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Properties

Label 28.0.18917407352...1232.1
Degree $28$
Signature $[0, 14]$
Discriminant $2^{77}\cdot 29^{24}$
Root discriminant $120.59$
Ramified primes $2, 29$
Class number $3088744$ (GRH)
Class group $[2, 2, 772186]$ (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![186649783, -18656684, 493240074, -376661420, 473363957, -435001108, 364501416, -54174868, 19222670, 11010404, 20038, -2935524, 10716934, -2924152, 1432704, 252872, 113927, 37484, 22712, -16648, -654, 4028, -690, -772, 191, 64, -14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 4*x^27 - 14*x^26 + 64*x^25 + 191*x^24 - 772*x^23 - 690*x^22 + 4028*x^21 - 654*x^20 - 16648*x^19 + 22712*x^18 + 37484*x^17 + 113927*x^16 + 252872*x^15 + 1432704*x^14 - 2924152*x^13 + 10716934*x^12 - 2935524*x^11 + 20038*x^10 + 11010404*x^9 + 19222670*x^8 - 54174868*x^7 + 364501416*x^6 - 435001108*x^5 + 473363957*x^4 - 376661420*x^3 + 493240074*x^2 - 18656684*x + 186649783)
 
gp: K = bnfinit(x^28 - 4*x^27 - 14*x^26 + 64*x^25 + 191*x^24 - 772*x^23 - 690*x^22 + 4028*x^21 - 654*x^20 - 16648*x^19 + 22712*x^18 + 37484*x^17 + 113927*x^16 + 252872*x^15 + 1432704*x^14 - 2924152*x^13 + 10716934*x^12 - 2935524*x^11 + 20038*x^10 + 11010404*x^9 + 19222670*x^8 - 54174868*x^7 + 364501416*x^6 - 435001108*x^5 + 473363957*x^4 - 376661420*x^3 + 493240074*x^2 - 18656684*x + 186649783, 1)
 

Normalized defining polynomial

\( x^{28} - 4 x^{27} - 14 x^{26} + 64 x^{25} + 191 x^{24} - 772 x^{23} - 690 x^{22} + 4028 x^{21} - 654 x^{20} - 16648 x^{19} + 22712 x^{18} + 37484 x^{17} + 113927 x^{16} + 252872 x^{15} + 1432704 x^{14} - 2924152 x^{13} + 10716934 x^{12} - 2935524 x^{11} + 20038 x^{10} + 11010404 x^{9} + 19222670 x^{8} - 54174868 x^{7} + 364501416 x^{6} - 435001108 x^{5} + 473363957 x^{4} - 376661420 x^{3} + 493240074 x^{2} - 18656684 x + 186649783 \)

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:  \(18917407352603402612306290142504402709970002327473311711232=2^{77}\cdot 29^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.59$
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:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(464=2^{4}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{464}(123,·)$, $\chi_{464}(1,·)$, $\chi_{464}(83,·)$, $\chi_{464}(451,·)$, $\chi_{464}(227,·)$, $\chi_{464}(257,·)$, $\chi_{464}(393,·)$, $\chi_{464}(339,·)$, $\chi_{464}(139,·)$, $\chi_{464}(401,·)$, $\chi_{464}(355,·)$, $\chi_{464}(25,·)$, $\chi_{464}(281,·)$, $\chi_{464}(219,·)$, $\chi_{464}(297,·)$, $\chi_{464}(459,·)$, $\chi_{464}(161,·)$, $\chi_{464}(291,·)$, $\chi_{464}(65,·)$, $\chi_{464}(81,·)$, $\chi_{464}(169,·)$, $\chi_{464}(107,·)$, $\chi_{464}(49,·)$, $\chi_{464}(371,·)$, $\chi_{464}(315,·)$, $\chi_{464}(233,·)$, $\chi_{464}(313,·)$, $\chi_{464}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} + \frac{7}{17} a^{14} + \frac{3}{17} a^{13} + \frac{5}{17} a^{12} - \frac{1}{17} a^{11} + \frac{1}{17} a^{10} + \frac{2}{17} a^{9} + \frac{2}{17} a^{8} + \frac{7}{17} a^{7} + \frac{8}{17} a^{6} + \frac{6}{17} a^{5} - \frac{3}{17} a^{4} - \frac{8}{17} a^{3} - \frac{3}{17} a^{2} + \frac{7}{17} a$, $\frac{1}{17} a^{16} + \frac{5}{17} a^{14} + \frac{1}{17} a^{13} - \frac{2}{17} a^{12} + \frac{8}{17} a^{11} - \frac{5}{17} a^{10} + \frac{5}{17} a^{9} - \frac{7}{17} a^{8} - \frac{7}{17} a^{7} + \frac{1}{17} a^{6} + \frac{6}{17} a^{5} - \frac{4}{17} a^{4} + \frac{2}{17} a^{3} - \frac{6}{17} a^{2} + \frac{2}{17} a$, $\frac{1}{17} a^{17} - \frac{1}{17} a$, $\frac{1}{17} a^{18} - \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} - \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} - \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} - \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} - \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} - \frac{1}{17} a^{7}$, $\frac{1}{289} a^{24} - \frac{3}{289} a^{23} + \frac{2}{289} a^{22} + \frac{7}{289} a^{21} + \frac{4}{289} a^{20} + \frac{7}{289} a^{19} + \frac{8}{289} a^{18} - \frac{4}{289} a^{17} + \frac{7}{289} a^{16} - \frac{5}{289} a^{15} + \frac{77}{289} a^{13} + \frac{63}{289} a^{12} - \frac{24}{289} a^{11} + \frac{11}{289} a^{10} + \frac{144}{289} a^{9} + \frac{8}{289} a^{8} - \frac{30}{289} a^{7} - \frac{35}{289} a^{6} - \frac{12}{289} a^{5} + \frac{8}{17} a^{4} - \frac{4}{289} a^{3} + \frac{84}{289} a^{2} + \frac{8}{17} a$, $\frac{1}{289} a^{25} - \frac{7}{289} a^{23} - \frac{4}{289} a^{22} + \frac{8}{289} a^{21} + \frac{2}{289} a^{20} - \frac{5}{289} a^{19} + \frac{3}{289} a^{18} - \frac{5}{289} a^{17} - \frac{1}{289} a^{16} + \frac{2}{289} a^{15} + \frac{111}{289} a^{14} + \frac{39}{289} a^{13} - \frac{5}{289} a^{12} + \frac{75}{289} a^{11} - \frac{10}{289} a^{10} + \frac{100}{289} a^{9} - \frac{142}{289} a^{8} + \frac{113}{289} a^{7} + \frac{19}{289} a^{6} + \frac{117}{289} a^{5} - \frac{140}{289} a^{4} - \frac{64}{289} a^{3} - \frac{122}{289} a^{2} - \frac{5}{17} a$, $\frac{1}{2517523115743114860912854031895682732511169} a^{26} - \frac{1902845959980428710742937602906022408549}{2517523115743114860912854031895682732511169} a^{25} + \frac{298466426949662400923947937988397288824}{2517523115743114860912854031895682732511169} a^{24} - \frac{54148951997188043399350761282923475462910}{2517523115743114860912854031895682732511169} a^{23} + \frac{35171131716220289029770735047446147131594}{2517523115743114860912854031895682732511169} a^{22} - \frac{69177437807910459172997982635918931025431}{2517523115743114860912854031895682732511169} a^{21} + \frac{1440630266349852081400788165161912453011}{2517523115743114860912854031895682732511169} a^{20} - \frac{44488127502424186571654405761549061862440}{2517523115743114860912854031895682732511169} a^{19} + \frac{9007459731121115602168990460592900158478}{2517523115743114860912854031895682732511169} a^{18} - \frac{57013729634849684344093260665924277957590}{2517523115743114860912854031895682732511169} a^{17} - \frac{3716047609902699252545139563084964575848}{148089595043712638877226707758569572500657} a^{16} - \frac{28946398441165980511786922968854640518924}{2517523115743114860912854031895682732511169} a^{15} + \frac{491890884926649077849749333937059182328858}{2517523115743114860912854031895682732511169} a^{14} + \frac{285509991467304696216297218343514813040810}{2517523115743114860912854031895682732511169} a^{13} + \frac{680517103657269085415398408747630835995236}{2517523115743114860912854031895682732511169} a^{12} + \frac{424752304111205910895766923325790763827167}{2517523115743114860912854031895682732511169} a^{11} - \frac{748135742858629619673010284279002151339181}{2517523115743114860912854031895682732511169} a^{10} + \frac{954193492628157838807614955984516469661433}{2517523115743114860912854031895682732511169} a^{9} + \frac{740724082691853596747096052570480584218013}{2517523115743114860912854031895682732511169} a^{8} + \frac{745225055310080003896546019628197672177924}{2517523115743114860912854031895682732511169} a^{7} - \frac{322764733545555571463056551496038563554152}{2517523115743114860912854031895682732511169} a^{6} + \frac{837268958722533719513007839190288555395490}{2517523115743114860912854031895682732511169} a^{5} + \frac{260814052804829357366270469629071803020366}{2517523115743114860912854031895682732511169} a^{4} + \frac{892882272428430346663646823852517817169486}{2517523115743114860912854031895682732511169} a^{3} - \frac{540771513355707326166726367421666089209025}{2517523115743114860912854031895682732511169} a^{2} + \frac{46036105183294553350280730450640270733330}{148089595043712638877226707758569572500657} a - \frac{3412091055371447141695847211891172826784}{8711152649630155228072159279915857205921}$, $\frac{1}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{27} + \frac{1079503494918406158582480685819}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{26} - \frac{3191505028462149772218173406331962744633804695371902546383929084466819}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{25} + \frac{7225576832484615255597092490173540316935080180523218355580201516472452}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{24} - \frac{12156722250050722483981800458131897141461912062045506429695310113279240}{1006935069638362611299344079915511950688176699939189220093859443002628257} a^{23} + \frac{300925267149832452490825161231429627705112007699580886195938012172040898}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{22} + \frac{238773699803171462832781337862060694939236004010174822151335809415619648}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{21} + \frac{74809646100419249013546058554939305136164423514378408172500666008219565}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{20} - \frac{20641695543769138155648097190658818163010244866166437771207768123833711}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{19} - \frac{345973258050903769759593405597986085072906049744404415401711271229931558}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{18} - \frac{14741807876550124738319266950089989603160396407299084862070458784498430}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{17} + \frac{397569762598646136199252608794418077967711267276500119583572620787916213}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{16} - \frac{41162790446356185795535081327731968171752237387571670190476726786322551}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{15} - \frac{625234773704484795500560100941053635782363907164358782840526383068783855}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{14} - \frac{1865198238971380736794777794860284581350903352957881376584781682487802045}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{13} - \frac{1910839242194844221407846297413210785269214083542488021035819823928737194}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{12} + \frac{3153768064305371325374826451131378701407062214817392834439123233369519142}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{11} - \frac{875330744186979621945804485335680001254060786850166267880432078302253030}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{10} + \frac{2866409244129780478233712228466771559486075145176781560737993157640950086}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{9} - \frac{3338952572323680696538260612174442389943927200897266996324802222111529037}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{8} - \frac{4004549405775318782496320355011364852300642161526198842527512107099979299}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{7} + \frac{2628565108879520410546105451629683144707414824186581399684270955327660017}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{6} + \frac{4722565557364611040020864667346330866138333999159385275106334773085319881}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{5} - \frac{4895937280390354795797202842749047454320189412694691329693176726573812081}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{4} + \frac{6872844535705319917684209517397550730085202860393733848324639861316310972}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{3} + \frac{464131462507832358370897225415411712897919151604245987816464034485668816}{17117896183852164392088849358563703161699003898966216741595610531044680369} a^{2} + \frac{111673064060306414099704596173981994250187143878706047660826293215088587}{1006935069638362611299344079915511950688176699939189220093859443002628257} a - \frac{19924601054102816702685851346083302532428684232685586574517361262903196}{59231474684609565370549651759735997099304511761128777652579967235448721}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{772186}$, which has order $3088744$ (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:  \( 297452739458.56445 \) (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

\(\Q(\sqrt{2}) \), 4.0.2048.2, 7.7.594823321.1, 14.14.742003380228915810271232.1

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 $28$ $28$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{28}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{7}$

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
2Data not computed
29Data not computed