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Properties

Label 24.0.41995760826...8272.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 3^{12}\cdot 7^{20}$
Root discriminant $128.62$
Ramified primes $2, 3, 7$
Class number Not computed
Class group Not computed
Galois group $C_{24}$ (as 24T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10207918728, 0, 54442233216, 0, 95273908128, 0, 78854821920, 0, 35556683400, 0, 9433773216, 0, 1548576792, 0, 161744688, 0, 10835370, 0, 459648, 0, 11844, 0, 168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 168*x^22 + 11844*x^20 + 459648*x^18 + 10835370*x^16 + 161744688*x^14 + 1548576792*x^12 + 9433773216*x^10 + 35556683400*x^8 + 78854821920*x^6 + 95273908128*x^4 + 54442233216*x^2 + 10207918728)
 
gp: K = bnfinit(x^24 + 168*x^22 + 11844*x^20 + 459648*x^18 + 10835370*x^16 + 161744688*x^14 + 1548576792*x^12 + 9433773216*x^10 + 35556683400*x^8 + 78854821920*x^6 + 95273908128*x^4 + 54442233216*x^2 + 10207918728, 1)
 

Normalized defining polynomial

\( x^{24} + 168 x^{22} + 11844 x^{20} + 459648 x^{18} + 10835370 x^{16} + 161744688 x^{14} + 1548576792 x^{12} + 9433773216 x^{10} + 35556683400 x^{8} + 78854821920 x^{6} + 95273908128 x^{4} + 54442233216 x^{2} + 10207918728 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(419957608266393538093113781921531638278568932278272=2^{93}\cdot 3^{12}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $128.62$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(83,·)$, $\chi_{672}(131,·)$, $\chi_{672}(227,·)$, $\chi_{672}(193,·)$, $\chi_{672}(457,·)$, $\chi_{672}(395,·)$, $\chi_{672}(635,·)$, $\chi_{672}(529,·)$, $\chi_{672}(467,·)$, $\chi_{672}(251,·)$, $\chi_{672}(121,·)$, $\chi_{672}(25,·)$, $\chi_{672}(289,·)$, $\chi_{672}(587,·)$, $\chi_{672}(419,·)$, $\chi_{672}(337,·)$, $\chi_{672}(169,·)$, $\chi_{672}(299,·)$, $\chi_{672}(625,·)$, $\chi_{672}(563,·)$, $\chi_{672}(361,·)$, $\chi_{672}(505,·)$, $\chi_{672}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{189} a^{6}$, $\frac{1}{189} a^{7}$, $\frac{1}{1134} a^{8}$, $\frac{1}{1134} a^{9}$, $\frac{1}{3402} a^{10}$, $\frac{1}{3402} a^{11}$, $\frac{1}{71442} a^{12}$, $\frac{1}{71442} a^{13}$, $\frac{1}{214326} a^{14}$, $\frac{1}{214326} a^{15}$, $\frac{1}{1285956} a^{16}$, $\frac{1}{1285956} a^{17}$, $\frac{1}{27005076} a^{18}$, $\frac{1}{27005076} a^{19}$, $\frac{1}{9154720764} a^{20} + \frac{25}{3051573588} a^{18} - \frac{1}{4036473} a^{16} + \frac{1}{896994} a^{14} + \frac{53}{8072946} a^{12} + \frac{22}{192213} a^{10} - \frac{1}{4746} a^{8} - \frac{16}{7119} a^{6} + \frac{50}{1017} a^{4} - \frac{53}{339} a^{2} + \frac{7}{113}$, $\frac{1}{9154720764} a^{21} + \frac{25}{3051573588} a^{19} - \frac{1}{4036473} a^{17} + \frac{1}{896994} a^{15} + \frac{53}{8072946} a^{13} + \frac{22}{192213} a^{11} - \frac{1}{4746} a^{9} - \frac{16}{7119} a^{7} + \frac{50}{1017} a^{5} - \frac{53}{339} a^{3} + \frac{7}{113} a$, $\frac{1}{636701674415436} a^{22} + \frac{1}{182331521883} a^{20} - \frac{494063}{70744630490604} a^{18} - \frac{93010}{842197982031} a^{16} - \frac{152879}{80209331622} a^{14} - \frac{5314}{93577553559} a^{12} - \frac{9847}{91877814} a^{10} + \frac{551}{13144761} a^{8} + \frac{16454}{70731333} a^{6} - \frac{320524}{7859037} a^{4} + \frac{603289}{7859037} a^{2} + \frac{727670}{2619679}$, $\frac{1}{636701674415436} a^{23} + \frac{1}{182331521883} a^{21} - \frac{494063}{70744630490604} a^{19} - \frac{93010}{842197982031} a^{17} - \frac{152879}{80209331622} a^{15} - \frac{5314}{93577553559} a^{13} - \frac{9847}{91877814} a^{11} + \frac{551}{13144761} a^{9} + \frac{16454}{70731333} a^{7} - \frac{320524}{7859037} a^{5} + \frac{603289}{7859037} a^{3} + \frac{727670}{2619679} a$

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:  $11$
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

$C_{24}$ (as 24T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 24
The 24 conjugacy class representatives for $C_{24}$
Character table for $C_{24}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{16})^+\), 6.6.1229312.1, 8.0.417644767346688.52, 12.12.49519263525896192.1

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 R $24$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ $24$ $24$

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
3Data not computed
$7$7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$
7.12.10.5$x^{12} + 56 x^{6} + 1323$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$