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Properties

Label 24.0.96060566590...0000.2
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{36}\cdot 5^{18}$
Root discriminant $34.75$
Ramified primes $2, 3, 5$
Class number $50$ (GRH)
Class group $[5, 10]$ (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![81, 0, 243, 0, 567, 0, 1242, 0, 2673, 0, 2106, 0, 1386, 0, 837, 0, 441, 0, 111, 0, 27, 0, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 6*x^22 + 27*x^20 + 111*x^18 + 441*x^16 + 837*x^14 + 1386*x^12 + 2106*x^10 + 2673*x^8 + 1242*x^6 + 567*x^4 + 243*x^2 + 81)
 
gp: K = bnfinit(x^24 + 6*x^22 + 27*x^20 + 111*x^18 + 441*x^16 + 837*x^14 + 1386*x^12 + 2106*x^10 + 2673*x^8 + 1242*x^6 + 567*x^4 + 243*x^2 + 81, 1)
 

Normalized defining polynomial

\( x^{24} + 6 x^{22} + 27 x^{20} + 111 x^{18} + 441 x^{16} + 837 x^{14} + 1386 x^{12} + 2106 x^{10} + 2673 x^{8} + 1242 x^{6} + 567 x^{4} + 243 x^{2} + 81 \)

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:  \(9606056659007943744000000000000000000=2^{24}\cdot 3^{36}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.75$
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$$\chi_{180}(1,·)$, $\chi_{180}(131,·)$, $\chi_{180}(133,·)$, $\chi_{180}(71,·)$, $\chi_{180}(73,·)$, $\chi_{180}(11,·)$, $\chi_{180}(13,·)$, $\chi_{180}(143,·)$, $\chi_{180}(83,·)$, $\chi_{180}(23,·)$, $\chi_{180}(157,·)$, $\chi_{180}(97,·)$, $\chi_{180}(37,·)$, $\chi_{180}(167,·)$, $\chi_{180}(169,·)$, $\chi_{180}(107,·)$, $\chi_{180}(109,·)$, $\chi_{180}(47,·)$, $\chi_{180}(49,·)$, $\chi_{180}(179,·)$, $\chi_{180}(119,·)$, $\chi_{180}(121,·)$, $\chi_{180}(59,·)$, $\chi_{180}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12}$, $\frac{1}{9} a^{13}$, $\frac{1}{9} a^{14}$, $\frac{1}{9} a^{15}$, $\frac{1}{9} a^{16}$, $\frac{1}{9} a^{17}$, $\frac{1}{346977} a^{18} - \frac{346}{115659} a^{16} - \frac{680}{115659} a^{14} - \frac{965}{115659} a^{12} - \frac{236}{38553} a^{10} + \frac{350}{38553} a^{8} - \frac{3472}{38553} a^{6} + \frac{6169}{12851} a^{4} - \frac{3624}{12851} a^{2} - \frac{3631}{12851}$, $\frac{1}{346977} a^{19} - \frac{346}{115659} a^{17} - \frac{680}{115659} a^{15} - \frac{965}{115659} a^{13} - \frac{236}{38553} a^{11} + \frac{350}{38553} a^{9} - \frac{3472}{38553} a^{7} + \frac{6169}{12851} a^{5} - \frac{3624}{12851} a^{3} - \frac{3631}{12851} a$, $\frac{1}{346977} a^{20} - \frac{449}{38553} a^{10} - \frac{3635}{12851}$, $\frac{1}{346977} a^{21} - \frac{449}{38553} a^{11} - \frac{3635}{12851} a$, $\frac{1}{346977} a^{22} - \frac{449}{38553} a^{12} - \frac{3635}{12851} a^{2}$, $\frac{1}{346977} a^{23} - \frac{449}{38553} a^{13} - \frac{3635}{12851} a^{3}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  \( \frac{665}{115659} a^{22} + \frac{1330}{38553} a^{20} + \frac{53902}{346977} a^{18} + \frac{24605}{38553} a^{16} + \frac{32585}{12851} a^{14} + \frac{61845}{12851} a^{12} + \frac{102410}{12851} a^{10} + \frac{154356}{12851} a^{8} + \frac{197505}{12851} a^{6} + \frac{91770}{12851} a^{4} + \frac{41895}{12851} a^{2} + \frac{17955}{12851} \) (order $10$)
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:  \( 27273221.26089917 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), 4.0.18000.1, \(\Q(\zeta_{5})\), \(\Q(\zeta_{36})^+\), 6.6.820125.1, 6.6.157464000.1, 8.0.324000000.3, 12.12.24794911296000000.1, 12.0.3099363912000000000.1, 12.0.84075626953125.1

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 ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
3Data not computed
5Data not computed