↑

Properties

Label 28.0.12021339863...4701.1
Degree $28$
Signature $[0, 14]$
Discriminant $13^{14}\cdot 29^{27}$
Root discriminant $92.71$
Ramified primes $13, 29$
Class number Not computed
Class group Not computed
Galois group $C_{28}$ (as 28T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32010844381, -31872138280, 31872138280, -30253900435, 30253900435, -24644009239, 24644009239, -15561328255, 15561328255, -7235537353, 7235537353, -2441900167, 2441900167, -598193557, 598193557, -106538461, 106538461, -13750612, 13750612, -1270375, 1270375, -81781, 81781, -3481, 3481, -88, 88, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 88*x^26 - 88*x^25 + 3481*x^24 - 3481*x^23 + 81781*x^22 - 81781*x^21 + 1270375*x^20 - 1270375*x^19 + 13750612*x^18 - 13750612*x^17 + 106538461*x^16 - 106538461*x^15 + 598193557*x^14 - 598193557*x^13 + 2441900167*x^12 - 2441900167*x^11 + 7235537353*x^10 - 7235537353*x^9 + 15561328255*x^8 - 15561328255*x^7 + 24644009239*x^6 - 24644009239*x^5 + 30253900435*x^4 - 30253900435*x^3 + 31872138280*x^2 - 31872138280*x + 32010844381)
 
gp: K = bnfinit(x^28 - x^27 + 88*x^26 - 88*x^25 + 3481*x^24 - 3481*x^23 + 81781*x^22 - 81781*x^21 + 1270375*x^20 - 1270375*x^19 + 13750612*x^18 - 13750612*x^17 + 106538461*x^16 - 106538461*x^15 + 598193557*x^14 - 598193557*x^13 + 2441900167*x^12 - 2441900167*x^11 + 7235537353*x^10 - 7235537353*x^9 + 15561328255*x^8 - 15561328255*x^7 + 24644009239*x^6 - 24644009239*x^5 + 30253900435*x^4 - 30253900435*x^3 + 31872138280*x^2 - 31872138280*x + 32010844381, 1)
 

Normalized defining polynomial

\( x^{28} - x^{27} + 88 x^{26} - 88 x^{25} + 3481 x^{24} - 3481 x^{23} + 81781 x^{22} - 81781 x^{21} + 1270375 x^{20} - 1270375 x^{19} + 13750612 x^{18} - 13750612 x^{17} + 106538461 x^{16} - 106538461 x^{15} + 598193557 x^{14} - 598193557 x^{13} + 2441900167 x^{12} - 2441900167 x^{11} + 7235537353 x^{10} - 7235537353 x^{9} + 15561328255 x^{8} - 15561328255 x^{7} + 24644009239 x^{6} - 24644009239 x^{5} + 30253900435 x^{4} - 30253900435 x^{3} + 31872138280 x^{2} - 31872138280 x + 32010844381 \)

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:  \(12021339863667063616623959462764825581356981533355604701=13^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.71$
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$$\chi_{377}(1,·)$, $\chi_{377}(259,·)$, $\chi_{377}(196,·)$, $\chi_{377}(326,·)$, $\chi_{377}(311,·)$, $\chi_{377}(12,·)$, $\chi_{377}(77,·)$, $\chi_{377}(142,·)$, $\chi_{377}(144,·)$, $\chi_{377}(337,·)$, $\chi_{377}(274,·)$, $\chi_{377}(339,·)$, $\chi_{377}(90,·)$, $\chi_{377}(155,·)$, $\chi_{377}(220,·)$, $\chi_{377}(350,·)$, $\chi_{377}(352,·)$, $\chi_{377}(272,·)$, $\chi_{377}(209,·)$, $\chi_{377}(92,·)$, $\chi_{377}(170,·)$, $\chi_{377}(363,·)$, $\chi_{377}(53,·)$, $\chi_{377}(246,·)$, $\chi_{377}(183,·)$, $\chi_{377}(248,·)$, $\chi_{377}(313,·)$, $\chi_{377}(298,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{8878212019} a^{15} + \frac{392185080}{8878212019} a^{14} + \frac{45}{8878212019} a^{13} - \frac{1284650678}{8878212019} a^{12} + \frac{810}{8878212019} a^{11} - \frac{3440312149}{8878212019} a^{10} + \frac{7425}{8878212019} a^{9} + \frac{4136398850}{8878212019} a^{8} + \frac{36450}{8878212019} a^{7} - \frac{383548868}{8878212019} a^{6} + \frac{91854}{8878212019} a^{5} - \frac{767097736}{8878212019} a^{4} + \frac{102060}{8878212019} a^{3} - \frac{575323302}{8878212019} a^{2} + \frac{32805}{8878212019} a + \frac{1922620253}{8878212019}$, $\frac{1}{8878212019} a^{16} + \frac{48}{8878212019} a^{14} - \frac{1176555240}{8878212019} a^{13} + \frac{936}{8878212019} a^{12} - \frac{1494594265}{8878212019} a^{11} + \frac{9504}{8878212019} a^{10} + \frac{4215722082}{8878212019} a^{9} + \frac{53460}{8878212019} a^{8} - \frac{1608364278}{8878212019} a^{7} + \frac{163296}{8878212019} a^{6} + \frac{3248937046}{8878212019} a^{5} + \frac{244944}{8878212019} a^{4} - \frac{4004806450}{8878212019} a^{3} + \frac{139968}{8878212019} a^{2} + \frac{820286384}{8878212019} a + \frac{13122}{8878212019}$, $\frac{1}{8878212019} a^{17} - \frac{2245015042}{8878212019} a^{14} - \frac{1224}{8878212019} a^{13} - \frac{1978845854}{8878212019} a^{12} - \frac{29376}{8878212019} a^{11} + \frac{664676873}{8878212019} a^{10} - \frac{302940}{8878212019} a^{9} + \frac{4043367359}{8878212019} a^{8} - \frac{1586304}{8878212019} a^{7} + \frac{3902858672}{8878212019} a^{6} - \frac{4164048}{8878212019} a^{5} - \frac{2696963198}{8878212019} a^{4} - \frac{4758912}{8878212019} a^{3} + \frac{1801168823}{8878212019} a^{2} - \frac{1561518}{8878212019} a - \frac{3503651954}{8878212019}$, $\frac{1}{8878212019} a^{18} - \frac{1377}{8878212019} a^{14} + \frac{1386498827}{8878212019} a^{13} - \frac{35802}{8878212019} a^{12} - \frac{906603002}{8878212019} a^{11} - \frac{408969}{8878212019} a^{10} - \frac{2117473}{8878212019} a^{9} - \frac{2453814}{8878212019} a^{8} + \frac{4220960449}{8878212019} a^{7} - \frac{7807590}{8878212019} a^{6} - \frac{3315860643}{8878212019} a^{5} - \frac{12045996}{8878212019} a^{4} - \frac{859431009}{8878212019} a^{3} - \frac{7026831}{8878212019} a^{2} - \frac{553896749}{8878212019} a - \frac{669222}{8878212019}$, $\frac{1}{8878212019} a^{19} - \frac{145579172}{8878212019} a^{14} + \frac{26163}{8878212019} a^{13} - \frac{3106394827}{8878212019} a^{12} + \frac{706401}{8878212019} a^{11} + \frac{3653271500}{8878212019} a^{10} + \frac{7770411}{8878212019} a^{9} + \frac{230060701}{8878212019} a^{8} + \frac{42384060}{8878212019} a^{7} + \frac{1230069261}{8878212019} a^{6} + \frac{114436962}{8878212019} a^{5} - \frac{645783220}{8878212019} a^{4} + \frac{133509789}{8878212019} a^{3} - \frac{2613213912}{8878212019} a^{2} + \frac{44503263}{8878212019} a + \frac{1740906719}{8878212019}$, $\frac{1}{8878212019} a^{20} + \frac{30780}{8878212019} a^{14} + \frac{3444667913}{8878212019} a^{13} + \frac{900315}{8878212019} a^{12} - \frac{2722567446}{8878212019} a^{11} + \frac{10969992}{8878212019} a^{10} - \frac{1986453517}{8878212019} a^{9} + \frac{68562450}{8878212019} a^{8} - \frac{1579898701}{8878212019} a^{7} + \frac{224386200}{8878212019} a^{6} + \frac{796181054}{8878212019} a^{5} + \frac{353408265}{8878212019} a^{4} + \frac{1948372621}{8878212019} a^{3} + \frac{209427120}{8878212019} a^{2} + \frac{987577957}{8878212019} a + \frac{20194758}{8878212019}$, $\frac{1}{8878212019} a^{21} - \frac{2521960666}{8878212019} a^{14} - \frac{484785}{8878212019} a^{13} + \frac{4147180787}{8878212019} a^{12} - \frac{13961808}{8878212019} a^{11} + \frac{386742090}{8878212019} a^{10} - \frac{159979050}{8878212019} a^{9} + \frac{2502062778}{8878212019} a^{8} - \frac{897544800}{8878212019} a^{7} - \frac{1591647176}{8878212019} a^{6} - \frac{2473857855}{8878212019} a^{5} - \frac{2827283839}{8878212019} a^{4} - \frac{2931979680}{8878212019} a^{3} - \frac{2594164388}{8878212019} a^{2} - \frac{989543142}{8878212019} a + \frac{3909931314}{8878212019}$, $\frac{1}{8878212019} a^{22} - \frac{592515}{8878212019} a^{14} + \frac{2218654510}{8878212019} a^{13} - \frac{18486468}{8878212019} a^{12} + \frac{1186117180}{8878212019} a^{11} - \frac{234635940}{8878212019} a^{10} + \frac{3910859757}{8878212019} a^{9} - \frac{1508373900}{8878212019} a^{8} - \frac{1132616202}{8878212019} a^{7} + \frac{3838871944}{8878212019} a^{6} - \frac{960268823}{8878212019} a^{5} + \frac{815267899}{8878212019} a^{4} + \frac{466764743}{8878212019} a^{3} + \frac{4040445547}{8878212019} a^{2} + \frac{771774383}{8878212019} a - \frac{471211020}{8878212019}$, $\frac{1}{8878212019} a^{23} - \frac{560054596}{8878212019} a^{14} + \frac{8176707}{8878212019} a^{13} - \frac{102909025}{8878212019} a^{12} + \frac{245301210}{8878212019} a^{11} - \frac{4040754597}{8878212019} a^{10} + \frac{2891049975}{8878212019} a^{9} + \frac{2413086503}{8878212019} a^{8} - \frac{1198592363}{8878212019} a^{7} - \frac{3824741500}{8878212019} a^{6} + \frac{1970868595}{8878212019} a^{5} + \frac{3616031408}{8878212019} a^{4} + \frac{2365042314}{8878212019} a^{3} + \frac{914171377}{8878212019} a^{2} + \frac{1209819517}{8878212019} a + \frac{198624367}{8878212019}$, $\frac{1}{8878212019} a^{24} + \frac{10328472}{8878212019} a^{14} - \frac{1535088262}{8878212019} a^{13} + \frac{335675340}{8878212019} a^{12} - \frac{3185344806}{8878212019} a^{11} + \frac{4382223120}{8878212019} a^{10} - \frac{3062975108}{8878212019} a^{9} + \frac{2123703168}{8878212019} a^{8} - \frac{844148981}{8878212019} a^{7} - \frac{56271809}{8878212019} a^{6} - \frac{2367757713}{8878212019} a^{5} - \frac{1689238494}{8878212019} a^{4} + \frac{2157260815}{8878212019} a^{3} - \frac{1830891089}{8878212019} a^{2} + \frac{3768978836}{8878212019} a + \frac{533608091}{8878212019}$, $\frac{1}{8878212019} a^{25} + \frac{1202770709}{8878212019} a^{14} - \frac{129105900}{8878212019} a^{13} - \frac{3612024271}{8878212019} a^{12} - \frac{3983839200}{8878212019} a^{11} + \frac{3214131748}{8878212019} a^{10} - \frac{3539505280}{8878212019} a^{9} + \frac{3701607377}{8878212019} a^{8} - \frac{3644171411}{8878212019} a^{7} + \frac{4294922164}{8878212019} a^{6} - \frac{432019549}{8878212019} a^{5} - \frac{2273803469}{8878212019} a^{4} + \frac{552486852}{8878212019} a^{3} + \frac{445680623}{8878212019} a^{2} - \frac{919859147}{8878212019} a - \frac{1312874477}{8878212019}$, $\frac{1}{8878212019} a^{26} - \frac{167837670}{8878212019} a^{14} + \frac{4410777957}{8878212019} a^{13} + \frac{3267638479}{8878212019} a^{12} - \frac{3305032471}{8878212019} a^{11} - \frac{3745985833}{8878212019} a^{10} - \frac{4267827853}{8878212019} a^{9} - \frac{1298006836}{8878212019} a^{8} + \frac{3913528936}{8878212019} a^{7} + \frac{543659647}{8878212019} a^{6} - \frac{1104143519}{8878212019} a^{5} + \frac{2503845244}{8878212019} a^{4} - \frac{4173505223}{8878212019} a^{3} + \frac{543659647}{8878212019} a^{2} - \frac{3431770786}{8878212019} a - \frac{726733619}{8878212019}$, $\frac{1}{8878212019} a^{27} - \frac{2347029241}{8878212019} a^{14} + \frac{1942121610}{8878212019} a^{13} - \frac{3030150179}{8878212019} a^{12} - \frac{970653418}{8878212019} a^{11} + \frac{267729965}{8878212019} a^{10} + \frac{1947010254}{8878212019} a^{9} - \frac{3368421739}{8878212019} a^{8} + \frac{1138650056}{8878212019} a^{7} + \frac{3824511722}{8878212019} a^{6} - \frac{2389091579}{8878212019} a^{5} - \frac{3194406760}{8878212019} a^{4} + \frac{3985275196}{8878212019} a^{3} - \frac{477893934}{8878212019} a^{2} + \frac{696578951}{8878212019} a - \frac{3080872801}{8878212019}$

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:  $13$
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:  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

\(\Q(\sqrt{29}) \), 4.0.4121741.1, 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 $28$ $28$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$13$13.14.7.2$x^{14} - 48268090 x^{2} + 125497034$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
13.14.7.2$x^{14} - 48268090 x^{2} + 125497034$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed