Properties

Label 8.2.221626368.1
Degree $8$
Signature $[2, 3]$
Discriminant $-221626368$
Root discriminant $11.05$
Ramified primes $2, 3, 167$
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 - 2*x^7 + 2*x^6 + 2*x^5 - 4*x^4 + 2*x^3 + 2*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^8 - 2*x^7 + 2*x^6 + 2*x^5 - 4*x^4 + 2*x^3 + 2*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 2, 2, -4, 2, 2, -2, 1]);
 

\( x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1 \)

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:  \(-221626368\)\(\medspace = -\,2^{14}\cdot 3^{4}\cdot 167\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.05$
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:  $2, 3, 167$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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:  \( \frac{1}{3} a^{7} - \frac{2}{3} a^{6} + \frac{1}{3} a^{5} + a^{4} - \frac{5}{3} a^{3} + \frac{1}{3} a^{2} + \frac{2}{3} a - 2 \),  \( \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} \),  \( \frac{1}{3} a^{7} - \frac{2}{3} a^{6} + \frac{1}{3} a^{5} + a^{4} - \frac{5}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - 1 \),  \( \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{4}{3} a^{3} + \frac{4}{3} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 19.0453050111 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{3}\cdot 19.0453050111 \cdot 1}{2\sqrt{221626368}}\approx 0.634668316618$

Galois group

$S_4\wr C_2$ (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

\(\Q(\sqrt{3}) \)

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.8.14.1$x^{8} + 2 x^{7} + 6$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
167Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.167.2t1.a.a$1$ $ 167 $ $x^{2} - x + 42$ $C_2$ (as 2T1) $1$ $-1$
1.2004.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 167 $ $x^{2} + 501$ $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
2.2004.4t3.d.a$2$ $ 2^{2} \cdot 3 \cdot 167 $ $x^{4} + 5 x^{2} + 48$ $D_{4}$ (as 4T3) $1$ $0$
4.334668.6t13.b.a$4$ $ 2^{2} \cdot 3 \cdot 167^{2}$ $x^{6} - 2 x^{5} + 9 x^{4} - 19 x^{3} + 27 x^{2} - 44 x + 72$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.670674672.12t34.b.a$4$ $ 2^{4} \cdot 3^{2} \cdot 167^{3}$ $x^{6} - 2 x^{5} + 9 x^{4} - 19 x^{3} + 27 x^{2} - 44 x + 72$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.48192192.12t34.b.a$4$ $ 2^{6} \cdot 3^{3} \cdot 167^{2}$ $x^{6} - 2 x^{5} + 9 x^{4} - 19 x^{3} + 27 x^{2} - 44 x + 72$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.24048.6t13.b.a$4$ $ 2^{4} \cdot 3^{2} \cdot 167 $ $x^{6} - 2 x^{5} + 9 x^{4} - 19 x^{3} + 27 x^{2} - 44 x + 72$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.860...032.12t201.a.a$6$ $ 2^{12} \cdot 3^{3} \cdot 167^{4}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.143...344.12t202.a.a$6$ $ 2^{12} \cdot 3^{3} \cdot 167^{5}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.18468864.8t47.a.a$6$ $ 2^{12} \cdot 3^{3} \cdot 167 $ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.3084300288.12t200.a.a$6$ $ 2^{12} \cdot 3^{3} \cdot 167^{2}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.515078148096.16t1294.a.a$9$ $ 2^{12} \cdot 3^{3} \cdot 167^{3}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.239...448.18t272.a.a$9$ $ 2^{12} \cdot 3^{3} \cdot 167^{6}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.414...144.18t273.a.a$9$ $ 2^{18} \cdot 3^{6} \cdot 167^{6}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.890...888.18t274.a.a$9$ $ 2^{18} \cdot 3^{6} \cdot 167^{3}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.443...072.36t1763.a.a$12$ $ 2^{24} \cdot 3^{6} \cdot 167^{7}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.158...648.24t2821.a.a$12$ $ 2^{24} \cdot 3^{6} \cdot 167^{5}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.213...824.36t1758.a.a$18$ $ 2^{30} \cdot 3^{9} \cdot 167^{9}$ $x^{8} - 2 x^{7} + 2 x^{6} + 2 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 6 x + 1$ $S_4\wr C_2$ (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 *.