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Properties

Label 24.24.9249282519...1872.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{24}\cdot 7^{20}\cdot 17^{21}$
Root discriminant $120.76$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (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![11796113, 0, -117961130, 0, 389271729, 0, -610027558, 0, 521209766, 0, -260208375, 0, 79004219, 0, -14954016, 0, 1774290, 0, -129353, 0, 5474, 0, -119, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 119*x^22 + 5474*x^20 - 129353*x^18 + 1774290*x^16 - 14954016*x^14 + 79004219*x^12 - 260208375*x^10 + 521209766*x^8 - 610027558*x^6 + 389271729*x^4 - 117961130*x^2 + 11796113)
 
gp: K = bnfinit(x^24 - 119*x^22 + 5474*x^20 - 129353*x^18 + 1774290*x^16 - 14954016*x^14 + 79004219*x^12 - 260208375*x^10 + 521209766*x^8 - 610027558*x^6 + 389271729*x^4 - 117961130*x^2 + 11796113, 1)
 

Normalized defining polynomial

\( x^{24} - 119 x^{22} + 5474 x^{20} - 129353 x^{18} + 1774290 x^{16} - 14954016 x^{14} + 79004219 x^{12} - 260208375 x^{10} + 521209766 x^{8} - 610027558 x^{6} + 389271729 x^{4} - 117961130 x^{2} + 11796113 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(92492825191368190136484582761598050250102639951872=2^{24}\cdot 7^{20}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.76$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(476=2^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{476}(1,·)$, $\chi_{476}(451,·)$, $\chi_{476}(137,·)$, $\chi_{476}(205,·)$, $\chi_{476}(467,·)$, $\chi_{476}(81,·)$, $\chi_{476}(19,·)$, $\chi_{476}(149,·)$, $\chi_{476}(87,·)$, $\chi_{476}(223,·)$, $\chi_{476}(225,·)$, $\chi_{476}(355,·)$, $\chi_{476}(421,·)$, $\chi_{476}(195,·)$, $\chi_{476}(423,·)$, $\chi_{476}(169,·)$, $\chi_{476}(429,·)$, $\chi_{476}(111,·)$, $\chi_{476}(305,·)$, $\chi_{476}(83,·)$, $\chi_{476}(373,·)$, $\chi_{476}(361,·)$, $\chi_{476}(59,·)$, $\chi_{476}(383,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{119} a^{8}$, $\frac{1}{119} a^{9}$, $\frac{1}{119} a^{10}$, $\frac{1}{119} a^{11}$, $\frac{1}{10829} a^{12} - \frac{1}{13}$, $\frac{1}{10829} a^{13} - \frac{1}{13} a$, $\frac{1}{10829} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{10829} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{184093} a^{16} + \frac{3}{13} a^{4}$, $\frac{1}{184093} a^{17} + \frac{3}{13} a^{5}$, $\frac{1}{5154604} a^{18} - \frac{1}{21658} a^{12} - \frac{23}{364} a^{6} + \frac{15}{52}$, $\frac{1}{5154604} a^{19} - \frac{1}{21658} a^{13} - \frac{23}{364} a^{7} + \frac{15}{52} a$, $\frac{1}{871128076} a^{20} + \frac{15}{435564038} a^{18} - \frac{6}{2393209} a^{16} - \frac{29}{3660202} a^{14} - \frac{58}{1830101} a^{12} + \frac{40}{20111} a^{10} - \frac{1535}{1045772} a^{8} + \frac{1007}{30758} a^{6} - \frac{24}{169} a^{4} + \frac{2671}{8788} a^{2} - \frac{1795}{4394}$, $\frac{1}{871128076} a^{21} + \frac{15}{435564038} a^{19} - \frac{6}{2393209} a^{17} - \frac{29}{3660202} a^{15} - \frac{58}{1830101} a^{13} + \frac{40}{20111} a^{11} - \frac{1535}{1045772} a^{9} + \frac{1007}{30758} a^{7} - \frac{24}{169} a^{5} + \frac{2671}{8788} a^{3} - \frac{1795}{4394} a$, $\frac{1}{267436319332} a^{22} - \frac{38}{66859079833} a^{20} + \frac{44}{734715163} a^{18} + \frac{1861}{2728942034} a^{16} + \frac{2952}{80263001} a^{14} - \frac{90}{3324503} a^{12} + \frac{806285}{321052004} a^{10} + \frac{27207}{11466143} a^{8} - \frac{25118}{363181} a^{6} - \frac{641557}{2697916} a^{4} + \frac{301086}{674479} a^{2} - \frac{21508}{51883}$, $\frac{1}{267436319332} a^{23} - \frac{38}{66859079833} a^{21} + \frac{44}{734715163} a^{19} + \frac{1861}{2728942034} a^{17} + \frac{2952}{80263001} a^{15} - \frac{90}{3324503} a^{13} + \frac{806285}{321052004} a^{11} + \frac{27207}{11466143} a^{9} - \frac{25118}{363181} a^{7} - \frac{641557}{2697916} a^{5} + \frac{301086}{674479} a^{3} - \frac{21508}{51883} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $23$
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:  \( 439337526752110400 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, 8.8.252217127391488.1, 12.12.683635509017782097.1

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 $24$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{24}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
7Data not computed
17Data not computed