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Properties

Label 24.0.11935913115...7296.3
Degree $24$
Signature $[0, 12]$
Discriminant $2^{48}\cdot 3^{12}\cdot 7^{20}$
Root discriminant $35.06$
Ramified primes $2, 3, 7$
Class number $12$ (GRH)
Class group $[2, 6]$ (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, -4, 0, 15, 0, -56, 0, 209, 0, -780, 0, 2911, 0, -780, 0, 209, 0, -56, 0, 15, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^22 + 15*x^20 - 56*x^18 + 209*x^16 - 780*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1)
 
gp: K = bnfinit(x^24 - 4*x^22 + 15*x^20 - 56*x^18 + 209*x^16 - 780*x^14 + 2911*x^12 - 780*x^10 + 209*x^8 - 56*x^6 + 15*x^4 - 4*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 4 x^{22} + 15 x^{20} - 56 x^{18} + 209 x^{16} - 780 x^{14} + 2911 x^{12} - 780 x^{10} + 209 x^{8} - 56 x^{6} + 15 x^{4} - 4 x^{2} + 1 \)

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:  \(11935913115234869169771450911000887296=2^{48}\cdot 3^{12}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.06$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(168=2^{3}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(67,·)$, $\chi_{168}(5,·)$, $\chi_{168}(71,·)$, $\chi_{168}(73,·)$, $\chi_{168}(139,·)$, $\chi_{168}(143,·)$, $\chi_{168}(145,·)$, $\chi_{168}(19,·)$, $\chi_{168}(149,·)$, $\chi_{168}(23,·)$, $\chi_{168}(25,·)$, $\chi_{168}(29,·)$, $\chi_{168}(95,·)$, $\chi_{168}(97,·)$, $\chi_{168}(163,·)$, $\chi_{168}(101,·)$, $\chi_{168}(167,·)$, $\chi_{168}(43,·)$, $\chi_{168}(47,·)$, $\chi_{168}(115,·)$, $\chi_{168}(53,·)$, $\chi_{168}(121,·)$, $\chi_{168}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2911} a^{14} - \frac{780}{2911}$, $\frac{1}{2911} a^{15} - \frac{780}{2911} a$, $\frac{1}{2911} a^{16} - \frac{780}{2911} a^{2}$, $\frac{1}{2911} a^{17} - \frac{780}{2911} a^{3}$, $\frac{1}{2911} a^{18} - \frac{780}{2911} a^{4}$, $\frac{1}{2911} a^{19} - \frac{780}{2911} a^{5}$, $\frac{1}{2911} a^{20} - \frac{780}{2911} a^{6}$, $\frac{1}{2911} a^{21} - \frac{780}{2911} a^{7}$, $\frac{1}{2911} a^{22} - \frac{780}{2911} a^{8}$, $\frac{1}{2911} a^{23} - \frac{780}{2911} a^{9}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( \frac{836}{2911} a^{22} - \frac{3135}{2911} a^{20} + \frac{11704}{2911} a^{18} - \frac{43681}{2911} a^{16} + \frac{163020}{2911} a^{14} - 209 a^{12} + 780 a^{10} - \frac{43681}{2911} a^{8} + \frac{11704}{2911} a^{6} - \frac{3135}{2911} a^{4} + \frac{836}{2911} a^{2} - \frac{209}{2911} \) (order $14$)
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:  \( 162312869.89332253 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{42}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{14})\), 6.6.4148928.1, 6.0.1229312.1, 6.0.33191424.1, \(\Q(\zeta_{7})\), 6.0.29042496.1, 6.6.8605184.1, 6.6.232339968.1, 8.0.12745506816.2, 12.0.70506920137457664.1, 12.0.843466573910016.3, 12.12.3454839086735425536.1, 12.0.74049191673856.1, 12.0.3454839086735425536.5, 12.0.53981860730241024.5, 12.0.3454839086735425536.3

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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$2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
2.12.24.307$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$$4$$3$$24$$C_6\times C_2$$[2, 3]^{3}$
$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}$