Properties

Label 8.2.181398528.1
Degree $8$
Signature $[2, 3]$
Discriminant $-181398528$
Root discriminant $10.77$
Ramified primes $2, 3$
Class number $1$
Class group trivial
Galois group $\textrm{GL(2,3)}$ (as 8T23)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 6*x^4 + 4*x^2 - 3)
 
gp: K = bnfinit(x^8 - 6*x^4 + 4*x^2 - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 4, 0, -6, 0, 0, 0, 1]);
 

\( x^{8} - 6 x^{4} + 4 x^{2} - 3 \)

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-181398528\)\(\medspace = -\,2^{10}\cdot 3^{11}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.77$
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:  $2, 3$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{1}{4} a^{7} + \frac{1}{4} a^{4} - \frac{7}{4} a^{3} + \frac{1}{2} a^{2} + a - \frac{5}{4} \),  \( \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{5}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2} \),  \( \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{2} a^{3} - \frac{1}{4} a + \frac{1}{4} \),  \( \frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{7}{4} a^{3} + \frac{1}{2} a^{2} + a + \frac{1}{4} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 16.6622023168 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 16.6622023168 \cdot 1}{2\sqrt{181398528}}\approx 0.613740983510$

Galois group

$\GL(2,3)$ (as 8T23):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 48
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$
Character table for $\textrm{GL(2,3)}$

Intermediate fields

4.2.3888.1

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

Sibling fields

Degree 16 sibling: Deg 16
Degree 24 sibling: Deg 24
Arithmetically equvalently sibling: 8.2.181398528.2

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 ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$2.8.10.2$x^{8} + 20 x^{2} + 20$$8$$1$$10$$\textrm{GL(2,3)}$$[4/3, 4/3, 3/2]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.10.3$x^{6} + 36$$3$$2$$10$$D_{6}$$[5/2]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
2.972.3t2.d.a$2$ $ 2^{2} \cdot 3^{5}$ $x^{3} - 6$ $S_3$ (as 3T2) $1$ $0$
2.1944.24t22.a.a$2$ $ 2^{3} \cdot 3^{5}$ $x^{8} - 6 x^{4} + 4 x^{2} - 3$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.1944.24t22.a.b$2$ $ 2^{3} \cdot 3^{5}$ $x^{8} - 6 x^{4} + 4 x^{2} - 3$ $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.11664.6t8.b.a$3$ $ 2^{4} \cdot 3^{6}$ $x^{4} - 2 x^{3} - 6 x + 3$ $S_4$ (as 4T5) $1$ $-1$
* 3.3888.4t5.b.a$3$ $ 2^{4} \cdot 3^{5}$ $x^{4} - 2 x^{3} - 6 x + 3$ $S_4$ (as 4T5) $1$ $1$
* 4.46656.8t23.a.a$4$ $ 2^{6} \cdot 3^{6}$ $x^{8} - 6 x^{4} + 4 x^{2} - 3$ $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.