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Properties

Label 24.0.74840859024...8272.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{48}\cdot 3^{47}$
Root discriminant $34.39$
Ramified primes $2, 3$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:D_8:C_2$ (as 24T118)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 3)
 
gp: K = bnfinit(x^24 + 3, 1)
 

Normalized defining polynomial

\( x^{24} + 3 \)

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:  \(7484085902451519080029070374597558272=2^{48}\cdot 3^{47}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.39$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{1}{2} a^{12} + \frac{1}{2} \) (order $6$)
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:  \( 2250001371.5010424 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:D_8:C_2$ (as 24T118):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 21 conjugacy class representatives for $C_3:D_8:C_2$
Character table for $C_3:D_8:C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 4.0.432.1, 6.0.177147.2, 8.0.143327232.1, 12.0.385610460475392.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Arithmetically equvalently siblings: data not computed

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{6}$

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$2.8.16.51$x^{8} + 3$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
2.8.16.51$x^{8} + 3$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
2.8.16.51$x^{8} + 3$$4$$2$$16$$Z_8 : Z_8^\times$$[2, 2, 3, 3]^{2}$
3Data not computed