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Properties

Label 8.2.57748079088.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{4}\cdot 3^{6}\cdot 7\cdot 29^{4}$
Root discriminant $22.14$
Ramified primes $2, 3, 7, 29$
Class number $1$
Class group Trivial
Galois group $S_4\wr C_2$ (as 8T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 4, -4, 10, 10, -4, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 5*x^6 - 4*x^5 + 10*x^4 + 10*x^3 - 4*x^2 + 4*x + 4)
 
gp: K = bnfinit(x^8 - x^7 + 5*x^6 - 4*x^5 + 10*x^4 + 10*x^3 - 4*x^2 + 4*x + 4, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-57748079088=-\,2^{4}\cdot 3^{6}\cdot 7\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.14$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{7} + \frac{1}{11} a^{6} + \frac{7}{22} a^{4} - \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11} a + \frac{3}{11}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 262.571913649 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_4\wr C_2$ (as 8T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{29}) \)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.6.3.1$x^{6} - 58 x^{4} + 841 x^{2} - 219501$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.7_29.2t1.1c1$1$ $ 7 \cdot 29 $ $x^{2} - x + 51$ $C_2$ (as 2T1) $1$ $-1$
* 1.29.2t1.1c1$1$ $ 29 $ $x^{2} - x - 7$ $C_2$ (as 2T1) $1$ $1$
2.3e2_7_29.4t3.2c1$2$ $ 3^{2} \cdot 7 \cdot 29 $ $x^{4} - 2 x^{3} + x + 16$ $D_{4}$ (as 4T3) $1$ $0$
4.2e2_3e2_7e2_29e3.12t34.1c1$4$ $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 29^{3}$ $x^{6} - x^{5} + 3 x^{3} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e4_3e2_7e3_29e2.12t34.1c1$4$ $ 2^{4} \cdot 3^{2} \cdot 7^{3} \cdot 29^{2}$ $x^{6} - x^{5} + 3 x^{3} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.2e2_3e2_7e2_29.6t13.1c1$4$ $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 29 $ $x^{6} - x^{5} + 3 x^{3} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e4_3e2_7_29e2.6t13.1c1$4$ $ 2^{4} \cdot 3^{2} \cdot 7 \cdot 29^{2}$ $x^{6} - x^{5} + 3 x^{3} - 2 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.2e4_3e4_7e4_29e3.12t201.1c1$6$ $ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 29^{3}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.2e4_3e6_7e5_29e3.12t202.1c1$6$ $ 2^{4} \cdot 3^{6} \cdot 7^{5} \cdot 29^{3}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.2e4_3e6_7_29e3.8t47.1c1$6$ $ 2^{4} \cdot 3^{6} \cdot 7 \cdot 29^{3}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.2e4_3e4_7e2_29e3.12t200.1c1$6$ $ 2^{4} \cdot 3^{4} \cdot 7^{2} \cdot 29^{3}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.2e6_3e6_7e3_29e3.16t1294.1c1$9$ $ 2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 29^{3}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.2e6_3e6_7e6_29e3.18t272.1c1$9$ $ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 29^{3}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.2e6_3e6_7e6_29e6.18t273.1c1$9$ $ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 29^{6}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.2e6_3e6_7e3_29e6.18t274.1c1$9$ $ 2^{6} \cdot 3^{6} \cdot 7^{3} \cdot 29^{6}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.2e8_3e10_7e7_29e6.36t1944.1c1$12$ $ 2^{8} \cdot 3^{10} \cdot 7^{7} \cdot 29^{6}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.2e8_3e10_7e5_29e6.24t2821.1c1$12$ $ 2^{8} \cdot 3^{10} \cdot 7^{5} \cdot 29^{6}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.2e12_3e14_7e9_29e9.36t1758.1c1$18$ $ 2^{12} \cdot 3^{14} \cdot 7^{9} \cdot 29^{9}$ $x^{8} - x^{7} + 5 x^{6} - 4 x^{5} + 10 x^{4} + 10 x^{3} - 4 x^{2} + 4 x + 4$ $S_4\wr C_2$ (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 *.