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Properties

Label 24.0.16878953293...9609.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{36}\cdot 13^{18}$
Root discriminant $35.57$
Ramified primes $3, 13$
Class number $18$ (GRH)
Class group $[3, 6]$ (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![729, 0, 0, 945, 0, 0, -692, 0, 0, 3403, 0, 0, 5609, 0, 0, -1277, 0, 0, 218, 0, 0, -17, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729)
 
gp: K = bnfinit(x^24 - 17*x^21 + 218*x^18 - 1277*x^15 + 5609*x^12 + 3403*x^9 - 692*x^6 + 945*x^3 + 729, 1)
 

Normalized defining polynomial

\( x^{24} - 17 x^{21} + 218 x^{18} - 1277 x^{15} + 5609 x^{12} + 3403 x^{9} - 692 x^{6} + 945 x^{3} + 729 \)

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:  \(16878953293629664473677397903764439609=3^{36}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.57$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(64,·)$, $\chi_{117}(1,·)$, $\chi_{117}(5,·)$, $\chi_{117}(70,·)$, $\chi_{117}(8,·)$, $\chi_{117}(73,·)$, $\chi_{117}(77,·)$, $\chi_{117}(14,·)$, $\chi_{117}(79,·)$, $\chi_{117}(83,·)$, $\chi_{117}(86,·)$, $\chi_{117}(25,·)$, $\chi_{117}(92,·)$, $\chi_{117}(31,·)$, $\chi_{117}(34,·)$, $\chi_{117}(38,·)$, $\chi_{117}(103,·)$, $\chi_{117}(40,·)$, $\chi_{117}(44,·)$, $\chi_{117}(109,·)$, $\chi_{117}(47,·)$, $\chi_{117}(112,·)$, $\chi_{117}(116,·)$, $\chi_{117}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{8} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{11} + \frac{1}{9} a^{5}$, $\frac{1}{1647} a^{18} - \frac{37}{1647} a^{15} + \frac{253}{1647} a^{12} - \frac{748}{1647} a^{9} + \frac{316}{1647} a^{6} - \frac{55}{1647} a^{3} - \frac{1}{61}$, $\frac{1}{1647} a^{19} - \frac{37}{1647} a^{16} + \frac{253}{1647} a^{13} - \frac{748}{1647} a^{10} + \frac{316}{1647} a^{7} - \frac{55}{1647} a^{4} - \frac{1}{61} a$, $\frac{1}{1647} a^{20} - \frac{37}{1647} a^{17} + \frac{70}{1647} a^{14} - \frac{748}{1647} a^{11} + \frac{133}{1647} a^{8} - \frac{55}{1647} a^{5} - \frac{70}{549} a^{2}$, $\frac{1}{333878023359} a^{21} + \frac{21560387}{111292674453} a^{18} + \frac{1698524242}{37097558151} a^{15} - \frac{14794673017}{111292674453} a^{12} + \frac{8631920026}{37097558151} a^{9} + \frac{29103961883}{111292674453} a^{6} + \frac{63648023030}{333878023359} a^{3} - \frac{1722997796}{12365852717}$, $\frac{1}{1001634070077} a^{22} - \frac{138037736}{1001634070077} a^{19} - \frac{51407798935}{1001634070077} a^{16} - \frac{95671899992}{1001634070077} a^{13} - \frac{178752124471}{1001634070077} a^{10} + \frac{23252714197}{1001634070077} a^{7} - \frac{333275577296}{1001634070077} a^{4} - \frac{13886131616}{37097558151} a$, $\frac{1}{3004902210231} a^{23} - \frac{746194427}{3004902210231} a^{20} - \frac{140198675821}{3004902210231} a^{17} + \frac{84342480544}{3004902210231} a^{14} - \frac{836777664133}{3004902210231} a^{11} + \frac{2704151200}{49260691971} a^{8} + \frac{590514436333}{3004902210231} a^{5} - \frac{12669893453}{37097558151} a^{2}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( \frac{186604117}{37097558151} a^{23} - \frac{9794458525}{111292674453} a^{20} + \frac{42314454700}{37097558151} a^{17} - \frac{259553185612}{37097558151} a^{14} + \frac{1178954688616}{37097558151} a^{11} + \frac{28434787913}{37097558151} a^{8} - \frac{14712814746}{12365852717} a^{5} + \frac{638512165729}{111292674453} a^{2} \) (order $18$)
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:  \( 533054958.55350393 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{13})\), 4.4.19773.1, 4.0.2197.1, \(\Q(\zeta_{9})\), 6.6.14414517.1, 6.0.43243551.1, 8.0.390971529.1, 12.0.1870004703089601.1, 12.12.4108400332687853397.1, 12.0.456488925854205933.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$13$13.12.9.2$x^{12} - 52 x^{8} + 676 x^{4} - 79092$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
13.12.9.2$x^{12} - 52 x^{8} + 676 x^{4} - 79092$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$