Properties

Label 24.0.497...896.1
Degree $24$
Signature $[0, 12]$
Discriminant $4.971\times 10^{33}$
Root discriminant $25.35$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (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 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1)
 
gp: K = bnfinit(x^24 - 13*x^20 + 143*x^16 - 336*x^12 + 663*x^8 - 26*x^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, -26, 0, 0, 0, 663, 0, 0, 0, -336, 0, 0, 0, 143, 0, 0, 0, -13, 0, 0, 0, 1]);
 

\( x^{24} - 13 x^{20} + 143 x^{16} - 336 x^{12} + 663 x^{8} - 26 x^{4} + 1 \)

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:  \(4971225787269833056964369392336896\)\(\medspace = 2^{48}\cdot 3^{12}\cdot 7^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $25.35$
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$$\chi_{168}(1,·)$, $\chi_{168}(67,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(11,·)$, $\chi_{168}(79,·)$, $\chi_{168}(107,·)$, $\chi_{168}(149,·)$, $\chi_{168}(23,·)$, $\chi_{168}(25,·)$, $\chi_{168}(71,·)$, $\chi_{168}(155,·)$, $\chi_{168}(29,·)$, $\chi_{168}(95,·)$, $\chi_{168}(163,·)$, $\chi_{168}(37,·)$, $\chi_{168}(43,·)$, $\chi_{168}(109,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(121,·)$, $\chi_{168}(127,·)$, $\chi_{168}(151,·)$, $\chi_{168}(85,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12} + \frac{1}{13}$, $\frac{1}{13} a^{13} + \frac{1}{13} a$, $\frac{1}{13} a^{14} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} + \frac{1}{13} a^{3}$, $\frac{1}{377} a^{16} + \frac{10}{377} a^{12} - \frac{12}{29} a^{8} + \frac{66}{377} a^{4} - \frac{120}{377}$, $\frac{1}{377} a^{17} + \frac{10}{377} a^{13} - \frac{12}{29} a^{9} + \frac{66}{377} a^{5} - \frac{120}{377} a$, $\frac{1}{377} a^{18} + \frac{10}{377} a^{14} - \frac{12}{29} a^{10} + \frac{66}{377} a^{6} - \frac{120}{377} a^{2}$, $\frac{1}{377} a^{19} + \frac{10}{377} a^{15} - \frac{12}{29} a^{11} + \frac{66}{377} a^{7} - \frac{120}{377} a^{3}$, $\frac{1}{16211} a^{20} - \frac{11}{16211} a^{16} - \frac{395}{16211} a^{12} - \frac{51}{16211} a^{8} + \frac{2}{16211} a^{4} - \frac{1656}{16211}$, $\frac{1}{16211} a^{21} - \frac{11}{16211} a^{17} - \frac{395}{16211} a^{13} - \frac{51}{16211} a^{9} + \frac{2}{16211} a^{5} - \frac{1656}{16211} a$, $\frac{1}{16211} a^{22} - \frac{11}{16211} a^{18} - \frac{395}{16211} a^{14} - \frac{51}{16211} a^{10} + \frac{2}{16211} a^{6} - \frac{1656}{16211} a^{2}$, $\frac{1}{16211} a^{23} - \frac{11}{16211} a^{19} - \frac{395}{16211} a^{15} - \frac{51}{16211} a^{11} + \frac{2}{16211} a^{7} - \frac{1656}{16211} a^{3}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{7213}{16211} a^{23} + \frac{93791}{16211} a^{19} - \frac{1031701}{16211} a^{15} + \frac{2426101}{16211} a^{11} - \frac{4783341}{16211} a^{7} + \frac{187582}{16211} a^{3} \) (order $24$)
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:  \( 24013423.00725467 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 24013423.00725467 \cdot 3}{24\sqrt{4971225787269833056964369392336896}}\approx 0.161172432351781$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{3})\), 6.0.153664.1, 6.0.64827.1, 6.6.4148928.1, 6.0.1229312.1, 6.6.1229312.1, 6.6.33191424.1, 6.0.33191424.1, \(\Q(\zeta_{24})\), 12.0.17213603549184.1, 12.0.96717311574016.1, 12.0.70506920137457664.2, 12.0.1101670627147776.1, 12.0.1101670627147776.2, 12.0.70506920137457664.1, 12.12.70506920137457664.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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
2Data not computed
$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$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$