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Properties

Label 28.0.47124132165...2064.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 3^{14}\cdot 7^{48}$
Root discriminant $97.35$
Ramified primes $2, 3, 7$
Class number $14413$ (GRH)
Class group $[14413]$ (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![88529281, 0, -270828656, 0, 528315102, 0, -607640670, 0, 500991365, 0, -277641777, 0, 114161607, 0, -34632564, 0, 8063678, 0, -1405327, 0, 185514, 0, -17304, 0, 1141, 0, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 42*x^26 + 1141*x^24 - 17304*x^22 + 185514*x^20 - 1405327*x^18 + 8063678*x^16 - 34632564*x^14 + 114161607*x^12 - 277641777*x^10 + 500991365*x^8 - 607640670*x^6 + 528315102*x^4 - 270828656*x^2 + 88529281)
 
gp: K = bnfinit(x^28 - 42*x^26 + 1141*x^24 - 17304*x^22 + 185514*x^20 - 1405327*x^18 + 8063678*x^16 - 34632564*x^14 + 114161607*x^12 - 277641777*x^10 + 500991365*x^8 - 607640670*x^6 + 528315102*x^4 - 270828656*x^2 + 88529281, 1)
 

Normalized defining polynomial

\( x^{28} - 42 x^{26} + 1141 x^{24} - 17304 x^{22} + 185514 x^{20} - 1405327 x^{18} + 8063678 x^{16} - 34632564 x^{14} + 114161607 x^{12} - 277641777 x^{10} + 500991365 x^{8} - 607640670 x^{6} + 528315102 x^{4} - 270828656 x^{2} + 88529281 \)

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:  \(47124132165319849265944537684819890333789452787530072064=2^{28}\cdot 3^{14}\cdot 7^{48}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.35$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(588=2^{2}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{588}(1,·)$, $\chi_{588}(323,·)$, $\chi_{588}(197,·)$, $\chi_{588}(449,·)$, $\chi_{588}(463,·)$, $\chi_{588}(337,·)$, $\chi_{588}(211,·)$, $\chi_{588}(533,·)$, $\chi_{588}(407,·)$, $\chi_{588}(127,·)$, $\chi_{588}(281,·)$, $\chi_{588}(71,·)$, $\chi_{588}(155,·)$, $\chi_{588}(29,·)$, $\chi_{588}(491,·)$, $\chi_{588}(547,·)$, $\chi_{588}(421,·)$, $\chi_{588}(295,·)$, $\chi_{588}(169,·)$, $\chi_{588}(43,·)$, $\chi_{588}(365,·)$, $\chi_{588}(239,·)$, $\chi_{588}(113,·)$, $\chi_{588}(505,·)$, $\chi_{588}(379,·)$, $\chi_{588}(253,·)$, $\chi_{588}(85,·)$, $\chi_{588}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{5}{19} a^{14} + \frac{6}{19} a^{12} + \frac{8}{19} a^{10} - \frac{2}{19} a^{8} - \frac{9}{19} a^{6} + \frac{7}{19} a^{4} + \frac{3}{19} a^{2} + \frac{4}{19}$, $\frac{1}{19} a^{17} - \frac{5}{19} a^{15} + \frac{6}{19} a^{13} + \frac{8}{19} a^{11} - \frac{2}{19} a^{9} - \frac{9}{19} a^{7} + \frac{7}{19} a^{5} + \frac{3}{19} a^{3} + \frac{4}{19} a$, $\frac{1}{19} a^{18} + \frac{1}{19}$, $\frac{1}{19} a^{19} + \frac{1}{19} a$, $\frac{1}{589} a^{20} - \frac{1}{589} a^{18} - \frac{3}{589} a^{16} - \frac{118}{589} a^{14} - \frac{56}{589} a^{12} + \frac{280}{589} a^{10} - \frac{241}{589} a^{8} - \frac{11}{589} a^{6} + \frac{169}{589} a^{4} + \frac{277}{589} a^{2} + \frac{101}{589}$, $\frac{1}{589} a^{21} - \frac{1}{589} a^{19} - \frac{3}{589} a^{17} - \frac{118}{589} a^{15} - \frac{56}{589} a^{13} + \frac{280}{589} a^{11} - \frac{241}{589} a^{9} - \frac{11}{589} a^{7} + \frac{169}{589} a^{5} + \frac{277}{589} a^{3} + \frac{101}{589} a$, $\frac{1}{11191} a^{22} - \frac{8}{11191} a^{20} + \frac{4}{11191} a^{18} - \frac{128}{11191} a^{16} - \frac{253}{11191} a^{14} + \frac{486}{11191} a^{12} - \frac{79}{361} a^{10} - \frac{5330}{11191} a^{8} - \frac{1831}{11191} a^{6} + \frac{1822}{11191} a^{4} + \frac{4548}{11191} a^{2} - \frac{2598}{11191}$, $\frac{1}{1085527} a^{23} - \frac{26}{35017} a^{21} - \frac{965}{1085527} a^{19} + \frac{9334}{1085527} a^{17} - \frac{411451}{1085527} a^{15} - \frac{1946}{1085527} a^{13} + \frac{457351}{1085527} a^{11} - \frac{285979}{1085527} a^{9} - \frac{23092}{1085527} a^{7} - \frac{464058}{1085527} a^{5} + \frac{62099}{1085527} a^{3} + \frac{334994}{1085527} a$, $\frac{1}{85756633} a^{24} + \frac{1231}{85756633} a^{22} + \frac{61988}{85756633} a^{20} - \frac{461698}{85756633} a^{18} - \frac{738535}{85756633} a^{16} + \frac{40294085}{85756633} a^{14} + \frac{37116755}{85756633} a^{12} + \frac{8516577}{85756633} a^{10} - \frac{37920798}{85756633} a^{8} - \frac{34831837}{85756633} a^{6} + \frac{31392711}{85756633} a^{4} + \frac{26637805}{85756633} a^{2} + \frac{400207}{884089}$, $\frac{1}{85756633} a^{25} - \frac{33}{85756633} a^{23} + \frac{61593}{85756633} a^{21} + \frac{93539}{4513507} a^{19} - \frac{452160}{85756633} a^{17} + \frac{35199770}{85756633} a^{15} - \frac{20700659}{85756633} a^{13} - \frac{10919398}{85756633} a^{11} + \frac{36584971}{85756633} a^{9} + \frac{10080927}{85756633} a^{7} - \frac{296312}{4513507} a^{5} + \frac{35939660}{85756633} a^{3} - \frac{22658195}{85756633} a$, $\frac{1}{35250318844011541583942685906411054825733} a^{26} - \frac{28200797211225457182037715603174}{35250318844011541583942685906411054825733} a^{24} - \frac{1140849516045863661674908354703668972}{35250318844011541583942685906411054825733} a^{22} + \frac{22208829950960441605121362864031230965}{35250318844011541583942685906411054825733} a^{20} - \frac{190662409898838612833557284111379089194}{35250318844011541583942685906411054825733} a^{18} - \frac{332016700550464291341562345119317820131}{35250318844011541583942685906411054825733} a^{16} - \frac{2588179646282983680083043796426676410047}{35250318844011541583942685906411054825733} a^{14} - \frac{14233175168879475589134638391000591240760}{35250318844011541583942685906411054825733} a^{12} + \frac{12675283670401564752556320897404280882658}{35250318844011541583942685906411054825733} a^{10} + \frac{9786057661042802397893456468514256424530}{35250318844011541583942685906411054825733} a^{8} + \frac{11760870878343908113748208665678516081307}{35250318844011541583942685906411054825733} a^{6} + \frac{15229237615356784040336218865847767598416}{35250318844011541583942685906411054825733} a^{4} + \frac{3833644202488757269142871495561890280556}{35250318844011541583942685906411054825733} a^{2} + \frac{764207001672853762081960125698085583}{3746446895951912167493111479053146437}$, $\frac{1}{3419280927869119533642440532921872318096101} a^{27} - \frac{6193960391109613745807365258443689}{3419280927869119533642440532921872318096101} a^{25} + \frac{718743577473890246174490432217030352}{3419280927869119533642440532921872318096101} a^{23} - \frac{76504981147352754895770131096845414185}{3419280927869119533642440532921872318096101} a^{21} + \frac{6045328474179169064206300006283445558264}{3419280927869119533642440532921872318096101} a^{19} + \frac{66869596788503560391740411664071734373701}{3419280927869119533642440532921872318096101} a^{17} - \frac{1501726740884060814627886272854816461098365}{3419280927869119533642440532921872318096101} a^{15} + \frac{624510773576716157233021837861916870755609}{3419280927869119533642440532921872318096101} a^{13} - \frac{1029219820619452932666456367801622852108323}{3419280927869119533642440532921872318096101} a^{11} - \frac{1609443791993271316647940335275350552982007}{3419280927869119533642440532921872318096101} a^{9} + \frac{1453876755635404837907687076707550018910312}{3419280927869119533642440532921872318096101} a^{7} + \frac{530367366060867935846065272639605938125147}{3419280927869119533642440532921872318096101} a^{5} + \frac{1692603367264980408077144771997026958428621}{3419280927869119533642440532921872318096101} a^{3} + \frac{180653357815721229206706743157653607463}{363405348907335480246831813468155204389} a$

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

Class group and class number

$C_{14413}$, which has order $14413$ (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:  \( \frac{19515167931632417211280112536418}{363405348907335480246831813468155204389} a^{27} - \frac{818862349177277708667083611866499}{363405348907335480246831813468155204389} a^{25} + \frac{22179378031795726660757159278556018}{363405348907335480246831813468155204389} a^{23} - \frac{334470832234416874032568530767423102}{363405348907335480246831813468155204389} a^{21} + \frac{3544099691572607419621554318020922356}{363405348907335480246831813468155204389} a^{19} - \frac{26338435134744256301290713254082195580}{363405348907335480246831813468155204389} a^{17} + \frac{146633443151068985988891386836299339630}{363405348907335480246831813468155204389} a^{15} - \frac{601152864970126829554122350998861297817}{363405348907335480246831813468155204389} a^{13} + \frac{1850165976862832005062879722090511973954}{363405348907335480246831813468155204389} a^{11} - \frac{4026510355365017148529329530367867920340}{363405348907335480246831813468155204389} a^{9} + \frac{6117215841455201664390683495742175632186}{363405348907335480246831813468155204389} a^{7} - \frac{5193184202491327687402342144147979018530}{363405348907335480246831813468155204389} a^{5} + \frac{28843197688330914379229185696003464528}{3746446895951912167493111479053146437} a^{3} - \frac{246640213453304031148697233697742735128}{363405348907335480246831813468155204389} a \) (order $12$)
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:  \( 5301892196902.783 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\), 7.7.13841287201.1, 14.14.6864701899232030692065067008.1, 14.0.418988153029298748294987.1, 14.0.3138866894939200133545984.1

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 R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
2Data not computed
3Data not computed
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$