Properties

Label 24.0.170...729.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.707\times 10^{30}$
Root discriminant $18.18$
Ramified primes $3, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1)
 
gp: K = bnfinit(x^24 - x^23 + x^21 - x^20 + x^18 - x^17 + x^15 - x^14 + x^12 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1]);
 

\( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)

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:  \(1706902865139206151939937338729\)\(\medspace = 3^{12}\cdot 13^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.18$
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:  $3, 13$
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:  \(39=3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{39}(1,·)$, $\chi_{39}(2,·)$, $\chi_{39}(4,·)$, $\chi_{39}(5,·)$, $\chi_{39}(7,·)$, $\chi_{39}(8,·)$, $\chi_{39}(10,·)$, $\chi_{39}(11,·)$, $\chi_{39}(14,·)$, $\chi_{39}(16,·)$, $\chi_{39}(17,·)$, $\chi_{39}(19,·)$, $\chi_{39}(20,·)$, $\chi_{39}(22,·)$, $\chi_{39}(23,·)$, $\chi_{39}(25,·)$, $\chi_{39}(28,·)$, $\chi_{39}(29,·)$, $\chi_{39}(31,·)$, $\chi_{39}(32,·)$, $\chi_{39}(34,·)$, $\chi_{39}(35,·)$, $\chi_{39}(37,·)$, $\chi_{39}(38,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -a \) (order $78$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{3} + 1 \),  \( a^{6} + 1 \),  \( a^{17} - a^{15} + a^{4} - 1 \),  \( a^{6} + a^{3} + 1 \),  \( a^{21} + a^{3} \),  \( a - 1 \),  \( a^{2} - 1 \),  \( a^{4} - 1 \),  \( a^{5} - 1 \),  \( a^{7} - 1 \),  \( a^{10} - 1 \) (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2851634.018949717 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 2851634.018949717 \cdot 2}{78\sqrt{1706902865139206151939937338729}}\approx 0.211876698285567$ (assuming GRH)

Galois group

$C_2\times C_{12}$ (as 24T2):

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{13})\), 4.0.2197.1, 4.4.19773.1, 6.0.771147.1, \(\Q(\zeta_{13})^+\), 6.0.10024911.1, 8.0.390971529.1, 12.0.100498840557921.1, \(\Q(\zeta_{13})\), \(\Q(\zeta_{39})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$