Properties

Label 8.0.658266368.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 137^{3}$
Root discriminant $12.66$
Ramified primes $2, 137$
Class number $1$
Class group Trivial
Galois Group $D_{8}$ (as 8T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 12, 2, -6, 17, -18, 12, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 12*x^6 - 18*x^5 + 17*x^4 - 6*x^3 + 2*x^2 + 12*x + 4)
gp: K = bnfinit(x^8 - 4*x^7 + 12*x^6 - 18*x^5 + 17*x^4 - 6*x^3 + 2*x^2 + 12*x + 4, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 4 x^{7} \) \(\mathstrut +\mathstrut 12 x^{6} \) \(\mathstrut -\mathstrut 18 x^{5} \) \(\mathstrut +\mathstrut 17 x^{4} \) \(\mathstrut -\mathstrut 6 x^{3} \) \(\mathstrut +\mathstrut 2 x^{2} \) \(\mathstrut +\mathstrut 12 x \) \(\mathstrut +\mathstrut 4 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 4]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(658266368=2^{8}\cdot 137^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.66$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 137$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{154} a^{7} + \frac{5}{154} a^{6} - \frac{10}{77} a^{5} + \frac{3}{14} a^{4} - \frac{71}{154} a^{3} + \frac{24}{77} a^{2} - \frac{2}{11} a + \frac{34}{77}$

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

Class group and class number

Trivial Abelian group, 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:  \( \frac{12}{77} a^{7} - \frac{111}{154} a^{6} + \frac{367}{154} a^{5} - \frac{61}{14} a^{4} + \frac{837}{154} a^{3} - \frac{271}{77} a^{2} + \frac{18}{11} a + \frac{123}{77} \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{1}{77} a^{7} + \frac{5}{77} a^{6} - \frac{20}{77} a^{5} + \frac{13}{14} a^{4} - \frac{71}{77} a^{3} + \frac{19}{154} a^{2} + \frac{18}{11} a - \frac{9}{77} \),  \( \frac{9}{77} a^{7} - \frac{32}{77} a^{6} + \frac{179}{154} a^{5} - \frac{23}{14} a^{4} + \frac{185}{154} a^{3} - \frac{137}{154} a^{2} - \frac{3}{11} a - \frac{4}{77} \),  \( \frac{3}{154} a^{7} + \frac{15}{154} a^{6} - \frac{30}{77} a^{5} + \frac{8}{7} a^{4} - \frac{213}{154} a^{3} + \frac{67}{154} a^{2} + \frac{16}{11} a + \frac{25}{77} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 38.7298997714 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_8$ (as 8T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.2192.1

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

Sibling fields

Galois closure: data not computed
Degree 8 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$V_4$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$V_4$$[2]^{2}$
$137$$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{137}$$x + 3$$1$$1$$0$Trivial$[\ ]$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.137.2t1.1c1$1$ $ 137 $ $x^{2} - x - 34$ $C_2$ (as 2T1) $1$ $1$
1.2e2_137.2t1.1c1$1$ $ 2^{2} \cdot 137 $ $x^{2} + 137$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_137.4t3.2c1$2$ $ 2^{2} \cdot 137 $ $x^{4} - 2 x^{3} + 7 x^{2} - 6 x + 10$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e2_137.8t6.2c1$2$ $ 2^{2} \cdot 137 $ $x^{8} - 4 x^{7} + 12 x^{6} - 18 x^{5} + 17 x^{4} - 6 x^{3} + 2 x^{2} + 12 x + 4$ $D_{8}$ (as 8T6) $1$ $0$
* 2.2e2_137.8t6.2c2$2$ $ 2^{2} \cdot 137 $ $x^{8} - 4 x^{7} + 12 x^{6} - 18 x^{5} + 17 x^{4} - 6 x^{3} + 2 x^{2} + 12 x + 4$ $D_{8}$ (as 8T6) $1$ $0$

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 *.