Properties

Label 24.0.572...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $5.726\times 10^{29}$
Root discriminant $17.37$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1)
 
gp: K = bnfinit(x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1]);
 

\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \)

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

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(572565594852444156646728515625\)\(\medspace = 3^{36}\cdot 5^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.37$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $24$
This field is Galois and abelian over $\Q$.
Conductor:  \(45=3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{45}(1,·)$, $\chi_{45}(2,·)$, $\chi_{45}(4,·)$, $\chi_{45}(7,·)$, $\chi_{45}(8,·)$, $\chi_{45}(11,·)$, $\chi_{45}(13,·)$, $\chi_{45}(14,·)$, $\chi_{45}(16,·)$, $\chi_{45}(17,·)$, $\chi_{45}(19,·)$, $\chi_{45}(22,·)$, $\chi_{45}(23,·)$, $\chi_{45}(26,·)$, $\chi_{45}(28,·)$, $\chi_{45}(29,·)$, $\chi_{45}(31,·)$, $\chi_{45}(32,·)$, $\chi_{45}(34,·)$, $\chi_{45}(37,·)$, $\chi_{45}(38,·)$, $\chi_{45}(41,·)$, $\chi_{45}(43,·)$, $\chi_{45}(44,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -a \) (order $90$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{9} + 1 \),  \( a^{6} - 1 \),  \( a^{12} - 1 \),  \( a^{22} - a^{17} - a^{2} \),  \( a^{20} - a^{15} + a^{5} \),  \( a^{2} - 1 \),  \( a - 1 \),  \( a^{22} + a^{20} - a^{16} + a^{13} + a^{12} - a^{10} + a^{7} + a^{4} - a \),  \( a^{4} - 1 \),  \( a^{21} - a^{18} - a^{14} - a^{9} + a^{6} - 1 \),  \( a^{22} - a^{18} - a^{16} + a^{13} + a^{4} - a \) (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2124832.236129185 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 2124832.236129185 \cdot 1}{90\sqrt{572565594852444156646728515625}}\approx 0.118121222580884$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 6.0.2460375.1, 6.6.820125.1, \(\Q(\zeta_{9})\), \(\Q(\zeta_{15})\), 12.0.6053445140625.1, \(\Q(\zeta_{45})^+\), 12.0.84075626953125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed