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Properties

Label 24.0.58876300618...4464.2
Degree $24$
Signature $[0, 12]$
Discriminant $2^{66}\cdot 7^{20}$
Root discriminant $34.05$
Ramified primes $2, 7$
Class number $52$ (GRH)
Class group $[52]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 128, 0, 224, 0, 384, 0, 656, 0, 1120, 0, 1912, 0, 560, 0, 164, 0, 48, 0, 14, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 4*x^22 + 14*x^20 + 48*x^18 + 164*x^16 + 560*x^14 + 1912*x^12 + 1120*x^10 + 656*x^8 + 384*x^6 + 224*x^4 + 128*x^2 + 64)
 
gp: K = bnfinit(x^24 + 4*x^22 + 14*x^20 + 48*x^18 + 164*x^16 + 560*x^14 + 1912*x^12 + 1120*x^10 + 656*x^8 + 384*x^6 + 224*x^4 + 128*x^2 + 64, 1)
 

Normalized defining polynomial

\( x^{24} + 4 x^{22} + 14 x^{20} + 48 x^{18} + 164 x^{16} + 560 x^{14} + 1912 x^{12} + 1120 x^{10} + 656 x^{8} + 384 x^{6} + 224 x^{4} + 128 x^{2} + 64 \)

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:  \(5887630061813314259984772021002174464=2^{66}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.05$
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:  \(112=2^{4}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(97,·)$, $\chi_{112}(5,·)$, $\chi_{112}(65,·)$, $\chi_{112}(9,·)$, $\chi_{112}(13,·)$, $\chi_{112}(109,·)$, $\chi_{112}(17,·)$, $\chi_{112}(85,·)$, $\chi_{112}(89,·)$, $\chi_{112}(25,·)$, $\chi_{112}(101,·)$, $\chi_{112}(29,·)$, $\chi_{112}(69,·)$, $\chi_{112}(33,·)$, $\chi_{112}(37,·)$, $\chi_{112}(81,·)$, $\chi_{112}(41,·)$, $\chi_{112}(45,·)$, $\chi_{112}(93,·)$, $\chi_{112}(53,·)$, $\chi_{112}(73,·)$, $\chi_{112}(57,·)$, $\chi_{112}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{1912} a^{14} - \frac{99}{239}$, $\frac{1}{1912} a^{15} - \frac{99}{239} a$, $\frac{1}{3824} a^{16} + \frac{70}{239} a^{2}$, $\frac{1}{3824} a^{17} + \frac{70}{239} a^{3}$, $\frac{1}{3824} a^{18} - \frac{99}{478} a^{4}$, $\frac{1}{3824} a^{19} - \frac{99}{478} a^{5}$, $\frac{1}{7648} a^{20} + \frac{35}{239} a^{6}$, $\frac{1}{7648} a^{21} + \frac{35}{239} a^{7}$, $\frac{1}{7648} a^{22} - \frac{99}{956} a^{8}$, $\frac{1}{7648} a^{23} - \frac{99}{956} a^{9}$

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

Class group and class number

$C_{52}$, which has order $52$ (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:  \( -\frac{41}{3824} a^{22} - \frac{287}{7648} a^{20} - \frac{123}{956} a^{18} - \frac{1681}{3824} a^{16} - \frac{1435}{956} a^{14} - \frac{41}{8} a^{12} - \frac{35}{2} a^{10} - \frac{1681}{956} a^{8} - \frac{246}{239} a^{6} - \frac{287}{478} a^{4} - \frac{82}{239} a^{2} - \frac{41}{239} \) (order $14$)
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

$C_2\times C_{12}$ (as 24T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\zeta_{16})^+\), 4.0.100352.5, 6.0.8605184.1, 6.6.1229312.1, \(\Q(\zeta_{7})\), 8.0.10070523904.2, 12.0.74049191673856.2, 12.12.49519263525896192.1, 12.0.2426443912768913408.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$2$2.12.33.375$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
2.12.33.375$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$