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Properties

Label 24.0.77844331621...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 5^{12}\cdot 13^{20}$
Root discriminant $37.91$
Ramified primes $2, 5, 13$
Class number $192$ (GRH)
Class group $[4, 4, 12]$ (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, 63, 0, 931, 0, 5922, 0, 19363, 0, 36003, 0, 40285, 0, 27783, 0, 11783, 0, 2997, 0, 436, 0, 33, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 33*x^22 + 436*x^20 + 2997*x^18 + 11783*x^16 + 27783*x^14 + 40285*x^12 + 36003*x^10 + 19363*x^8 + 5922*x^6 + 931*x^4 + 63*x^2 + 1)
 
gp: K = bnfinit(x^24 + 33*x^22 + 436*x^20 + 2997*x^18 + 11783*x^16 + 27783*x^14 + 40285*x^12 + 36003*x^10 + 19363*x^8 + 5922*x^6 + 931*x^4 + 63*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{24} + 33 x^{22} + 436 x^{20} + 2997 x^{18} + 11783 x^{16} + 27783 x^{14} + 40285 x^{12} + 36003 x^{10} + 19363 x^{8} + 5922 x^{6} + 931 x^{4} + 63 x^{2} + 1 \)

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:  \(77844331621911754501328896000000000000=2^{24}\cdot 5^{12}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.91$
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$$\chi_{260}(1,·)$, $\chi_{260}(259,·)$, $\chi_{260}(69,·)$, $\chi_{260}(51,·)$, $\chi_{260}(129,·)$, $\chi_{260}(9,·)$, $\chi_{260}(139,·)$, $\chi_{260}(231,·)$, $\chi_{260}(79,·)$, $\chi_{260}(81,·)$, $\chi_{260}(131,·)$, $\chi_{260}(211,·)$, $\chi_{260}(29,·)$, $\chi_{260}(159,·)$, $\chi_{260}(101,·)$, $\chi_{260}(209,·)$, $\chi_{260}(199,·)$, $\chi_{260}(49,·)$, $\chi_{260}(179,·)$, $\chi_{260}(181,·)$, $\chi_{260}(121,·)$, $\chi_{260}(251,·)$, $\chi_{260}(61,·)$, $\chi_{260}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{69271526267} a^{22} + \frac{7322951344}{69271526267} a^{20} + \frac{12927355857}{69271526267} a^{18} - \frac{34372883738}{69271526267} a^{16} - \frac{34094544437}{69271526267} a^{14} - \frac{16198948921}{69271526267} a^{12} + \frac{1247984216}{69271526267} a^{10} + \frac{28407819091}{69271526267} a^{8} - \frac{18122206093}{69271526267} a^{6} - \frac{16123745477}{69271526267} a^{4} + \frac{32481864334}{69271526267} a^{2} - \frac{7567604524}{69271526267}$, $\frac{1}{69271526267} a^{23} + \frac{7322951344}{69271526267} a^{21} + \frac{12927355857}{69271526267} a^{19} - \frac{34372883738}{69271526267} a^{17} - \frac{34094544437}{69271526267} a^{15} - \frac{16198948921}{69271526267} a^{13} + \frac{1247984216}{69271526267} a^{11} + \frac{28407819091}{69271526267} a^{9} - \frac{18122206093}{69271526267} a^{7} - \frac{16123745477}{69271526267} a^{5} + \frac{32481864334}{69271526267} a^{3} - \frac{7567604524}{69271526267} a$

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

Class group and class number

$C_{4}\times C_{4}\times C_{12}$, which has order $192$ (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:  \( -\frac{360845}{3788849} a^{23} - \frac{12252940}{3788849} a^{21} - \frac{168794505}{3788849} a^{19} - \frac{1233987035}{3788849} a^{17} - \frac{5304248225}{3788849} a^{15} - \frac{14132751566}{3788849} a^{13} - \frac{23887975392}{3788849} a^{11} - \frac{25388167555}{3788849} a^{9} - \frac{16240133884}{3788849} a^{7} - \frac{5707052397}{3788849} a^{5} - \frac{917581164}{3788849} a^{3} - \frac{43158198}{3788849} a \) (order $4$)
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:  \( 7346081.887826216 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(i, \sqrt{65})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), 6.0.1827904.1, 6.6.46411625.1, 6.0.2970344000.1, 6.0.228488000.1, 6.6.3570125.1, 6.0.23762752.1, \(\Q(\zeta_{13})^+\), 8.0.4569760000.1, 12.0.8822943478336000000.1, 12.0.52206766144000000.1, 12.0.564668382613504.1, 12.0.8822943478336000000.2, 12.12.2154038935140625.1, 12.0.8822943478336000000.4, 12.0.8822943478336000000.3

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$