459370128870521784}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{15} + \frac{257216952044246870565767496915728223107026123346262488931061346118723081094565104353651595398456339509592213355290841441132621470095828771396}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{14} + \frac{19882567381283704380433446493309326381737686238971602873748804354180346437867856089627600478620235773459724303297981087702792134140878182123}{80316918692037458386464622512071827264112522223962831129218643982058313121315818137374011391683529102730570202108099163522434692515526946107} a^{13} + \frac{724379505225263639707224651774019969373073268017665293042281322173757176060549824484935592198256376061586824080709351827138486429699385716985}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{12} - \frac{38769694264730161651081639634022829647838692148892732659661124044610228994976223383703992310977944462622738226057940906888986845398153294916}{80316918692037458386464622512071827264112522223962831129218643982058313121315818137374011391683529102730570202108099163522434692515526946107} a^{11} + \frac{187153723433313938559188885635642512729976638747766086252508855664516307945478696215124262168036345869791982633713062379135818355586484785959}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{10} - \frac{639080041325792241195431519833342164723440390937522212161430820440881905419500983260513641519492887392957161624736052433552844264346424088148}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{9} + \frac{471830418219131662460793666655086420980242080231214166046168849752549625415578718623255441977668014022558942531806790528202421649050436852137}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{8} - \frac{97579802100654143259501616390051276380411854088384184085919377587961739295134679102936601552303074857997391919565678849396005577818667948252}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{7} + \frac{713496773920592776606522314492491693612748115246743776285442118430607732658690884954154858329041423085154878644019187339828066589831478308940}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{6} - \frac{698821897725616625032991974009070384200996570604706511826497149830744868552728382650636772107893430216775586188976015109856079813109944265550}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{5} - \frac{475968129434492219871298544251261794585409177191948589941056218612950825466411868736650432135474530980804374590383021712403289376593842791248}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{4} - \frac{177806541868095431011748623184145996671270761676474148017622799915370274058715646018770032456330388528480227122650298967700280946378838954828}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{3} + \frac{648817938352736062388507157352513383284362456507212333424358315114446350587838795472154032669834684088858491790896831991409264701994616994072}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a^{2} - \frac{301542335577683960493012970566373487127614953127678755948499892896597971174176388158328231732399561961919277810474001211711054710452507429490}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033} a + \frac{762499040813286847683371249934269594040697950751166448727900503165129767898634788380435578362034049982110443320775711885544735746091587678978}{1526021455148711709342827827729364718018137922255293791455154235659107949305000544610106216441987052951880833840053884106926259157795011976033}$

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

\(\Q(\sqrt{197}) \), 4.0.7645373.1, 7.7.58451728309129.1, 14.14.673071094837873811440793512277.1

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 $28$ $28$ $28$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ $28$ $28$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$

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
197Data not computed