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Properties

Label 24.0.42586662987...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 5^{18}\cdot 13^{16}$
Root discriminant $36.97$
Ramified primes $2, 5, 13$
Class number $48$ (GRH)
Class group $[4, 12]$ (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, -14, 0, 187, 0, -2493, 0, 33233, 0, -22063, 0, 12278, 0, -6558, 0, 3373, 0, -478, 0, 67, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 9*x^22 + 67*x^20 - 478*x^18 + 3373*x^16 - 6558*x^14 + 12278*x^12 - 22063*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1)
 
gp: K = bnfinit(x^24 - 9*x^22 + 67*x^20 - 478*x^18 + 3373*x^16 - 6558*x^14 + 12278*x^12 - 22063*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 9 x^{22} + 67 x^{20} - 478 x^{18} + 3373 x^{16} - 6558 x^{14} + 12278 x^{12} - 22063 x^{10} + 33233 x^{8} - 2493 x^{6} + 187 x^{4} - 14 x^{2} + 1 \)

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:  \(42586662987723509824000000000000000000=2^{24}\cdot 5^{18}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.97$
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:  $2, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(260=2^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(3,·)$, $\chi_{260}(133,·)$, $\chi_{260}(243,·)$, $\chi_{260}(9,·)$, $\chi_{260}(139,·)$, $\chi_{260}(79,·)$, $\chi_{260}(81,·)$, $\chi_{260}(131,·)$, $\chi_{260}(87,·)$, $\chi_{260}(217,·)$, $\chi_{260}(27,·)$, $\chi_{260}(29,·)$, $\chi_{260}(159,·)$, $\chi_{260}(209,·)$, $\chi_{260}(107,·)$, $\chi_{260}(237,·)$, $\chi_{260}(157,·)$, $\chi_{260}(113,·)$, $\chi_{260}(211,·)$, $\chi_{260}(53,·)$, $\chi_{260}(183,·)$, $\chi_{260}(61,·)$, $\chi_{260}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{12} + \frac{1}{5} a^{8} + \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{13} + \frac{1}{5} a^{9} + \frac{1}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{554516275} a^{18} - \frac{50811331}{554516275} a^{16} + \frac{179665711}{554516275} a^{14} + \frac{203399784}{554516275} a^{12} + \frac{141023521}{554516275} a^{10} + \frac{69836499}{554516275} a^{8} + \frac{33642206}{554516275} a^{6} - \frac{161758786}{554516275} a^{4} - \frac{194030859}{554516275} a^{2} + \frac{95644604}{554516275}$, $\frac{1}{554516275} a^{19} - \frac{50811331}{554516275} a^{17} + \frac{179665711}{554516275} a^{15} + \frac{203399784}{554516275} a^{13} + \frac{141023521}{554516275} a^{11} + \frac{69836499}{554516275} a^{9} + \frac{33642206}{554516275} a^{7} - \frac{161758786}{554516275} a^{5} - \frac{194030859}{554516275} a^{3} + \frac{95644604}{554516275} a$, $\frac{1}{554516275} a^{20} - \frac{2792773}{22180651} a^{10} + \frac{200675724}{554516275}$, $\frac{1}{554516275} a^{21} - \frac{2792773}{22180651} a^{11} + \frac{200675724}{554516275} a$, $\frac{1}{554516275} a^{22} - \frac{2792773}{22180651} a^{12} + \frac{200675724}{554516275} a^{2}$, $\frac{1}{554516275} a^{23} - \frac{2792773}{22180651} a^{13} + \frac{200675724}{554516275} a^{3}$

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

Class group and class number

$C_{4}\times C_{12}$, which has order $48$ (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:  \( -\frac{3851}{554516275} a^{21} - \frac{2646912}{22180651} a^{11} - \frac{2024590874}{554516275} a \) (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:  \( 352314418.0293608 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), 3.3.169.1, \(\Q(i, \sqrt{5})\), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{5})\), 6.0.228488000.1, 6.6.3570125.1, 6.0.1827904.1, \(\Q(\zeta_{20})\), 12.0.52206766144000000.1, 12.12.6525845768000000000.1, 12.0.1593224064453125.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$13$13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
13.12.8.1$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$