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Properties

Label 28.28.1159427335...5069.1
Degree $28$
Signature $[28, 0]$
Discriminant $11^{14}\cdot 29^{27}$
Root discriminant $85.28$
Ramified primes $11, 29$
Class number $1$ (GRH)
Class group Trivial (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![13304911, 125401190, -125401190, -1492836655, 1492836655, 4117054541, -4117054541, -4965626443, 4965626443, 3360164459, -3360164459, -1433472727, 1433472727, 410233883, -410233883, -81421213, 81421213, 11366636, -11366636, -1113601, 1113601, 74993, -74993, -3307, 3307, 86, -86, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911)
 
gp: K = bnfinit(x^28 - x^27 - 86*x^26 + 86*x^25 + 3307*x^24 - 3307*x^23 - 74993*x^22 + 74993*x^21 + 1113601*x^20 - 1113601*x^19 - 11366636*x^18 + 11366636*x^17 + 81421213*x^16 - 81421213*x^15 - 410233883*x^14 + 410233883*x^13 + 1433472727*x^12 - 1433472727*x^11 - 3360164459*x^10 + 3360164459*x^9 + 4965626443*x^8 - 4965626443*x^7 - 4117054541*x^6 + 4117054541*x^5 + 1492836655*x^4 - 1492836655*x^3 - 125401190*x^2 + 125401190*x + 13304911, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} - 86 x^{26} + 86 x^{25} + 3307 x^{24} - 3307 x^{23} - 74993 x^{22} + 74993 x^{21} + 1113601 x^{20} - 1113601 x^{19} - 11366636 x^{18} + 11366636 x^{17} + 81421213 x^{16} - 81421213 x^{15} - 410233883 x^{14} + 410233883 x^{13} + 1433472727 x^{12} - 1433472727 x^{11} - 3360164459 x^{10} + 3360164459 x^{9} + 4965626443 x^{8} - 4965626443 x^{7} - 4117054541 x^{6} + 4117054541 x^{5} + 1492836655 x^{4} - 1492836655 x^{3} - 125401190 x^{2} + 125401190 x + 13304911 \)

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:  \(1159427335739550761098088865697701264851767903254485069=11^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.28$
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:  $11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(319=11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{319}(1,·)$, $\chi_{319}(131,·)$, $\chi_{319}(100,·)$, $\chi_{319}(263,·)$, $\chi_{319}(265,·)$, $\chi_{319}(10,·)$, $\chi_{319}(76,·)$, $\chi_{319}(98,·)$, $\chi_{319}(142,·)$, $\chi_{319}(144,·)$, $\chi_{319}(210,·)$, $\chi_{319}(67,·)$, $\chi_{319}(21,·)$, $\chi_{319}(23,·)$, $\chi_{319}(153,·)$, $\chi_{319}(122,·)$, $\chi_{319}(199,·)$, $\chi_{319}(32,·)$, $\chi_{319}(34,·)$, $\chi_{319}(164,·)$, $\chi_{319}(230,·)$, $\chi_{319}(43,·)$, $\chi_{319}(78,·)$, $\chi_{319}(45,·)$, $\chi_{319}(111,·)$, $\chi_{319}(307,·)$, $\chi_{319}(186,·)$, $\chi_{319}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2977199} a^{15} + \frac{742231}{2977199} a^{14} - \frac{45}{2977199} a^{13} - \frac{1401712}{2977199} a^{12} + \frac{810}{2977199} a^{11} - \frac{689344}{2977199} a^{10} - \frac{7425}{2977199} a^{9} + \frac{1309616}{2977199} a^{8} + \frac{36450}{2977199} a^{7} - \frac{141429}{2977199} a^{6} - \frac{91854}{2977199} a^{5} + \frac{282858}{2977199} a^{4} + \frac{102060}{2977199} a^{3} + \frac{1276456}{2977199} a^{2} - \frac{32805}{2977199} a - \frac{1371484}{2977199}$, $\frac{1}{2977199} a^{16} - \frac{48}{2977199} a^{14} - \frac{750506}{2977199} a^{13} + \frac{936}{2977199} a^{12} - \frac{502256}{2977199} a^{11} - \frac{9504}{2977199} a^{10} - \frac{1397757}{2977199} a^{9} + \frac{53460}{2977199} a^{8} - \frac{654066}{2977199} a^{7} - \frac{163296}{2977199} a^{6} - \frac{687968}{2977199} a^{5} + \frac{244944}{2977199} a^{4} + \frac{1031952}{2977199} a^{3} - \frac{139968}{2977199} a^{2} - \frac{16951}{2977199} a + \frac{13122}{2977199}$, $\frac{1}{2977199} a^{17} - \frac{849806}{2977199} a^{14} - \frac{1224}{2977199} a^{13} + \frac{691145}{2977199} a^{12} + \frac{29376}{2977199} a^{11} + \frac{1240119}{2977199} a^{10} - \frac{302940}{2977199} a^{9} - \frac{313677}{2977199} a^{8} - \frac{1390895}{2977199} a^{7} + \frac{1455037}{2977199} a^{6} - \frac{1186849}{2977199} a^{5} - \frac{276859}{2977199} a^{4} - \frac{1195486}{2977199} a^{3} - \frac{1268242}{2977199} a^{2} + \frac{1415681}{2977199} a - \frac{332854}{2977199}$, $\frac{1}{2977199} a^{18} - \frac{1377}{2977199} a^{14} + \frac{1153462}{2977199} a^{13} + \frac{35802}{2977199} a^{12} - \frac{1127189}{2977199} a^{11} - \frac{408969}{2977199} a^{10} - \frac{1438546}{2977199} a^{9} - \frac{523385}{2977199} a^{8} - \frac{871858}{2977199} a^{7} + \frac{1124007}{2977199} a^{6} + \frac{823398}{2977199} a^{5} + \frac{137200}{2977199} a^{4} + \frac{1148049}{2977199} a^{3} - \frac{1072433}{2977199} a^{2} + \frac{272752}{2977199} a + \frac{669222}{2977199}$, $\frac{1}{2977199} a^{19} - \frac{950907}{2977199} a^{14} - \frac{26163}{2977199} a^{13} + \frac{917538}{2977199} a^{12} + \frac{706401}{2977199} a^{11} - \frac{938753}{2977199} a^{10} + \frac{1161186}{2977199} a^{9} + \frac{1263979}{2977199} a^{8} + \frac{703274}{2977199} a^{7} - \frac{406400}{2977199} a^{6} - \frac{1303400}{2977199} a^{5} + \frac{630446}{2977199} a^{4} - \frac{464166}{2977199} a^{3} + \frac{1405254}{2977199} a^{2} + \frac{154722}{2977199} a - \frac{989302}{2977199}$, $\frac{1}{2977199} a^{20} - \frac{30780}{2977199} a^{14} - \frac{192491}{2977199} a^{13} + \frac{900315}{2977199} a^{12} + \frac{1178575}{2977199} a^{11} + \frac{938804}{2977199} a^{10} - \frac{281667}{2977199} a^{9} + \frac{86873}{2977199} a^{8} - \frac{397008}{2977199} a^{7} - \frac{1096275}{2977199} a^{6} + \frac{1083130}{2977199} a^{5} - \frac{878416}{2977199} a^{4} + \frac{240672}{2977199} a^{3} - \frac{1023190}{2977199} a^{2} - \frac{402315}{2977199} a - \frac{645635}{2977199}$, $\frac{1}{2977199} a^{21} - \frac{1347437}{2977199} a^{14} - \frac{484785}{2977199} a^{13} - \frac{926076}{2977199} a^{12} - \frac{924187}{2977199} a^{11} + \frac{207286}{2977199} a^{10} + \frac{789696}{2977199} a^{9} + \frac{1286211}{2977199} a^{8} + \frac{1407901}{2977199} a^{7} + \frac{563448}{2977199} a^{6} + \frac{194514}{2977199} a^{5} + \frac{1280036}{2977199} a^{4} - \frac{561335}{2977199} a^{3} - \frac{1181838}{2977199} a^{2} - \frac{1113074}{2977199} a - \frac{572899}{2977199}$, $\frac{1}{2977199} a^{22} - \frac{592515}{2977199} a^{14} + \frac{960438}{2977199} a^{13} + \frac{623274}{2977199} a^{12} - \frac{1000777}{2977199} a^{11} + \frac{562781}{2977199} a^{10} - \frac{44874}{2977199} a^{9} - \frac{1065993}{2977199} a^{8} - \frac{209805}{2977199} a^{7} + \frac{1057832}{2977199} a^{6} - \frac{1058533}{2977199} a^{5} + \frac{689228}{2977199} a^{4} + \frac{1416572}{2977199} a^{3} + \frac{181903}{2977199} a^{2} - \frac{770131}{2977199} a + \frac{813578}{2977199}$, $\frac{1}{2977199} a^{23} + \frac{1056720}{2977199} a^{14} + \frac{754890}{2977199} a^{13} + \frac{909777}{2977199} a^{12} + \frac{1170892}{2977199} a^{11} + \frac{1180174}{2977199} a^{10} - \frac{189746}{2977199} a^{9} - \frac{1301328}{2977199} a^{8} - \frac{1349163}{2977199} a^{7} - \frac{10885}{50461} a^{6} - \frac{985862}{2977199} a^{5} + \frac{583936}{2977199} a^{4} - \frac{603285}{2977199} a^{3} - \frac{145654}{2977199} a^{2} - \frac{1485925}{2977199} a - \frac{1352409}{2977199}$, $\frac{1}{2977199} a^{24} - \frac{1396875}{2977199} a^{14} + \frac{826993}{2977199} a^{13} - \frac{748147}{2977199} a^{12} - \frac{306913}{2977199} a^{11} + \frac{213808}{2977199} a^{10} - \frac{74693}{2977199} a^{9} - \frac{1403115}{2977199} a^{8} + \frac{914447}{2977199} a^{7} + \frac{431616}{2977199} a^{6} - \frac{1076181}{2977199} a^{5} - \frac{461042}{2977199} a^{4} + \frac{44921}{2977199} a^{3} + \frac{640292}{2977199} a^{2} + \frac{819234}{2977199} a + \frac{894071}{2977199}$, $\frac{1}{2977199} a^{25} + \frac{1157766}{2977199} a^{14} - \frac{1086343}{2977199} a^{13} + \frac{686616}{2977199} a^{12} + \frac{346938}{2977199} a^{11} - \frac{93327}{2977199} a^{10} - \frac{638674}{2977199} a^{9} + \frac{1066907}{2977199} a^{8} + \frac{468068}{2977199} a^{7} + \frac{1260686}{2977199} a^{6} - \frac{671989}{2977199} a^{5} + \frac{1325585}{2977199} a^{4} - \frac{448522}{2977199} a^{3} - \frac{141264}{2977199} a^{2} + \frac{1456704}{2977199} a + \frac{117612}{2977199}$, $\frac{1}{2977199} a^{26} - \frac{1114526}{2977199} a^{14} - \frac{803496}{2977199} a^{13} - \frac{1446575}{2977199} a^{12} - \frac{66102}{2977199} a^{11} + \frac{670900}{2977199} a^{10} - \frac{671255}{2977199} a^{9} - \frac{503068}{2977199} a^{8} - \frac{491388}{2977199} a^{7} + \frac{1025023}{2977199} a^{6} + \frac{1215469}{2977199} a^{5} - \frac{865347}{2977199} a^{4} + \frac{311887}{2977199} a^{3} + \frac{1025023}{2977199} a^{2} + \frac{503599}{2977199} a + \frac{1207283}{2977199}$, $\frac{1}{2977199} a^{27} - \frac{638533}{2977199} a^{14} - \frac{987862}{2977199} a^{13} - \frac{1040150}{2977199} a^{12} + \frac{1345663}{2977199} a^{11} + \frac{514542}{2977199} a^{10} + \frac{754602}{2977199} a^{9} - \frac{991112}{2977199} a^{8} - \frac{1359831}{2977199} a^{7} - \frac{258329}{2977199} a^{6} - \frac{571737}{2977199} a^{5} + \frac{282284}{2977199} a^{4} - \frac{293610}{2977199} a^{3} + \frac{1270101}{2977199} a^{2} - \frac{814427}{2977199} a - \frac{1066004}{2977199}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $27$
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:  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:  \( 1525104928454641200 \) (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

\(\Q(\sqrt{29}) \), 4.4.2951069.1, 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 $28$ $28$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
11Data not computed
29Data not computed