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Properties

Label 24.0.22071588705...7584.2
Degree $24$
Signature $[0, 12]$
Discriminant $2^{36}\cdot 13^{22}$
Root discriminant $29.69$
Ramified primes $2, 13$
Class number $13$ (GRH)
Class group $[13]$ (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![4096, 0, 2048, 0, 1024, 0, 512, 0, 256, 0, 128, 0, 64, 0, 32, 0, 16, 0, 8, 0, 4, 0, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 2*x^22 + 4*x^20 + 8*x^18 + 16*x^16 + 32*x^14 + 64*x^12 + 128*x^10 + 256*x^8 + 512*x^6 + 1024*x^4 + 2048*x^2 + 4096)
 
gp: K = bnfinit(x^24 + 2*x^22 + 4*x^20 + 8*x^18 + 16*x^16 + 32*x^14 + 64*x^12 + 128*x^10 + 256*x^8 + 512*x^6 + 1024*x^4 + 2048*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{24} + 2 x^{22} + 4 x^{20} + 8 x^{18} + 16 x^{16} + 32 x^{14} + 64 x^{12} + 128 x^{10} + 256 x^{8} + 512 x^{6} + 1024 x^{4} + 2048 x^{2} + 4096 \)

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:  \(220715887053399008657112652614467584=2^{36}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.69$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(104=2^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{104}(1,·)$, $\chi_{104}(5,·)$, $\chi_{104}(97,·)$, $\chi_{104}(9,·)$, $\chi_{104}(77,·)$, $\chi_{104}(17,·)$, $\chi_{104}(21,·)$, $\chi_{104}(89,·)$, $\chi_{104}(25,·)$, $\chi_{104}(101,·)$, $\chi_{104}(29,·)$, $\chi_{104}(69,·)$, $\chi_{104}(33,·)$, $\chi_{104}(37,·)$, $\chi_{104}(81,·)$, $\chi_{104}(41,·)$, $\chi_{104}(45,·)$, $\chi_{104}(93,·)$, $\chi_{104}(49,·)$, $\chi_{104}(53,·)$, $\chi_{104}(73,·)$, $\chi_{104}(57,·)$, $\chi_{104}(61,·)$, $\chi_{104}(85,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$

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

Class group and class number

$C_{13}$, which has order $13$ (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:  \( -\frac{1}{2} a^{2} \) (order $26$)
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:  \( 34260599.803001165 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), 3.3.169.1, \(\Q(\sqrt{2}, \sqrt{13})\), 4.0.140608.2, 4.0.2197.1, 6.6.14623232.1, \(\Q(\zeta_{13})^+\), 6.6.190102016.1, 8.0.19770609664.2, 12.12.36138776487264256.1, 12.0.469804094334435328.1, \(\Q(\zeta_{13})\)

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
2Data not computed
13Data not computed