Properties

Label 24.0.291...576.2
Degree $24$
Signature $[0, 12]$
Discriminant $2.914\times 10^{33}$
Root discriminant $24.79$
Ramified primes $2, 3, 7$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 2*x^22 - 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 - 512*x^6 + 2048*x^2 + 4096)
 
gp: K = bnfinit(x^24 + 2*x^22 - 8*x^18 - 16*x^16 + 64*x^12 - 256*x^8 - 512*x^6 + 2048*x^2 + 4096, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 0, 2048, 0, 0, 0, -512, 0, -256, 0, 0, 0, 64, 0, 0, 0, -16, 0, -8, 0, 0, 0, 2, 0, 1]);
 

\( x^{24} + 2 x^{22} - 8 x^{18} - 16 x^{16} + 64 x^{12} - 256 x^{8} - 512 x^{6} + 2048 x^{2} + 4096 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2914041287899137980901233132568576\)\(\medspace = 2^{36}\cdot 3^{12}\cdot 7^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.79$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $24$
This field is Galois and abelian over $\Q$.
Conductor:  \(168=2^{3}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(131,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(139,·)$, $\chi_{168}(11,·)$, $\chi_{168}(145,·)$, $\chi_{168}(19,·)$, $\chi_{168}(115,·)$, $\chi_{168}(89,·)$, $\chi_{168}(25,·)$, $\chi_{168}(155,·)$, $\chi_{168}(107,·)$, $\chi_{168}(97,·)$, $\chi_{168}(67,·)$, $\chi_{168}(163,·)$, $\chi_{168}(17,·)$, $\chi_{168}(41,·)$, $\chi_{168}(43,·)$, $\chi_{168}(113,·)$, $\chi_{168}(83,·)$, $\chi_{168}(73,·)$, $\chi_{168}(121,·)$, $\chi_{168}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$

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

Class group and class number

$C_{7}$, which has order $7$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{1}{2048} a^{22} + \frac{1}{1024} a^{20} - \frac{1}{256} a^{16} - \frac{1}{128} a^{14} + \frac{1}{64} a^{12} + \frac{1}{32} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4} + 1 \) (order $42$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 19752911.509995345 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 19752911.509995345 \cdot 7}{42\sqrt{2914041287899137980901233132568576}}\approx 0.230882192371594$ (assuming GRH)

Galois group

$C_2^2\times C_6$ (as 24T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-42}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{14})\), 6.0.64827.1, 6.0.1229312.1, 6.6.33191424.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 6.6.8605184.1, 6.0.232339968.1, 8.0.796594176.1, 12.0.1101670627147776.1, \(\Q(\zeta_{21})\), 12.0.53981860730241024.3, 12.0.74049191673856.1, 12.0.53981860730241024.4, 12.0.53981860730241024.6, 12.12.53981860730241024.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$