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Properties

Label 24.0.12733857775...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{48}\cdot 3^{32}\cdot 5^{12}$
Root discriminant $38.70$
Ramified primes $2, 3, 5$
Class number $63$ (GRH)
Class group $[3, 21]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 483, 0, 0, 0, 5607, 0, 0, 0, 9016, 0, 0, 0, 3567, 0, 0, 0, 126, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 126*x^20 + 3567*x^16 + 9016*x^12 + 5607*x^8 + 483*x^4 + 1)
 
gp: K = bnfinit(x^24 + 126*x^20 + 3567*x^16 + 9016*x^12 + 5607*x^8 + 483*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{24} + 126 x^{20} + 3567 x^{16} + 9016 x^{12} + 5607 x^{8} + 483 x^{4} + 1 \)

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:  \(127338577759142414150270976000000000000=2^{48}\cdot 3^{32}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.70$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(360=2^{3}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{360}(1,·)$, $\chi_{360}(259,·)$, $\chi_{360}(199,·)$, $\chi_{360}(139,·)$, $\chi_{360}(109,·)$, $\chi_{360}(271,·)$, $\chi_{360}(19,·)$, $\chi_{360}(151,·)$, $\chi_{360}(331,·)$, $\chi_{360}(79,·)$, $\chi_{360}(349,·)$, $\chi_{360}(31,·)$, $\chi_{360}(289,·)$, $\chi_{360}(91,·)$, $\chi_{360}(229,·)$, $\chi_{360}(241,·)$, $\chi_{360}(169,·)$, $\chi_{360}(301,·)$, $\chi_{360}(49,·)$, $\chi_{360}(211,·)$, $\chi_{360}(181,·)$, $\chi_{360}(121,·)$, $\chi_{360}(61,·)$, $\chi_{360}(319,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{9} a^{12} - \frac{1}{3} a^{4} - \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{3} a^{5} - \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{3} a^{6} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{3} a^{7} - \frac{1}{9} a^{3}$, $\frac{1}{477} a^{16} - \frac{23}{477} a^{12} - \frac{10}{159} a^{8} + \frac{32}{477} a^{4} - \frac{229}{477}$, $\frac{1}{477} a^{17} - \frac{23}{477} a^{13} - \frac{10}{159} a^{9} + \frac{32}{477} a^{5} - \frac{229}{477} a$, $\frac{1}{477} a^{18} - \frac{23}{477} a^{14} - \frac{10}{159} a^{10} + \frac{32}{477} a^{6} - \frac{229}{477} a^{2}$, $\frac{1}{477} a^{19} - \frac{23}{477} a^{15} - \frac{10}{159} a^{11} + \frac{32}{477} a^{7} - \frac{229}{477} a^{3}$, $\frac{1}{210415671} a^{20} + \frac{26320}{210415671} a^{16} + \frac{5688997}{210415671} a^{12} + \frac{58299548}{210415671} a^{8} + \frac{46834}{2963601} a^{4} - \frac{7902811}{210415671}$, $\frac{1}{210415671} a^{21} + \frac{26320}{210415671} a^{17} + \frac{5688997}{210415671} a^{13} + \frac{58299548}{210415671} a^{9} + \frac{46834}{2963601} a^{5} - \frac{7902811}{210415671} a$, $\frac{1}{210415671} a^{22} + \frac{26320}{210415671} a^{18} + \frac{5688997}{210415671} a^{14} + \frac{58299548}{210415671} a^{10} + \frac{46834}{2963601} a^{6} - \frac{7902811}{210415671} a^{2}$, $\frac{1}{210415671} a^{23} + \frac{26320}{210415671} a^{19} + \frac{5688997}{210415671} a^{15} + \frac{58299548}{210415671} a^{11} + \frac{46834}{2963601} a^{7} - \frac{7902811}{210415671} a^{3}$

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

Class group and class number

$C_{3}\times C_{21}$, which has order $63$ (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:  \( \frac{2610308}{11074509} a^{23} + \frac{6205096}{208953} a^{19} + \frac{9307358687}{11074509} a^{15} + \frac{23433189976}{11074509} a^{11} + \frac{202920278}{155979} a^{7} + \frac{1150393573}{11074509} a^{3} \) (order $8$)
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:  \( 81723202.08229 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

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 $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 6.0.419904.1, 6.6.3359232.1, 6.0.3359232.1, 6.6.419904000.1, 6.0.419904000.3, 6.6.820125.1, 6.0.52488000.1, 8.0.40960000.1, 12.0.722204136308736.1, 12.0.11284439629824000000.2, 12.0.2754990144000000.1, 12.12.176319369216000000.1, 12.0.11284439629824000000.1, 12.0.11284439629824000000.3, 12.0.176319369216000000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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
$3$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$