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Properties

Label 24.0.79022433020...7792.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 7^{20}$
Root discriminant $74.26$
Ramified primes $2, 7$
Class number $52292$ (GRH)
Class group $[2, 26146]$ (GRH)
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![19208, 0, 307328, 0, 1613472, 0, 4006240, 0, 5419400, 0, 4313568, 0, 2124248, 0, 665616, 0, 133770, 0, 17024, 0, 1316, 0, 56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 56*x^22 + 1316*x^20 + 17024*x^18 + 133770*x^16 + 665616*x^14 + 2124248*x^12 + 4313568*x^10 + 5419400*x^8 + 4006240*x^6 + 1613472*x^4 + 307328*x^2 + 19208)
 
gp: K = bnfinit(x^24 + 56*x^22 + 1316*x^20 + 17024*x^18 + 133770*x^16 + 665616*x^14 + 2124248*x^12 + 4313568*x^10 + 5419400*x^8 + 4006240*x^6 + 1613472*x^4 + 307328*x^2 + 19208, 1)
 

Normalized defining polynomial

\( x^{24} + 56 x^{22} + 1316 x^{20} + 17024 x^{18} + 133770 x^{16} + 665616 x^{14} + 2124248 x^{12} + 4313568 x^{10} + 5419400 x^{8} + 4006240 x^{6} + 1613472 x^{4} + 307328 x^{2} + 19208 \)

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:  \(790224330201082600125157415256880139617697792=2^{93}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.26$
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:  $2, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(224=2^{5}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{224}(1,·)$, $\chi_{224}(5,·)$, $\chi_{224}(193,·)$, $\chi_{224}(9,·)$, $\chi_{224}(13,·)$, $\chi_{224}(45,·)$, $\chi_{224}(81,·)$, $\chi_{224}(213,·)$, $\chi_{224}(121,·)$, $\chi_{224}(25,·)$, $\chi_{224}(157,·)$, $\chi_{224}(69,·)$, $\chi_{224}(101,·)$, $\chi_{224}(65,·)$, $\chi_{224}(113,·)$, $\chi_{224}(169,·)$, $\chi_{224}(173,·)$, $\chi_{224}(125,·)$, $\chi_{224}(177,·)$, $\chi_{224}(181,·)$, $\chi_{224}(137,·)$, $\chi_{224}(57,·)$, $\chi_{224}(61,·)$, $\chi_{224}(117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{14} a^{8}$, $\frac{1}{14} a^{9}$, $\frac{1}{14} a^{10}$, $\frac{1}{14} a^{11}$, $\frac{1}{98} a^{12}$, $\frac{1}{98} a^{13}$, $\frac{1}{98} a^{14}$, $\frac{1}{98} a^{15}$, $\frac{1}{196} a^{16}$, $\frac{1}{196} a^{17}$, $\frac{1}{1372} a^{18}$, $\frac{1}{1372} a^{19}$, $\frac{1}{155036} a^{20} + \frac{25}{155036} a^{18} - \frac{9}{5537} a^{16} + \frac{27}{11074} a^{14} + \frac{53}{11074} a^{12} + \frac{22}{791} a^{10} - \frac{27}{1582} a^{8} - \frac{48}{791} a^{6} + \frac{50}{113} a^{4} - \frac{53}{113} a^{2} + \frac{7}{113}$, $\frac{1}{155036} a^{21} + \frac{25}{155036} a^{19} - \frac{9}{5537} a^{17} + \frac{27}{11074} a^{15} + \frac{53}{11074} a^{13} + \frac{22}{791} a^{11} - \frac{27}{1582} a^{9} - \frac{48}{791} a^{7} + \frac{50}{113} a^{5} - \frac{53}{113} a^{3} + \frac{7}{113} a$, $\frac{1}{3594199588} a^{22} + \frac{3}{9263401} a^{20} - \frac{494063}{3594199588} a^{18} - \frac{93010}{128364271} a^{16} - \frac{152879}{36675506} a^{14} - \frac{5314}{128364271} a^{12} - \frac{9847}{378098} a^{10} + \frac{551}{162281} a^{8} + \frac{16454}{2619679} a^{6} - \frac{961572}{2619679} a^{4} + \frac{603289}{2619679} a^{2} + \frac{727670}{2619679}$, $\frac{1}{3594199588} a^{23} + \frac{3}{9263401} a^{21} - \frac{494063}{3594199588} a^{19} - \frac{93010}{128364271} a^{17} - \frac{152879}{36675506} a^{15} - \frac{5314}{128364271} a^{13} - \frac{9847}{378098} a^{11} + \frac{551}{162281} a^{9} + \frac{16454}{2619679} a^{7} - \frac{961572}{2619679} a^{5} + \frac{603289}{2619679} a^{3} + \frac{727670}{2619679} a$

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

Class group and class number

$C_{2}\times C_{26146}$, which has order $52292$ (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:  $11$
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:  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:  \( 39019312.21180779 \) (assuming GRH)
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, 8.0.5156108238848.1, 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 $24$ $24$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ $24$ $24$

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
$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}$