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Properties

Label 24.0.14864652697...6832.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 3^{36}$
Root discriminant $76.24$
Ramified primes $2, 3$
Class number $74034$ (GRH)
Class group $[9, 8226]$ (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![648, 0, 15552, 0, 115344, 0, 395712, 0, 737748, 0, 807840, 0, 542232, 0, 227664, 0, 59940, 0, 9744, 0, 936, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 48*x^22 + 936*x^20 + 9744*x^18 + 59940*x^16 + 227664*x^14 + 542232*x^12 + 807840*x^10 + 737748*x^8 + 395712*x^6 + 115344*x^4 + 15552*x^2 + 648)
 
gp: K = bnfinit(x^24 + 48*x^22 + 936*x^20 + 9744*x^18 + 59940*x^16 + 227664*x^14 + 542232*x^12 + 807840*x^10 + 737748*x^8 + 395712*x^6 + 115344*x^4 + 15552*x^2 + 648, 1)
 

Normalized defining polynomial

\( x^{24} + 48 x^{22} + 936 x^{20} + 9744 x^{18} + 59940 x^{16} + 227664 x^{14} + 542232 x^{12} + 807840 x^{10} + 737748 x^{8} + 395712 x^{6} + 115344 x^{4} + 15552 x^{2} + 648 \)

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:  \(1486465269728735333725176976133731985582456832=2^{93}\cdot 3^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.24$
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$$\chi_{288}(1,·)$, $\chi_{288}(5,·)$, $\chi_{288}(193,·)$, $\chi_{288}(265,·)$, $\chi_{288}(269,·)$, $\chi_{288}(77,·)$, $\chi_{288}(145,·)$, $\chi_{288}(149,·)$, $\chi_{288}(217,·)$, $\chi_{288}(25,·)$, $\chi_{288}(29,·)$, $\chi_{288}(197,·)$, $\chi_{288}(97,·)$, $\chi_{288}(101,·)$, $\chi_{288}(241,·)$, $\chi_{288}(169,·)$, $\chi_{288}(173,·)$, $\chi_{288}(221,·)$, $\chi_{288}(49,·)$, $\chi_{288}(53,·)$, $\chi_{288}(73,·)$, $\chi_{288}(121,·)$, $\chi_{288}(125,·)$, $\chi_{288}(245,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{6} a^{8}$, $\frac{1}{6} a^{9}$, $\frac{1}{6} a^{10}$, $\frac{1}{6} a^{11}$, $\frac{1}{18} a^{12}$, $\frac{1}{18} a^{13}$, $\frac{1}{18} a^{14}$, $\frac{1}{18} a^{15}$, $\frac{1}{612} a^{16} + \frac{8}{17}$, $\frac{1}{612} a^{17} + \frac{8}{17} a$, $\frac{1}{1836} a^{18} - \frac{3}{17} a^{2}$, $\frac{1}{1836} a^{19} - \frac{3}{17} a^{3}$, $\frac{1}{1836} a^{20} - \frac{3}{17} a^{4}$, $\frac{1}{1836} a^{21} - \frac{3}{17} a^{5}$, $\frac{1}{8458452} a^{22} - \frac{29}{156638} a^{20} - \frac{29}{313276} a^{18} + \frac{329}{1409742} a^{16} - \frac{148}{41463} a^{14} + \frac{424}{41463} a^{12} + \frac{547}{9214} a^{10} - \frac{758}{13821} a^{8} - \frac{14357}{234957} a^{6} - \frac{25851}{78319} a^{4} - \frac{28659}{78319} a^{2} + \frac{24525}{78319}$, $\frac{1}{8458452} a^{23} - \frac{29}{156638} a^{21} - \frac{29}{313276} a^{19} + \frac{329}{1409742} a^{17} - \frac{148}{41463} a^{15} + \frac{424}{41463} a^{13} + \frac{547}{9214} a^{11} - \frac{758}{13821} a^{9} - \frac{14357}{234957} a^{7} - \frac{25851}{78319} a^{5} - \frac{28659}{78319} a^{3} + \frac{24525}{78319} a$

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

Class group and class number

$C_{9}\times C_{8226}$, which has order $74034$ (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:  \( -1 \) (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:  \( 165705493.8155171 \) (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, 8.0.173946175488.1, 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 $24$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ $24$

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