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Properties

Label 24.0.11859329208...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{32}\cdot 5^{18}$
Root discriminant $28.93$
Ramified primes $2, 3, 5$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -9, 0, 75, 0, -622, 0, 5157, 0, -3582, 0, 1918, 0, -927, 0, 417, 0, -109, 0, 27, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^22 + 27*x^20 - 109*x^18 + 417*x^16 - 927*x^14 + 1918*x^12 - 3582*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1)
 
gp: K = bnfinit(x^24 - 6*x^22 + 27*x^20 - 109*x^18 + 417*x^16 - 927*x^14 + 1918*x^12 - 3582*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 6 x^{22} + 27 x^{20} - 109 x^{18} + 417 x^{16} - 927 x^{14} + 1918 x^{12} - 3582 x^{10} + 5157 x^{8} - 622 x^{6} + 75 x^{4} - 9 x^{2} + 1 \)

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:  \(118593292086517824000000000000000000=2^{24}\cdot 3^{32}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.93$
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:  \(180=2^{2}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(67,·)$, $\chi_{180}(133,·)$, $\chi_{180}(7,·)$, $\chi_{180}(73,·)$, $\chi_{180}(139,·)$, $\chi_{180}(13,·)$, $\chi_{180}(79,·)$, $\chi_{180}(19,·)$, $\chi_{180}(151,·)$, $\chi_{180}(91,·)$, $\chi_{180}(157,·)$, $\chi_{180}(31,·)$, $\chi_{180}(97,·)$, $\chi_{180}(163,·)$, $\chi_{180}(37,·)$, $\chi_{180}(103,·)$, $\chi_{180}(169,·)$, $\chi_{180}(43,·)$, $\chi_{180}(109,·)$, $\chi_{180}(49,·)$, $\chi_{180}(121,·)$, $\chi_{180}(61,·)$, $\chi_{180}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2082671} a^{18} - \frac{305717}{2082671} a^{16} + \frac{940293}{2082671} a^{14} - \frac{807635}{2082671} a^{12} + \frac{854232}{2082671} a^{10} - \frac{439510}{2082671} a^{8} + \frac{76434}{2082671} a^{6} + \frac{395442}{2082671} a^{4} - \frac{538377}{2082671} a^{2} - \frac{405150}{2082671}$, $\frac{1}{2082671} a^{19} - \frac{305717}{2082671} a^{17} + \frac{940293}{2082671} a^{15} - \frac{807635}{2082671} a^{13} + \frac{854232}{2082671} a^{11} - \frac{439510}{2082671} a^{9} + \frac{76434}{2082671} a^{7} + \frac{395442}{2082671} a^{5} - \frac{538377}{2082671} a^{3} - \frac{405150}{2082671} a$, $\frac{1}{2082671} a^{20} + \frac{440131}{2082671} a^{10} - \frac{632838}{2082671}$, $\frac{1}{2082671} a^{21} + \frac{440131}{2082671} a^{11} - \frac{632838}{2082671} a$, $\frac{1}{2082671} a^{22} + \frac{440131}{2082671} a^{12} - \frac{632838}{2082671} a^{2}$, $\frac{1}{2082671} a^{23} + \frac{440131}{2082671} a^{13} - \frac{632838}{2082671} a^{3}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{785142}{2082671} a^{23} - \frac{4623614}{2082671} a^{21} + \frac{20675406}{2082671} a^{19} - \frac{83224782}{2082671} a^{17} + \frac{317895272}{2082671} a^{15} - \frac{691448388}{2082671} a^{13} + \frac{1425032730}{2082671} a^{11} - \frac{2645056160}{2082671} a^{9} + \frac{3736613901}{2082671} a^{7} - \frac{38471958}{2082671} a^{5} + \frac{4623614}{2082671} a^{3} - \frac{523428}{2082671} a \) (order $20$)
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:  \( 68395599.77210574 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), 6.0.52488000.1, 6.6.820125.1, 6.0.419904.1, \(\Q(\zeta_{20})\), 12.0.2754990144000000.1, 12.0.84075626953125.1, 12.12.344373768000000000.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$