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Properties

Label 8.0.6329421.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{5}\cdot 7\cdot 61^{2}$
Root discriminant $7.08$
Ramified primes $3, 7, 61$
Class number $1$
Class group Trivial
Galois group $C_2 \wr C_2\wr C_2$ (as 8T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 2, 0, -1, 3, -1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - x^6 + 3*x^5 - x^4 + 2*x^2 - x + 1)
 
gp: K = bnfinit(x^8 - x^7 - x^6 + 3*x^5 - x^4 + 2*x^2 - x + 1, 1)
 

Normalized defining polynomial

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

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6329421=3^{5}\cdot 7\cdot 61^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7.08$
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:  $3, 7, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

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:  $3$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -a^{6} + a^{5} + a^{4} - 2 a^{3} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a^{7} - a^{6} - a^{5} + 3 a^{4} - a^{3} + 2 a - 1 \),  \( a^{6} - a^{5} - a^{4} + 2 a^{3} + a \),  \( a^{7} - a^{6} - 2 a^{5} + 3 a^{4} - a^{2} + a - 2 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3.30447851178 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 8T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$
Character table for $C_2 \wr C_2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1

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

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
$61$$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{61}$$x + 2$$1$$1$$0$Trivial$[\ ]$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.1.1$x^{2} - 61$$2$$1$$1$$C_2$$[\ ]_{2}$
61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_7.2t1.1c1$1$ $ 3 \cdot 7 $ $x^{2} - x - 5$ $C_2$ (as 2T1) $1$ $1$
1.7_61.2t1.1c1$1$ $ 7 \cdot 61 $ $x^{2} - x + 107$ $C_2$ (as 2T1) $1$ $-1$
1.3_61.2t1.1c1$1$ $ 3 \cdot 61 $ $x^{2} - x + 46$ $C_2$ (as 2T1) $1$ $-1$
1.61.2t1.1c1$1$ $ 61 $ $x^{2} - x - 15$ $C_2$ (as 2T1) $1$ $1$
1.3_7_61.2t1.1c1$1$ $ 3 \cdot 7 \cdot 61 $ $x^{2} - x - 320$ $C_2$ (as 2T1) $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
2.3e2_7_61.4t3.2c1$2$ $ 3^{2} \cdot 7 \cdot 61 $ $x^{4} - 33 x^{2} - 48$ $D_{4}$ (as 4T3) $1$ $0$
2.3e2_7.4t3.1c1$2$ $ 3^{2} \cdot 7 $ $x^{4} - x^{3} - 3 x^{2} - x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.3_7e2_61.4t3.2c1$2$ $ 3 \cdot 7^{2} \cdot 61 $ $x^{4} - 2 x^{3} + 26 x^{2} - 25 x + 193$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3_61.4t3.2c1$2$ $ 3 \cdot 61 $ $x^{4} - 2 x^{3} - 2 x^{2} + 3 x + 3$ $D_{4}$ (as 4T3) $1$ $0$
2.3e2_7_61.4t3.1c1$2$ $ 3^{2} \cdot 7 \cdot 61 $ $x^{4} - 9 x^{2} - 300$ $D_{4}$ (as 4T3) $1$ $0$
2.3e2_7_61e2.4t3.1c1$2$ $ 3^{2} \cdot 7 \cdot 61^{2}$ $x^{4} - x^{3} - 48 x^{2} - 136 x - 719$ $D_{4}$ (as 4T3) $1$ $0$
4.3e3_7e3_61.8t35.1c1$4$ $ 3^{3} \cdot 7^{3} \cdot 61 $ $x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 2 x^{2} - x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_7_61e3.8t35.1c1$4$ $ 3^{3} \cdot 7 \cdot 61^{3}$ $x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 2 x^{2} - x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_7e2_61e2.8t29.2c1$4$ $ 3^{3} \cdot 7^{2} \cdot 61^{2}$ $x^{8} - x^{7} - 9 x^{6} + 4 x^{5} + 29 x^{4} - 6 x^{3} - 24 x^{2} + 6 x - 3$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $2$
* 4.3e3_7_61.8t35.1c1$4$ $ 3^{3} \cdot 7 \cdot 61 $ $x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 2 x^{2} - x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_7e3_61e3.8t35.1c1$4$ $ 3^{3} \cdot 7^{3} \cdot 61^{3}$ $x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} + 2 x^{2} - x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_7e2_61e2.8t29.1c1$4$ $ 3^{3} \cdot 7^{2} \cdot 61^{2}$ $x^{8} - x^{7} - 9 x^{6} + 4 x^{5} + 29 x^{4} - 6 x^{3} - 24 x^{2} + 6 x - 3$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $-2$

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 *.