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Properties

Label 28.0.10976959488...5625.1
Degree $28$
Signature $[0, 14]$
Discriminant $5^{14}\cdot 7^{50}$
Root discriminant $72.21$
Ramified primes $5, 7$
Class number $127$ (GRH)
Class group $[127]$ (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![6241, -9401, -47341, 51912, 333200, 343196, -202517, -124511, 1045310, 598339, -787227, 282121, 940065, -564725, -209631, 382109, -109165, -78533, 38654, 6370, -3381, -2414, 2618, 154, 49, -112, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 112*x^25 + 49*x^24 + 154*x^23 + 2618*x^22 - 2414*x^21 - 3381*x^20 + 6370*x^19 + 38654*x^18 - 78533*x^17 - 109165*x^16 + 382109*x^15 - 209631*x^14 - 564725*x^13 + 940065*x^12 + 282121*x^11 - 787227*x^10 + 598339*x^9 + 1045310*x^8 - 124511*x^7 - 202517*x^6 + 343196*x^5 + 333200*x^4 + 51912*x^3 - 47341*x^2 - 9401*x + 6241)
 
gp: K = bnfinit(x^28 - 112*x^25 + 49*x^24 + 154*x^23 + 2618*x^22 - 2414*x^21 - 3381*x^20 + 6370*x^19 + 38654*x^18 - 78533*x^17 - 109165*x^16 + 382109*x^15 - 209631*x^14 - 564725*x^13 + 940065*x^12 + 282121*x^11 - 787227*x^10 + 598339*x^9 + 1045310*x^8 - 124511*x^7 - 202517*x^6 + 343196*x^5 + 333200*x^4 + 51912*x^3 - 47341*x^2 - 9401*x + 6241, 1)
 

Normalized defining polynomial

\( x^{28} - 112 x^{25} + 49 x^{24} + 154 x^{23} + 2618 x^{22} - 2414 x^{21} - 3381 x^{20} + 6370 x^{19} + 38654 x^{18} - 78533 x^{17} - 109165 x^{16} + 382109 x^{15} - 209631 x^{14} - 564725 x^{13} + 940065 x^{12} + 282121 x^{11} - 787227 x^{10} + 598339 x^{9} + 1045310 x^{8} - 124511 x^{7} - 202517 x^{6} + 343196 x^{5} + 333200 x^{4} + 51912 x^{3} - 47341 x^{2} - 9401 x + 6241 \)

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:  \(10976959488814771402711672000547176203926080322265625=5^{14}\cdot 7^{50}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.21$
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:  $5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(245=5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{245}(64,·)$, $\chi_{245}(1,·)$, $\chi_{245}(69,·)$, $\chi_{245}(6,·)$, $\chi_{245}(71,·)$, $\chi_{245}(204,·)$, $\chi_{245}(139,·)$, $\chi_{245}(76,·)$, $\chi_{245}(141,·)$, $\chi_{245}(209,·)$, $\chi_{245}(146,·)$, $\chi_{245}(211,·)$, $\chi_{245}(216,·)$, $\chi_{245}(111,·)$, $\chi_{245}(29,·)$, $\chi_{245}(34,·)$, $\chi_{245}(99,·)$, $\chi_{245}(36,·)$, $\chi_{245}(134,·)$, $\chi_{245}(104,·)$, $\chi_{245}(41,·)$, $\chi_{245}(106,·)$, $\chi_{245}(174,·)$, $\chi_{245}(239,·)$, $\chi_{245}(176,·)$, $\chi_{245}(244,·)$, $\chi_{245}(181,·)$, $\chi_{245}(169,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{79} a^{25} + \frac{18}{79} a^{24} + \frac{24}{79} a^{23} - \frac{24}{79} a^{22} + \frac{1}{79} a^{21} + \frac{25}{79} a^{20} + \frac{3}{79} a^{19} + \frac{15}{79} a^{18} + \frac{18}{79} a^{17} - \frac{18}{79} a^{16} - \frac{13}{79} a^{15} + \frac{24}{79} a^{14} - \frac{25}{79} a^{12} - \frac{20}{79} a^{11} - \frac{3}{79} a^{10} - \frac{14}{79} a^{9} + \frac{28}{79} a^{8} - \frac{28}{79} a^{7} + \frac{16}{79} a^{6} - \frac{20}{79} a^{5} - \frac{32}{79} a^{4} + \frac{12}{79} a^{3} - \frac{39}{79} a^{2} + \frac{21}{79} a$, $\frac{1}{4018632481368851789} a^{26} + \frac{3239476338922207}{4018632481368851789} a^{25} - \frac{195882915028923777}{4018632481368851789} a^{24} + \frac{490868610797004764}{4018632481368851789} a^{23} + \frac{820445626015755631}{4018632481368851789} a^{22} + \frac{547186193643139741}{4018632481368851789} a^{21} + \frac{1657128850102375895}{4018632481368851789} a^{20} - \frac{1563180184559251119}{4018632481368851789} a^{19} - \frac{504094417684495801}{4018632481368851789} a^{18} - \frac{956322695715883803}{4018632481368851789} a^{17} - \frac{20556602989015122}{4018632481368851789} a^{16} - \frac{805302035253504623}{4018632481368851789} a^{15} - \frac{406666225010574870}{4018632481368851789} a^{14} - \frac{439453651870557614}{4018632481368851789} a^{13} + \frac{381926130695883568}{4018632481368851789} a^{12} - \frac{26027789688864041}{4018632481368851789} a^{11} + \frac{950717775402564522}{4018632481368851789} a^{10} + \frac{696778930572627866}{4018632481368851789} a^{9} - \frac{1689448167599777143}{4018632481368851789} a^{8} - \frac{267933989931331965}{4018632481368851789} a^{7} - \frac{140458249300051060}{4018632481368851789} a^{6} + \frac{1139388482937010310}{4018632481368851789} a^{5} - \frac{1359575463243245052}{4018632481368851789} a^{4} + \frac{914576869071983462}{4018632481368851789} a^{3} - \frac{941203200193414588}{4018632481368851789} a^{2} + \frac{1880659101033394374}{4018632481368851789} a + \frac{15909029554905028}{50868765586947491}$, $\frac{1}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{27} - \frac{125680552037074103959107613277563501656834623316412597547431}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{26} + \frac{4021208787332828330128999547484439910296225650950371400187646260860656312900}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{25} - \frac{78091403777154895404818488915490845485791379086739064687788123320488918407987}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{24} - \frac{584437252186351674243628610211475828548015831331093294278854083719957997197}{22884067419745146013489705951164536312989586732979459030531699722618850262993} a^{23} + \frac{690433203508800392037725424418267778555395778448909957620113729200546859971178}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{22} + \frac{620902644999679679819313945762243873354866901708321202292081797786082791561375}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{21} + \frac{47227082135224250433714337579968174551896811401049794951701505558911143679700}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{20} - \frac{439797600673562312400480956766750294783349399240442810791114275105541359484177}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{19} + \frac{47503397223324782828371712717896326396440113754433241648850484650066229301}{230562597393172622760577320512944569407751224576632733504910633603735387167} a^{18} + \frac{356870085177790388833182471842635466456580604855853693413041057381629039940498}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{17} + \frac{201940982958658752177320138977982716135711163752199049582319784597142728982782}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{16} + \frac{198449283062233740402366495350762610041211627221895130152299608237409695603348}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{15} - \frac{182814552040276247615983217601789575908342900352337157751261367274788699340506}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{14} + \frac{82539807293957761944146405164452608883619777856250227838147617254681085351405}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{13} + \frac{469304764021582824046159392667590920784977799747005081833524857281816685148085}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{12} + \frac{534531076322338530134419030326259175661017931462315683471700997631172819560563}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{11} - \frac{207119658519081093119351223114476609078197542317184130422897993065461389709608}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{10} - \frac{134313629935629045003934886719608174902586304943831678275510755447138705599849}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{9} - \frac{852593695373338222795978780494051534531401864382729814730392673746642424582723}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{8} + \frac{224747290233044304073734142125101042975258136723679088120584592171850888856617}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{7} - \frac{580521483401994462888740348757324377776804492846554162802512869875027265641455}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{6} - \frac{769338399496106300518918933123384141059970998201471556304446269635855825721678}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{5} + \frac{318367648366291282961285522368377213135443071691310212424702587340161642190589}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{4} - \frac{456473391565423534228061934886804808902492394424593824734644351164994084894044}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{3} - \frac{807076285115533515791688556052060810652365030756256786733350508568590769476496}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a^{2} - \frac{376370395152988694980546195891119268146219808705640249926232604343347192216799}{1807841326159866535065686770141998368726177351905377263412004278086889170776447} a + \frac{48740468951004722271190089727073116097739743342074470277419254047990635606}{22884067419745146013489705951164536312989586732979459030531699722618850262993}$

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

Class group and class number

$C_{127}$, which has order $127$ (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:  \( 1494941023922.2002 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{-7})\), 7.7.13841287201.1, 14.14.14967283701606751125078125.1, 14.0.1341068619663964900807.1, 14.0.104770985911247257875546875.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
5Data not computed
7Data not computed