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Properties

Label 28.0.79097329061...8896.2
Degree $28$
Signature $[0, 14]$
Discriminant $2^{42}\cdot 7^{50}$
Root discriminant $91.34$
Ramified primes $2, 7$
Class number $953$ (GRH)
Class group $[953]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![35681291, 55406190, 45073133, 69822746, 114888018, 123489366, 111233549, 77662530, 55894391, 42138278, 26391246, 13724228, 8125894, 2977282, 2075488, 513058, 394401, 22134, 110236, -19068, 26257, -5556, 4186, -868, 378, -56, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 28*x^26 - 56*x^25 + 378*x^24 - 868*x^23 + 4186*x^22 - 5556*x^21 + 26257*x^20 - 19068*x^19 + 110236*x^18 + 22134*x^17 + 394401*x^16 + 513058*x^15 + 2075488*x^14 + 2977282*x^13 + 8125894*x^12 + 13724228*x^11 + 26391246*x^10 + 42138278*x^9 + 55894391*x^8 + 77662530*x^7 + 111233549*x^6 + 123489366*x^5 + 114888018*x^4 + 69822746*x^3 + 45073133*x^2 + 55406190*x + 35681291)
 
gp: K = bnfinit(x^28 + 28*x^26 - 56*x^25 + 378*x^24 - 868*x^23 + 4186*x^22 - 5556*x^21 + 26257*x^20 - 19068*x^19 + 110236*x^18 + 22134*x^17 + 394401*x^16 + 513058*x^15 + 2075488*x^14 + 2977282*x^13 + 8125894*x^12 + 13724228*x^11 + 26391246*x^10 + 42138278*x^9 + 55894391*x^8 + 77662530*x^7 + 111233549*x^6 + 123489366*x^5 + 114888018*x^4 + 69822746*x^3 + 45073133*x^2 + 55406190*x + 35681291, 1)
 

Normalized defining polynomial

\( x^{28} + 28 x^{26} - 56 x^{25} + 378 x^{24} - 868 x^{23} + 4186 x^{22} - 5556 x^{21} + 26257 x^{20} - 19068 x^{19} + 110236 x^{18} + 22134 x^{17} + 394401 x^{16} + 513058 x^{15} + 2075488 x^{14} + 2977282 x^{13} + 8125894 x^{12} + 13724228 x^{11} + 26391246 x^{10} + 42138278 x^{9} + 55894391 x^{8} + 77662530 x^{7} + 111233549 x^{6} + 123489366 x^{5} + 114888018 x^{4} + 69822746 x^{3} + 45073133 x^{2} + 55406190 x + 35681291 \)

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:  \(7909732906157957559057675256932747228387119659164368896=2^{42}\cdot 7^{50}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.34$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(392=2^{3}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(83,·)$, $\chi_{392}(195,·)$, $\chi_{392}(211,·)$, $\chi_{392}(321,·)$, $\chi_{392}(363,·)$, $\chi_{392}(265,·)$, $\chi_{392}(139,·)$, $\chi_{392}(337,·)$, $\chi_{392}(209,·)$, $\chi_{392}(323,·)$, $\chi_{392}(251,·)$, $\chi_{392}(153,·)$, $\chi_{392}(267,·)$, $\chi_{392}(281,·)$, $\chi_{392}(27,·)$, $\chi_{392}(225,·)$, $\chi_{392}(155,·)$, $\chi_{392}(97,·)$, $\chi_{392}(41,·)$, $\chi_{392}(43,·)$, $\chi_{392}(99,·)$, $\chi_{392}(113,·)$, $\chi_{392}(307,·)$, $\chi_{392}(169,·)$, $\chi_{392}(57,·)$, $\chi_{392}(379,·)$, $\chi_{392}(377,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{67} a^{21} + \frac{5}{67} a^{20} - \frac{8}{67} a^{19} + \frac{18}{67} a^{18} - \frac{3}{67} a^{17} - \frac{30}{67} a^{16} - \frac{24}{67} a^{15} - \frac{6}{67} a^{14} - \frac{18}{67} a^{13} + \frac{9}{67} a^{12} - \frac{21}{67} a^{11} - \frac{16}{67} a^{10} - \frac{12}{67} a^{9} - \frac{6}{67} a^{8} + \frac{15}{67} a^{7} - \frac{8}{67} a^{6} - \frac{22}{67} a^{5} + \frac{7}{67} a^{4} + \frac{20}{67} a^{3} + \frac{21}{67} a^{2} - \frac{24}{67} a + \frac{5}{67}$, $\frac{1}{67} a^{22} - \frac{33}{67} a^{20} - \frac{9}{67} a^{19} - \frac{26}{67} a^{18} - \frac{15}{67} a^{17} - \frac{8}{67} a^{16} - \frac{20}{67} a^{15} + \frac{12}{67} a^{14} + \frac{32}{67} a^{13} + \frac{1}{67} a^{12} + \frac{22}{67} a^{11} + \frac{1}{67} a^{10} - \frac{13}{67} a^{9} - \frac{22}{67} a^{8} - \frac{16}{67} a^{7} + \frac{18}{67} a^{6} - \frac{17}{67} a^{5} - \frac{15}{67} a^{4} - \frac{12}{67} a^{3} + \frac{5}{67} a^{2} - \frac{9}{67} a - \frac{25}{67}$, $\frac{1}{67} a^{23} + \frac{22}{67} a^{20} - \frac{22}{67} a^{19} - \frac{24}{67} a^{18} + \frac{27}{67} a^{17} - \frac{5}{67} a^{16} + \frac{24}{67} a^{15} - \frac{32}{67} a^{14} + \frac{10}{67} a^{13} - \frac{16}{67} a^{12} - \frac{22}{67} a^{11} - \frac{5}{67} a^{10} - \frac{16}{67} a^{9} - \frac{13}{67} a^{8} - \frac{23}{67} a^{7} - \frac{13}{67} a^{6} - \frac{4}{67} a^{5} + \frac{18}{67} a^{4} - \frac{5}{67} a^{3} + \frac{14}{67} a^{2} - \frac{13}{67} a + \frac{31}{67}$, $\frac{1}{67} a^{24} + \frac{2}{67} a^{20} + \frac{18}{67} a^{19} + \frac{33}{67} a^{18} - \frac{6}{67} a^{17} + \frac{14}{67} a^{16} + \frac{27}{67} a^{15} + \frac{8}{67} a^{14} - \frac{22}{67} a^{13} - \frac{19}{67} a^{12} - \frac{12}{67} a^{11} + \frac{1}{67} a^{10} - \frac{17}{67} a^{9} - \frac{25}{67} a^{8} - \frac{8}{67} a^{7} - \frac{29}{67} a^{6} + \frac{33}{67} a^{5} - \frac{25}{67} a^{4} - \frac{24}{67} a^{3} - \frac{6}{67} a^{2} + \frac{23}{67} a + \frac{24}{67}$, $\frac{1}{67} a^{25} + \frac{8}{67} a^{20} - \frac{18}{67} a^{19} + \frac{25}{67} a^{18} + \frac{20}{67} a^{17} + \frac{20}{67} a^{16} - \frac{11}{67} a^{15} - \frac{10}{67} a^{14} + \frac{17}{67} a^{13} - \frac{30}{67} a^{12} - \frac{24}{67} a^{11} + \frac{15}{67} a^{10} - \frac{1}{67} a^{9} + \frac{4}{67} a^{8} + \frac{8}{67} a^{7} - \frac{18}{67} a^{6} + \frac{19}{67} a^{5} + \frac{29}{67} a^{4} + \frac{21}{67} a^{3} - \frac{19}{67} a^{2} + \frac{5}{67} a - \frac{10}{67}$, $\frac{1}{1576931733195372221} a^{26} - \frac{4150341623905808}{1576931733195372221} a^{25} - \frac{2657632206156344}{1576931733195372221} a^{24} - \frac{9987786518363792}{1576931733195372221} a^{23} + \frac{7298027809862549}{1576931733195372221} a^{22} + \frac{8938447327198090}{1576931733195372221} a^{21} - \frac{727513707888720124}{1576931733195372221} a^{20} + \frac{1019341427621525}{50868765586947491} a^{19} - \frac{784960489963949978}{1576931733195372221} a^{18} + \frac{626185983900685521}{1576931733195372221} a^{17} + \frac{706437562704659449}{1576931733195372221} a^{16} - \frac{107967479057790500}{1576931733195372221} a^{15} + \frac{653185799082094023}{1576931733195372221} a^{14} - \frac{391805805173717165}{1576931733195372221} a^{13} - \frac{169048510140108072}{1576931733195372221} a^{12} - \frac{699977320758685137}{1576931733195372221} a^{11} + \frac{187314346129190730}{1576931733195372221} a^{10} + \frac{613624780344167676}{1576931733195372221} a^{9} + \frac{370382401215298051}{1576931733195372221} a^{8} - \frac{400509350844624041}{1576931733195372221} a^{7} + \frac{112752594295864907}{1576931733195372221} a^{6} - \frac{237221264050375398}{1576931733195372221} a^{5} + \frac{593561602952866629}{1576931733195372221} a^{4} - \frac{32458951282831486}{1576931733195372221} a^{3} - \frac{27753969742466388}{1576931733195372221} a^{2} + \frac{106705831083758353}{1576931733195372221} a + \frac{47058262620976743}{1576931733195372221}$, $\frac{1}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{27} - \frac{992450841725750345945855327169159256055382874015347892307873112682117411987}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{26} - \frac{50658354248148756775209558033816893226284934788675170962335928660468654172339328262146875084}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{25} + \frac{1439983800377099459357209469518213106517350589519079236200141751977135559568590377551715367}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{24} - \frac{85281653303325391222822910510260758860757179363910345881349787804223898330491520755671131593}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{23} + \frac{26972093597742992820998525936375155304734594439376650162039163890252964461660085542155685518}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{22} + \frac{7425396455483253785322667265603728808745672330810559366023377740171472424785513492604447537}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{21} - \frac{1445277810225183169192277564030550772358226912322816220969762386827325833008618092007937905920}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{20} - \frac{56069798533543573909934838132374768015360842643410418657887461726189638486166818688282506701}{225427495622095322424615683363122128590246253634581179380934934845567391226525873986570687709} a^{19} - \frac{4452254239817801559634191139504061970491046735927498474436615870817498866274913546562664261520}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{18} + \frac{3787252064894057077712891402600293753409660662797600896193036496287943512128259461696743009660}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{17} - \frac{4806392834307865073877585728629900307183719779664307598319296385787047074657562259506173860975}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{16} + \frac{6471785029952929799493861080652037247939062198928513397667966120895672324795258741117564179663}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{15} - \frac{5330079847416887000536508598149306074913327538807234168929186642135303695206181156306237064776}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{14} + \frac{997762657897104178067642049966739982447837213215998417149242500232359571750527652867236413813}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{13} - \frac{6536203948543123647119212363569100805387716789667152456016605276579641605195957915268491192975}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{12} + \frac{3454739465163675984481011140611618021993023601478151130945694870398674046928284057243964930813}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{11} + \frac{298926295828636849096244191800755378233085731934899988249320131044526078845883947528014250733}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{10} - \frac{1124270460604884510031239827713545440979260981916817890304577568673321135050828579039308465921}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{9} - \frac{17032282123718730116925530666602415785951731092795566352205193380480082849708712393503576985}{40492338355711492231767428378898612910312329741332276189068741647863311560796872807239238811} a^{8} - \frac{16059032500375212875351894194694500611842824082316636259361544622309643980279445441630490620}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{7} + \frac{120279673698112297489602212018662420754758601954134865250527470356565395562768777502485229049}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{6} - \frac{130945931365242057709693565950185536917111759978085360073175001541198246077359564461703597660}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{5} - \frac{2521839289787284648674834935045302486731317257665530305881861487323200483086281183977379702523}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{4} - \frac{738929970888092490527685161101750862682322213031889445123922504565028973865916730707681110655}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{3} + \frac{7466123940570416762880340710905625665384085002379308358000226880933442450297758003442725554790}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a^{2} + \frac{868308448918460705469493219852121300168815234165061886722481497881441311864569025694549960602}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503} a + \frac{989058700045149679923461049488405461801781482426473754891012758695339173580441212838040015898}{15103642206680386602449250785329182615546498993516939018522640634653015212177233557100236076503}$

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

Class group and class number

$C_{953}$, which has order $953$ (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:  \( 5768582549095.03 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-2}, \sqrt{-7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, 14.0.401774962552217617093885952.1, 14.14.2812424737865523319657201664.1

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 R ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
$2$2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
2.14.21.6$x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$$2$$7$$21$$C_{14}$$[3]^{7}$
7Data not computed