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Properties

Label 24.24.1835142308...1072.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{93}\cdot 3^{32}$
Root discriminant $63.48$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (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![8, 0, -576, 0, 8592, 0, -54976, 0, 183060, 0, -336864, 0, 341336, 0, -184752, 0, 55620, 0, -9584, 0, 936, 0, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8)
 
gp: K = bnfinit(x^24 - 48*x^22 + 936*x^20 - 9584*x^18 + 55620*x^16 - 184752*x^14 + 341336*x^12 - 336864*x^10 + 183060*x^8 - 54976*x^6 + 8592*x^4 - 576*x^2 + 8, 1)
 

Normalized defining polynomial

\( x^{24} - 48 x^{22} + 936 x^{20} - 9584 x^{18} + 55620 x^{16} - 184752 x^{14} + 341336 x^{12} - 336864 x^{10} + 183060 x^{8} - 54976 x^{6} + 8592 x^{4} - 576 x^{2} + 8 \)

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:  \(18351423083070806589199715754737431920771072=2^{93}\cdot 3^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.48$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(288=2^{5}\cdot 3^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{288}(1,·)$, $\chi_{288}(133,·)$, $\chi_{288}(193,·)$, $\chi_{288}(265,·)$, $\chi_{288}(13,·)$, $\chi_{288}(205,·)$, $\chi_{288}(145,·)$, $\chi_{288}(277,·)$, $\chi_{288}(217,·)$, $\chi_{288}(25,·)$, $\chi_{288}(157,·)$, $\chi_{288}(229,·)$, $\chi_{288}(97,·)$, $\chi_{288}(37,·)$, $\chi_{288}(241,·)$, $\chi_{288}(169,·)$, $\chi_{288}(109,·)$, $\chi_{288}(253,·)$, $\chi_{288}(49,·)$, $\chi_{288}(181,·)$, $\chi_{288}(73,·)$, $\chi_{288}(121,·)$, $\chi_{288}(61,·)$, $\chi_{288}(85,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{34} a^{14} - \frac{3}{17} a^{12} + \frac{1}{17} a^{10} + \frac{5}{34} a^{8} + \frac{2}{17} a^{6} + \frac{5}{17} a^{4} + \frac{4}{17} a^{2} - \frac{7}{17}$, $\frac{1}{34} a^{15} - \frac{3}{17} a^{13} + \frac{1}{17} a^{11} + \frac{5}{34} a^{9} + \frac{2}{17} a^{7} + \frac{5}{17} a^{5} + \frac{4}{17} a^{3} - \frac{7}{17} a$, $\frac{1}{68} a^{16} - \frac{4}{17}$, $\frac{1}{68} a^{17} - \frac{4}{17} a$, $\frac{1}{68} a^{18} - \frac{4}{17} a^{2}$, $\frac{1}{68} a^{19} - \frac{4}{17} a^{3}$, $\frac{1}{68} a^{20} - \frac{4}{17} a^{4}$, $\frac{1}{68} a^{21} - \frac{4}{17} a^{5}$, $\frac{1}{29305626052} a^{22} + \frac{12782034}{7326406513} a^{20} + \frac{90656021}{14652813026} a^{18} - \frac{38946948}{7326406513} a^{16} - \frac{31856446}{7326406513} a^{14} - \frac{357119348}{7326406513} a^{12} + \frac{1518502401}{14652813026} a^{10} - \frac{59039979}{7326406513} a^{8} + \frac{139399973}{7326406513} a^{6} - \frac{3436165571}{7326406513} a^{4} + \frac{3159080086}{7326406513} a^{2} - \frac{1618059094}{7326406513}$, $\frac{1}{29305626052} a^{23} + \frac{12782034}{7326406513} a^{21} + \frac{90656021}{14652813026} a^{19} - \frac{38946948}{7326406513} a^{17} - \frac{31856446}{7326406513} a^{15} - \frac{357119348}{7326406513} a^{13} + \frac{1518502401}{14652813026} a^{11} - \frac{59039979}{7326406513} a^{9} + \frac{139399973}{7326406513} a^{7} - \frac{3436165571}{7326406513} a^{5} + \frac{3159080086}{7326406513} a^{3} - \frac{1618059094}{7326406513} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 71277883593115.94 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{16})^+\), 6.6.3359232.1, \(\Q(\zeta_{32})^+\), 12.12.369768517790072832.1

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 R $24$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ $24$

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
3Data not computed