Properties

Label 8.0.342102016.2
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 17^{4}$
Root discriminant $11.66$
Ramified primes $2, 17$
Class number $1$
Class group Trivial
Galois Group $D_4$ (as 8T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -8, 9, -2, 3, 20, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 3*x^6 + 20*x^5 + 3*x^4 - 2*x^3 + 9*x^2 - 8*x + 8)
gp: K = bnfinit(x^8 - 2*x^7 + 3*x^6 + 20*x^5 + 3*x^4 - 2*x^3 + 9*x^2 - 8*x + 8, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 3 x^{6} \) \(\mathstrut +\mathstrut 20 x^{5} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 8 \)

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:  \(342102016=2^{12}\cdot 17^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.66$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 17$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{376} a^{7} + \frac{1}{47} a^{6} + \frac{9}{94} a^{5} + \frac{1}{94} a^{4} + \frac{43}{376} a^{3} - \frac{17}{47} a^{2} + \frac{3}{94} a + \frac{14}{47}$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{35}{376} a^{7} - \frac{49}{376} a^{6} + \frac{19}{188} a^{5} + \frac{399}{188} a^{4} + \frac{471}{376} a^{3} - \frac{577}{376} a^{2} + \frac{11}{94} a + \frac{20}{47} \),  \( \frac{43}{376} a^{7} - \frac{173}{376} a^{6} + \frac{105}{94} a^{5} + \frac{43}{94} a^{4} - \frac{595}{376} a^{3} + \frac{497}{376} a^{2} - \frac{153}{94} a + \frac{38}{47} \),  \( \frac{67}{188} a^{7} - \frac{61}{94} a^{6} + \frac{39}{47} a^{5} + \frac{349}{47} a^{4} + \frac{437}{188} a^{3} - \frac{279}{94} a^{2} - \frac{34}{47} a + \frac{137}{47} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 62.7463864602 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_4$ (as 8T4):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 8
The 5 conjugacy class representatives for $D_4$
Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.1088.2 x2, 4.0.2312.1 x2

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

Sibling fields

Degree 4 siblings: 4.0.1088.2, 4.0.2312.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.17.2t1.1c1$1$ $ 17 $ $x^{2} - x - 4$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3_17.2t1.1c1$1$ $ 2^{3} \cdot 17 $ $x^{2} - 34$ $C_2$ (as 2T1) $1$ $1$
*2 2.2e3_17.4t3.5c1$2$ $ 2^{3} \cdot 17 $ $x^{8} - 2 x^{7} + 3 x^{6} + 20 x^{5} + 3 x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 8$ $D_4$ (as 8T4) $1$ $-2$

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 *.