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Properties

Label 24.0.87118133512...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{36}\cdot 5^{18}\cdot 7^{16}$
Root discriminant $34.61$
Ramified primes $2, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (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![4096, 0, -12288, 0, 31744, 0, -80384, 0, 203008, 0, -92544, 0, 33920, 0, -11456, 0, 3536, 0, -528, 0, 76, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 10*x^22 + 76*x^20 - 528*x^18 + 3536*x^16 - 11456*x^14 + 33920*x^12 - 92544*x^10 + 203008*x^8 - 80384*x^6 + 31744*x^4 - 12288*x^2 + 4096)
 
gp: K = bnfinit(x^24 - 10*x^22 + 76*x^20 - 528*x^18 + 3536*x^16 - 11456*x^14 + 33920*x^12 - 92544*x^10 + 203008*x^8 - 80384*x^6 + 31744*x^4 - 12288*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{24} - 10 x^{22} + 76 x^{20} - 528 x^{18} + 3536 x^{16} - 11456 x^{14} + 33920 x^{12} - 92544 x^{10} + 203008 x^{8} - 80384 x^{6} + 31744 x^{4} - 12288 x^{2} + 4096 \)

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:  \(8711813351237484544000000000000000000=2^{36}\cdot 5^{18}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.61$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(67,·)$, $\chi_{280}(179,·)$, $\chi_{280}(193,·)$, $\chi_{280}(233,·)$, $\chi_{280}(9,·)$, $\chi_{280}(11,·)$, $\chi_{280}(81,·)$, $\chi_{280}(211,·)$, $\chi_{280}(121,·)$, $\chi_{280}(249,·)$, $\chi_{280}(267,·)$, $\chi_{280}(219,·)$, $\chi_{280}(107,·)$, $\chi_{280}(163,·)$, $\chi_{280}(113,·)$, $\chi_{280}(169,·)$, $\chi_{280}(43,·)$, $\chi_{280}(99,·)$, $\chi_{280}(177,·)$, $\chi_{280}(51,·)$, $\chi_{280}(137,·)$, $\chi_{280}(57,·)$, $\chi_{280}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{84424192} a^{18} + \frac{2495}{42212096} a^{16} - \frac{40833}{21106048} a^{14} + \frac{24303}{10553024} a^{12} - \frac{43903}{5276512} a^{10} + \frac{28725}{1319128} a^{8} + \frac{47471}{1319128} a^{6} + \frac{48407}{659564} a^{4} + \frac{75253}{329782} a^{2} - \frac{54614}{164891}$, $\frac{1}{84424192} a^{19} + \frac{2495}{42212096} a^{17} - \frac{40833}{21106048} a^{15} + \frac{24303}{10553024} a^{13} - \frac{43903}{5276512} a^{11} + \frac{28725}{1319128} a^{9} + \frac{47471}{1319128} a^{7} + \frac{48407}{659564} a^{5} + \frac{75253}{329782} a^{3} - \frac{54614}{164891} a$, $\frac{1}{168848384} a^{20} - \frac{7135}{659564} a^{10} + \frac{61964}{164891}$, $\frac{1}{168848384} a^{21} - \frac{7135}{659564} a^{11} + \frac{61964}{164891} a$, $\frac{1}{337696768} a^{22} - \frac{7135}{1319128} a^{12} + \frac{30982}{164891} a^{2}$, $\frac{1}{337696768} a^{23} - \frac{7135}{1319128} a^{13} + \frac{30982}{164891} a^{3}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( \frac{19365}{168848384} a^{22} - \frac{187195}{168848384} a^{20} + \frac{703595}{84424192} a^{18} - \frac{1216713}{21106048} a^{16} + \frac{2033325}{5276512} a^{14} - \frac{12438785}{10553024} a^{12} + \frac{9108005}{2638256} a^{10} - \frac{1536290}{164891} a^{8} + \frac{26090063}{1319128} a^{6} - \frac{961795}{659564} a^{4} + \frac{187195}{329782} a^{2} - \frac{32275}{164891} \) (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:  \( 198346402.31462404 \) (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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{5})\), 4.4.8000.1, 6.0.1229312.1, 6.6.300125.1, 6.0.153664000.1, 8.0.64000000.1, 12.0.23612624896000000.1, 12.0.11259376953125.1, 12.12.2951578112000000000.1

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 R ${\href{/LocalNumberField/11.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
5Data not computed
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$