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Properties

Label 28.0.30928748566...2253.1
Degree $28$
Signature $[0, 14]$
Discriminant $13^{21}\cdot 29^{24}$
Root discriminant $122.73$
Ramified primes $13, 29$
Class number $263816$ (GRH)
Class group $[2, 14, 9422]$ (GRH)
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![118082201, 287060780, 372018822, -71603500, -258957409, -138342771, 337407873, 60669907, -10828500, 50763183, -29709892, -2683804, 13616855, -6022955, -239571, 932048, -755172, 274375, 155561, -109438, -1765, 20510, -1826, -2347, 434, 144, -34, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201)
 
gp: K = bnfinit(x^28 - 3*x^27 - 34*x^26 + 144*x^25 + 434*x^24 - 2347*x^23 - 1826*x^22 + 20510*x^21 - 1765*x^20 - 109438*x^19 + 155561*x^18 + 274375*x^17 - 755172*x^16 + 932048*x^15 - 239571*x^14 - 6022955*x^13 + 13616855*x^12 - 2683804*x^11 - 29709892*x^10 + 50763183*x^9 - 10828500*x^8 + 60669907*x^7 + 337407873*x^6 - 138342771*x^5 - 258957409*x^4 - 71603500*x^3 + 372018822*x^2 + 287060780*x + 118082201, 1)
 

Normalized defining polynomial

\( x^{28} - 3 x^{27} - 34 x^{26} + 144 x^{25} + 434 x^{24} - 2347 x^{23} - 1826 x^{22} + 20510 x^{21} - 1765 x^{20} - 109438 x^{19} + 155561 x^{18} + 274375 x^{17} - 755172 x^{16} + 932048 x^{15} - 239571 x^{14} - 6022955 x^{13} + 13616855 x^{12} - 2683804 x^{11} - 29709892 x^{10} + 50763183 x^{9} - 10828500 x^{8} + 60669907 x^{7} + 337407873 x^{6} - 138342771 x^{5} - 258957409 x^{4} - 71603500 x^{3} + 372018822 x^{2} + 287060780 x + 118082201 \)

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:  \(30928748566898619200779449873164521915364034557154874272253=13^{21}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.73$
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:  $13, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(377=13\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{377}(320,·)$, $\chi_{377}(1,·)$, $\chi_{377}(194,·)$, $\chi_{377}(326,·)$, $\chi_{377}(343,·)$, $\chi_{377}(268,·)$, $\chi_{377}(83,·)$, $\chi_{377}(25,·)$, $\chi_{377}(281,·)$, $\chi_{377}(285,·)$, $\chi_{377}(161,·)$, $\chi_{377}(226,·)$, $\chi_{377}(291,·)$, $\chi_{377}(103,·)$, $\chi_{377}(168,·)$, $\chi_{377}(233,·)$, $\chi_{377}(170,·)$, $\chi_{377}(239,·)$, $\chi_{377}(112,·)$, $\chi_{377}(339,·)$, $\chi_{377}(372,·)$, $\chi_{377}(53,·)$, $\chi_{377}(248,·)$, $\chi_{377}(313,·)$, $\chi_{377}(255,·)$, $\chi_{377}(252,·)$, $\chi_{377}(190,·)$, $\chi_{377}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{19} - \frac{1}{3} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{20} - \frac{1}{3} a^{16} + \frac{1}{3} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{19} - \frac{1}{3} a^{17} - \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{51} a^{24} + \frac{1}{17} a^{23} + \frac{5}{51} a^{22} + \frac{4}{51} a^{21} + \frac{5}{17} a^{20} + \frac{5}{51} a^{19} + \frac{8}{51} a^{18} + \frac{8}{17} a^{17} + \frac{7}{17} a^{16} + \frac{5}{51} a^{15} + \frac{25}{51} a^{14} + \frac{6}{17} a^{13} - \frac{11}{51} a^{12} + \frac{25}{51} a^{11} - \frac{14}{51} a^{10} - \frac{5}{17} a^{9} - \frac{25}{51} a^{8} - \frac{7}{17} a^{7} + \frac{4}{51} a^{6} + \frac{25}{51} a^{5} - \frac{1}{51} a^{4} + \frac{4}{51} a^{3} + \frac{1}{3} a^{2} - \frac{2}{17} a - \frac{10}{51}$, $\frac{1}{51} a^{25} - \frac{4}{51} a^{23} + \frac{2}{17} a^{22} + \frac{1}{17} a^{21} - \frac{2}{17} a^{20} - \frac{7}{51} a^{19} - \frac{8}{17} a^{16} + \frac{10}{51} a^{15} - \frac{2}{17} a^{14} - \frac{14}{51} a^{13} + \frac{8}{17} a^{12} + \frac{13}{51} a^{11} - \frac{7}{51} a^{10} + \frac{20}{51} a^{9} + \frac{20}{51} a^{8} - \frac{1}{51} a^{7} - \frac{4}{51} a^{6} + \frac{3}{17} a^{5} - \frac{10}{51} a^{4} - \frac{4}{17} a^{3} - \frac{2}{17} a^{2} - \frac{3}{17} a - \frac{7}{17}$, $\frac{1}{9741} a^{26} - \frac{88}{9741} a^{25} + \frac{4}{573} a^{24} + \frac{149}{9741} a^{23} - \frac{1100}{9741} a^{22} + \frac{40}{3247} a^{21} + \frac{1363}{9741} a^{20} + \frac{1300}{3247} a^{19} + \frac{498}{3247} a^{18} + \frac{4373}{9741} a^{17} + \frac{1319}{3247} a^{16} - \frac{3416}{9741} a^{15} - \frac{1460}{9741} a^{14} + \frac{1321}{3247} a^{13} + \frac{1784}{9741} a^{12} + \frac{919}{3247} a^{11} + \frac{3946}{9741} a^{10} - \frac{3925}{9741} a^{9} + \frac{598}{3247} a^{8} - \frac{29}{573} a^{7} - \frac{4859}{9741} a^{6} - \frac{838}{9741} a^{5} - \frac{3233}{9741} a^{4} - \frac{1994}{9741} a^{3} + \frac{1267}{9741} a^{2} + \frac{4487}{9741} a + \frac{2420}{9741}$, $\frac{1}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{27} + \frac{285547730201556799754820048681483382670168867631097447015516629918757410116645631582104851204058509975308195159921766221}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{26} + \frac{115114864804783077974171578191113289354006169587543679253599591746757767386985264441170212991235826019151980165845510362352}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{25} - \frac{394624714387628810343556049396933624409165396802458484881906268703750264665618972779604792789292825874785875447750242444080}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{24} + \frac{2745818774512725560571766025478789928927411366423442790362884351791658024985705914495787430215520937998071293769656944099077}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{23} - \frac{102687841051440593122933436809067983543703766934267345566185835383367783330885508223292647500912954214357694755136253628438}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{22} - \frac{6414621886718377529044920292256583084198620469718256394038415103250948039955457090370241891425163078746572647465320905652339}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{21} + \frac{9222996305121262966024275719173939688440877545441300564433247950020288179019019425123485219582917798000749234477831535331385}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{20} - \frac{226338539714295739610473879858962185412563282820644955973976599140244426892881161650018232658539903219577303308369773000819}{3666194222395525417934330354066209407723592720729655100735633186395385700598739942506446099475264199958972120639030642420623} a^{19} + \frac{17799028372883537068410535695472670290234890387421246237594191779961158734688798948302204921484665117308724636235732563805489}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{18} - \frac{215677325565214038483752324016397002037979798672284293810960567632486786043741596069726503437019185118337722692208067232817}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{17} - \frac{16049030527637229505548406001996151040031828329254343452960386750818211704241169132536379692470718203145098870979710602152165}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{16} + \frac{21472683681227295669515022288925379196131033993681059414037588116765529992871109543524058415525781473100367662711919316489046}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{15} + \frac{14035987944991433320205218340090354126261985415675711819015208756678411084810700724757396745770734078432566104977508872911001}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{14} + \frac{6508058808156658112005765630736824934681596914044541028291892642909473159623742965603309127227962938350445784004131147917629}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{13} - \frac{14541424099053445014767159321682838608029175478345842428381830922585235983970090726326400226357676615162106321763916613886741}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{12} + \frac{8792026578678916296680849049133324184126625432685794984282574737068411881066759857617443977278181773968607301082306761616546}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{11} + \frac{11213916264523184551760601554488576317547703618771118754814730855870855866043190013740255827960564260397444115531206263893663}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{10} - \frac{13657466273072011599396280757739545771473352233377095462183760075982418387847475996711448069999055978437683536623195187162269}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{9} + \frac{5265867777141046351314143528442639392981725202761523091329493025442965227183556698597921659408314877950216967953323987316630}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{8} + \frac{14607748393441145803805420703285277040820572990190753449569232326062122059066913695216216145911308826189360134542966617198330}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{7} + \frac{5062316417881038943482404505323324147773141225188868300931594393983254316107400960282059726755978768082184930972280475715484}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{6} - \frac{27885606504538029389285776427287048448485832855694706468021055848117421555105357083476315972786366409518810885843894309179614}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{5} - \frac{26643658961868999191031524962182124668156582432553165065252206542748307855917168432341161313853554349057635294864102192049275}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{4} + \frac{1976839213200117699338498105594364992275645008706854701077758420759297661297159473285138201277127877738962444481889223384042}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197} a^{3} + \frac{17648021385864394065660079844020912352349291200107427799685969772010153641118344461099364724685253965850759368791547377049853}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a^{2} - \frac{31069731036521352433866473821422030662407516743605910271090076294886654261448360338136142314674611663146036003533405773153804}{62325301780723932104883616019125559931301076252404136712505764168721556910178579022609583691079491399302526050863520921150591} a - \frac{1127898207366143360965824160636891948250781618738150271821585883613386305781782107050891170189305538623608865854199375567304}{20775100593574644034961205339708519977100358750801378904168588056240518970059526340869861230359830466434175350287840307050197}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{9422}$, which has order $263816$ (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:  \( 1523489837639.8035 \) (assuming GRH)
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

\(\Q(\sqrt{13}) \), 4.0.2197.1, 7.7.594823321.1, 14.14.22201352938819688612162197.1

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 $28$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{4}$ $28$ $28$ $28$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{7}$

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
13Data not computed
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$