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Properties

Label 28.28.1263858821...8125.1
Degree $28$
Signature $[28, 0]$
Discriminant $5^{14}\cdot 7^{14}\cdot 29^{27}$
Root discriminant $152.13$
Ramified primes $5, 7, 29$
Class number Not computed
Class group Not computed
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![-135847856289239, 799274837483108, -799274837483108, -1780718978270827, 1780718978270827, 1200607208822609, -1200607208822609, -408362479450039, 408362479450039, 83267147522159, -83267147522159, -11086013209879, 11086013209879, 1010545858331, -1010545858331, -64703836621, 64703836621, 2938505300, -2938505300, -94192291, 94192291, 2083823, -2083823, -30277, 30277, 260, -260, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 260*x^26 + 260*x^25 + 30277*x^24 - 30277*x^23 - 2083823*x^22 + 2083823*x^21 + 94192291*x^20 - 94192291*x^19 - 2938505300*x^18 + 2938505300*x^17 + 64703836621*x^16 - 64703836621*x^15 - 1010545858331*x^14 + 1010545858331*x^13 + 11086013209879*x^12 - 11086013209879*x^11 - 83267147522159*x^10 + 83267147522159*x^9 + 408362479450039*x^8 - 408362479450039*x^7 - 1200607208822609*x^6 + 1200607208822609*x^5 + 1780718978270827*x^4 - 1780718978270827*x^3 - 799274837483108*x^2 + 799274837483108*x - 135847856289239)
 
gp: K = bnfinit(x^28 - x^27 - 260*x^26 + 260*x^25 + 30277*x^24 - 30277*x^23 - 2083823*x^22 + 2083823*x^21 + 94192291*x^20 - 94192291*x^19 - 2938505300*x^18 + 2938505300*x^17 + 64703836621*x^16 - 64703836621*x^15 - 1010545858331*x^14 + 1010545858331*x^13 + 11086013209879*x^12 - 11086013209879*x^11 - 83267147522159*x^10 + 83267147522159*x^9 + 408362479450039*x^8 - 408362479450039*x^7 - 1200607208822609*x^6 + 1200607208822609*x^5 + 1780718978270827*x^4 - 1780718978270827*x^3 - 799274837483108*x^2 + 799274837483108*x - 135847856289239, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 260 x^{26} + 260 x^{25} + 30277 x^{24} - 30277 x^{23} - 2083823 x^{22} + 2083823 x^{21} + 94192291 x^{20} - 94192291 x^{19} - 2938505300 x^{18} + 2938505300 x^{17} + 64703836621 x^{16} - 64703836621 x^{15} - 1010545858331 x^{14} + 1010545858331 x^{13} + 11086013209879 x^{12} - 11086013209879 x^{11} - 83267147522159 x^{10} + 83267147522159 x^{9} + 408362479450039 x^{8} - 408362479450039 x^{7} - 1200607208822609 x^{6} + 1200607208822609 x^{5} + 1780718978270827 x^{4} - 1780718978270827 x^{3} - 799274837483108 x^{2} + 799274837483108 x - 135847856289239 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12638588217709750986410836410788356243350663111637823486328125=5^{14}\cdot 7^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.13$
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:  $5, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1015=5\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1015}(1,·)$, $\chi_{1015}(386,·)$, $\chi_{1015}(71,·)$, $\chi_{1015}(69,·)$, $\chi_{1015}(454,·)$, $\chi_{1015}(769,·)$, $\chi_{1015}(524,·)$, $\chi_{1015}(909,·)$, $\chi_{1015}(526,·)$, $\chi_{1015}(141,·)$, $\chi_{1015}(594,·)$, $\chi_{1015}(596,·)$, $\chi_{1015}(981,·)$, $\chi_{1015}(279,·)$, $\chi_{1015}(664,·)$, $\chi_{1015}(281,·)$, $\chi_{1015}(666,·)$, $\chi_{1015}(804,·)$, $\chi_{1015}(806,·)$, $\chi_{1015}(104,·)$, $\chi_{1015}(839,·)$, $\chi_{1015}(876,·)$, $\chi_{1015}(559,·)$, $\chi_{1015}(244,·)$, $\chi_{1015}(631,·)$, $\chi_{1015}(699,·)$, $\chi_{1015}(701,·)$, $\chi_{1015}(36,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3320390384119} a^{15} + \frac{11322477335}{3320390384119} a^{14} - \frac{135}{3320390384119} a^{13} - \frac{1426632144210}{3320390384119} a^{12} + \frac{7290}{3320390384119} a^{11} + \frac{890093071896}{3320390384119} a^{10} - \frac{200475}{3320390384119} a^{9} - \frac{114274705032}{3320390384119} a^{8} + \frac{2952450}{3320390384119} a^{7} - \frac{1216451023892}{3320390384119} a^{6} - \frac{22320522}{3320390384119} a^{5} + \frac{657925375114}{3320390384119} a^{4} + \frac{74401740}{3320390384119} a^{3} + \frac{179863098053}{3320390384119} a^{2} - \frac{71744535}{3320390384119} a + \frac{1018136946550}{3320390384119}$, $\frac{1}{3320390384119} a^{16} - \frac{144}{3320390384119} a^{14} + \frac{101902296015}{3320390384119} a^{13} + \frac{8424}{3320390384119} a^{12} + \frac{1358992902721}{3320390384119} a^{11} - \frac{256608}{3320390384119} a^{10} - \frac{1387653708303}{3320390384119} a^{9} + \frac{4330260}{3320390384119} a^{8} - \frac{574271434550}{3320390384119} a^{7} - \frac{39680928}{3320390384119} a^{6} - \frac{610930705463}{3320390384119} a^{5} + \frac{178564176}{3320390384119} a^{4} + \frac{1088992982524}{3320390384119} a^{3} - \frac{306110016}{3320390384119} a^{2} + \frac{22890615663}{3320390384119} a + \frac{86093442}{3320390384119}$, $\frac{1}{3320390384119} a^{17} - \frac{1588051351864}{3320390384119} a^{14} - \frac{11016}{3320390384119} a^{13} - \frac{1532222432260}{3320390384119} a^{12} + \frac{793152}{3320390384119} a^{11} + \frac{610914048199}{3320390384119} a^{10} - \frac{24538140}{3320390384119} a^{9} - \frac{427877038563}{3320390384119} a^{8} + \frac{385471872}{3320390384119} a^{7} + \frac{200812212396}{3320390384119} a^{6} - \frac{3035590992}{3320390384119} a^{5} - \frac{461074140511}{3320390384119} a^{4} + \frac{10407740544}{3320390384119} a^{3} - \frac{639946337657}{3320390384119} a^{2} - \frac{10245119598}{3320390384119} a + \frac{514543401964}{3320390384119}$, $\frac{1}{3320390384119} a^{18} - \frac{12393}{3320390384119} a^{14} - \frac{93779966165}{3320390384119} a^{13} + \frac{966654}{3320390384119} a^{12} - \frac{696000286194}{3320390384119} a^{11} - \frac{33126489}{3320390384119} a^{10} + \frac{648168123995}{3320390384119} a^{9} + \frac{596276802}{3320390384119} a^{8} - \frac{1157422141848}{3320390384119} a^{7} - \frac{5691733110}{3320390384119} a^{6} + \frac{1306978194229}{3320390384119} a^{5} + \frac{26344593252}{3320390384119} a^{4} - \frac{999470146592}{3320390384119} a^{3} - \frac{46103038191}{3320390384119} a^{2} + \frac{979812445727}{3320390384119} a + \frac{13172296626}{3320390384119}$, $\frac{1}{3320390384119} a^{19} + \frac{769285513492}{3320390384119} a^{14} - \frac{706401}{3320390384119} a^{13} + \frac{130631952951}{3320390384119} a^{12} + \frac{57218481}{3320390384119} a^{11} + \frac{1234752087805}{3320390384119} a^{10} - \frac{1888209873}{3320390384119} a^{9} + \frac{442852415389}{3320390384119} a^{8} + \frac{30897979740}{3320390384119} a^{7} + \frac{401783000933}{3320390384119} a^{6} - \frac{250273635894}{3320390384119} a^{5} + \frac{1111310629065}{3320390384119} a^{4} + \frac{875957725629}{3320390384119} a^{3} - \frac{1279151511412}{3320390384119} a^{2} - \frac{875957725629}{3320390384119} a + \frac{287718941950}{3320390384119}$, $\frac{1}{3320390384119} a^{20} - \frac{831060}{3320390384119} a^{14} + \frac{1052074366682}{3320390384119} a^{13} + \frac{72925515}{3320390384119} a^{12} + \frac{1282717508116}{3320390384119} a^{11} - \frac{2665708056}{3320390384119} a^{10} + \frac{783998548896}{3320390384119} a^{9} + \frac{49982026050}{3320390384119} a^{8} - \frac{94564077826}{3320390384119} a^{7} - \frac{490732619400}{3320390384119} a^{6} + \frac{1012041698905}{3320390384119} a^{5} - \frac{1001678757454}{3320390384119} a^{4} + \frac{158541717161}{3320390384119} a^{3} - \frac{801763618841}{3320390384119} a^{2} + \frac{1391664688201}{3320390384119} a + \frac{1192480265142}{3320390384119}$, $\frac{1}{3320390384119} a^{21} + \frac{723739798536}{3320390384119} a^{14} - \frac{39267585}{3320390384119} a^{13} - \frac{512201899035}{3320390384119} a^{12} + \frac{3392719344}{3320390384119} a^{11} + \frac{1642164023717}{3320390384119} a^{10} - \frac{116624727450}{3320390384119} a^{9} + \frac{574838599892}{3320390384119} a^{8} - \frac{1357459906519}{3320390384119} a^{7} - \frac{117573195280}{3320390384119} a^{6} + \frac{370970533940}{3320390384119} a^{5} + \frac{295450314033}{3320390384119} a^{4} + \frac{1263519511417}{3320390384119} a^{3} + \frac{1083620345239}{3320390384119} a^{2} + \frac{1335493922184}{3320390384119} a + \frac{1129605182349}{3320390384119}$, $\frac{1}{3320390384119} a^{22} - \frac{47993715}{3320390384119} a^{14} + \frac{901349763874}{3320390384119} a^{13} + \frac{4492211724}{3320390384119} a^{12} - \frac{1641037322751}{3320390384119} a^{11} - \frac{171049600260}{3320390384119} a^{10} + \frac{1212335256549}{3320390384119} a^{9} - \frac{21576664819}{3320390384119} a^{8} - \frac{1657964867220}{3320390384119} a^{7} + \frac{140793609115}{3320390384119} a^{6} - \frac{12594028929}{3320390384119} a^{5} - \frac{675809323752}{3320390384119} a^{4} - \frac{1209955562340}{3320390384119} a^{3} - \frac{111699370894}{3320390384119} a^{2} - \frac{1283561798937}{3320390384119} a + \frac{463858956965}{3320390384119}$, $\frac{1}{3320390384119} a^{23} + \frac{202175566097}{3320390384119} a^{14} - \frac{1986939801}{3320390384119} a^{13} + \frac{124761610057}{3320390384119} a^{12} + \frac{178824582090}{3320390384119} a^{11} + \frac{135438069647}{3320390384119} a^{10} + \frac{318054472913}{3320390384119} a^{9} - \frac{1184449646374}{3320390384119} a^{8} - \frac{936949056252}{3320390384119} a^{7} - \frac{1330339273822}{3320390384119} a^{6} + \frac{565513227455}{3320390384119} a^{5} + \frac{55735138780}{3320390384119} a^{4} + \frac{1284542765281}{3320390384119} a^{3} - \frac{810975492338}{3320390384119} a^{2} + \frac{421921690843}{3320390384119} a + \frac{1317184386364}{3320390384119}$, $\frac{1}{3320390384119} a^{24} - \frac{2509818696}{3320390384119} a^{14} + \frac{855339960200}{3320390384119} a^{13} + \frac{244707322860}{3320390384119} a^{12} + \frac{528891771353}{3320390384119} a^{11} + \frac{377249188917}{3320390384119} a^{10} + \frac{1277135093187}{3320390384119} a^{9} - \frac{578788239558}{3320390384119} a^{8} - \frac{1360328520604}{3320390384119} a^{7} + \frac{1365311551701}{3320390384119} a^{6} + \frac{1344978767370}{3320390384119} a^{5} - \frac{193856796076}{3320390384119} a^{4} + \frac{425529719227}{3320390384119} a^{3} + \frac{1258027915807}{3320390384119} a^{2} + \frac{717766929519}{3320390384119} a + \frac{1319172595296}{3320390384119}$, $\frac{1}{3320390384119} a^{25} + \frac{956309416881}{3320390384119} a^{14} - \frac{94118201100}{3320390384119} a^{13} + \frac{1088301766023}{3320390384119} a^{12} - \frac{1248514821957}{3320390384119} a^{11} + \frac{149463429974}{3320390384119} a^{10} + \frac{964647065930}{3320390384119} a^{9} + \frac{963176643170}{3320390384119} a^{8} + \frac{368183203293}{3320390384119} a^{7} + \frac{624171007241}{3320390384119} a^{6} + \frac{969283980380}{3320390384119} a^{5} + \frac{1429985895882}{3320390384119} a^{4} + \frac{701282378406}{3320390384119} a^{3} - \frac{712161871925}{3320390384119} a^{2} + \frac{314424542306}{3320390384119} a + \frac{796332244132}{3320390384119}$, $\frac{1}{3320390384119} a^{26} - \frac{122353661430}{3320390384119} a^{14} + \frac{694848064317}{3320390384119} a^{13} - \frac{1011237204496}{3320390384119} a^{12} + \frac{1473621017384}{3320390384119} a^{11} + \frac{840771346027}{3320390384119} a^{10} + \frac{1073137214704}{3320390384119} a^{9} - \frac{213474999945}{3320390384119} a^{8} - \frac{995248318987}{3320390384119} a^{7} + \frac{993318496117}{3320390384119} a^{6} + \frac{1320943773243}{3320390384119} a^{5} + \frac{557075283984}{3320390384119} a^{4} - \frac{348688129795}{3320390384119} a^{3} - \frac{1021304687304}{3320390384119} a^{2} + \frac{1631693415692}{3320390384119} a + \frac{1507122079485}{3320390384119}$, $\frac{1}{3320390384119} a^{27} + \frac{1466621097367}{3320390384119} a^{14} - \frac{927029576951}{3320390384119} a^{13} + \frac{521539992535}{3320390384119} a^{12} - \frac{386050157284}{3320390384119} a^{11} + \frac{55487375004}{3320390384119} a^{10} - \frac{1339982692142}{3320390384119} a^{9} + \frac{1042550547009}{3320390384119} a^{8} - \frac{1131223111107}{3320390384119} a^{7} + \frac{215566904174}{3320390384119} a^{6} + \frac{813551606191}{3320390384119} a^{5} - \frac{357207683619}{3320390384119} a^{4} - \frac{769428966621}{3320390384119} a^{3} + \frac{1650862411796}{3320390384119} a^{2} + \frac{671315065948}{3320390384119} a - \frac{1023487707993}{3320390384119}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $27$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{29}) \), 4.4.29876525.2, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ $28$ R R $28$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed