Properties

Label 8.6.82278203392.1
Degree $8$
Signature $[6, 1]$
Discriminant $-82278203392$
Root discriminant \(23.14\)
Ramified primes see page
Class number $1$
Class group trivial
Galois group $S_4\wr C_2$ (as 8T47)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 12*x^5 - 30*x^4 + 32*x^3 + 28*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^8 - 12*x^5 - 30*x^4 + 32*x^3 + 28*x^2 - 12*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 28, 32, -30, -12, 0, 0, 1]);
 

\( x^{8} - 12x^{5} - 30x^{4} + 32x^{3} + 28x^{2} - 12x + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[6, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:   \(-82278203392\) \(\medspace = -\,2^{14}\cdot 7^{3}\cdot 11^{4}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(23.14\)
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:   \(2\), \(7\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{124}a^{7}-\frac{15}{124}a^{6}-\frac{23}{124}a^{5}+\frac{23}{124}a^{4}-\frac{3}{124}a^{3}-\frac{47}{124}a^{2}-\frac{11}{124}a-\frac{33}{124}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

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:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{47}{62}a^{7}+\frac{4}{31}a^{6}+\frac{2}{31}a^{5}-\frac{281}{31}a^{4}-\frac{1505}{62}a^{3}+\frac{616}{31}a^{2}+\frac{718}{31}a-\frac{109}{31}$, $a$, $\frac{1}{124}a^{7}+\frac{4}{31}a^{6}-\frac{23}{124}a^{5}-\frac{2}{31}a^{4}-\frac{251}{124}a^{3}-\frac{101}{62}a^{2}+\frac{1105}{124}a-\frac{125}{62}$, $\frac{127}{124}a^{7}+\frac{17}{124}a^{6}-\frac{7}{124}a^{5}-\frac{1543}{124}a^{4}-\frac{4039}{124}a^{3}+\frac{3579}{124}a^{2}+\frac{4431}{124}a-\frac{657}{124}$, $\frac{103}{31}a^{7}+\frac{51}{124}a^{6}+\frac{5}{62}a^{5}-\frac{4939}{124}a^{4}-\frac{6477}{62}a^{3}+\frac{11543}{124}a^{2}+\frac{3238}{31}a-\frac{3335}{124}$, $\frac{27}{124}a^{7}-\frac{1}{62}a^{6}-\frac{1}{124}a^{5}-\frac{85}{31}a^{4}-\frac{825}{124}a^{3}+\frac{202}{31}a^{2}+\frac{695}{124}a-\frac{27}{62}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1214.43857717 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{1}\cdot 1214.43857717 \cdot 1}{2\sqrt{82278203392}}\approx 0.851261517669$

Galois group

Group 1152.157849 (as 8T47):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 1152
The 20 conjugacy class representatives for $S_4\wr C_2$
Character table for $S_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/padicField/3.8.0.1}{8} }$ ${\href{/padicField/5.8.0.1}{8} }$ R R ${\href{/padicField/13.8.0.1}{8} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }$ ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(2\) Copy content Toggle raw display 2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
\(7\) Copy content Toggle raw display 7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(11\) Copy content Toggle raw display 11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.56.2t1.b.a$1$ $ 2^{3} \cdot 7 $ \(\Q(\sqrt{-14}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
2.56.4t3.b.a$2$ $ 2^{3} \cdot 7 $ 4.0.392.1 $D_{4}$ (as 4T3) $1$ $0$
4.3035648.12t34.a.a$4$ $ 2^{9} \cdot 7^{2} \cdot 11^{2}$ 6.0.1328096.1 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.1285596928.12t34.i.a$4$ $ 2^{8} \cdot 7^{3} \cdot 11^{4}$ 6.0.1328096.1 $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.189728.6t13.a.a$4$ $ 2^{5} \cdot 7^{2} \cdot 11^{2}$ 6.0.1328096.1 $C_3^2:D_4$ (as 6T13) $1$ $0$
4.26236672.6t13.a.a$4$ $ 2^{8} \cdot 7 \cdot 11^{4}$ 6.0.1328096.1 $C_3^2:D_4$ (as 6T13) $1$ $2$
6.1469253632.12t201.a.a$6$ $ 2^{11} \cdot 7^{2} \cdot 11^{4}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $-2$
6.10284775424.12t202.a.a$6$ $ 2^{11} \cdot 7^{3} \cdot 11^{4}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $-4$
* 6.10284775424.8t47.a.a$6$ $ 2^{11} \cdot 7^{3} \cdot 11^{4}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $4$
6.71993427968.12t200.a.a$6$ $ 2^{11} \cdot 7^{4} \cdot 11^{4}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $2$
9.243...584.16t1294.a.a$9$ $ 2^{13} \cdot 7^{5} \cdot 11^{6}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $3$
9.348...512.18t272.a.a$9$ $ 2^{13} \cdot 7^{4} \cdot 11^{6}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $-3$
9.446...536.18t273.b.a$9$ $ 2^{20} \cdot 7^{4} \cdot 11^{6}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $-3$
9.312...752.18t274.a.a$9$ $ 2^{20} \cdot 7^{5} \cdot 11^{6}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $3$
12.740...432.36t1763.a.a$12$ $ 2^{22} \cdot 7^{7} \cdot 11^{8}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.151...968.24t2821.a.a$12$ $ 2^{22} \cdot 7^{5} \cdot 11^{8}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $2$
18.108...024.36t1758.a.a$18$ $ 2^{33} \cdot 7^{9} \cdot 11^{12}$ 8.6.82278203392.1 $S_4\wr C_2$ (as 8T47) $1$ $0$

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 *.