Properties

Label 8.0.20560078125.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{6}\cdot 5^{7}\cdot 19^{2}$
Root discriminant $19.46$
Ramified primes $3, 5, 19$
Class number $4$
Class group $[2, 2]$
Galois Group $C_8:C_2$ (as 8T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![211, 73, 133, 46, 55, 1, 13, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 13*x^6 + x^5 + 55*x^4 + 46*x^3 + 133*x^2 + 73*x + 211)
gp: K = bnfinit(x^8 - 2*x^7 + 13*x^6 + x^5 + 55*x^4 + 46*x^3 + 133*x^2 + 73*x + 211, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 13 x^{6} \) \(\mathstrut +\mathstrut x^{5} \) \(\mathstrut +\mathstrut 55 x^{4} \) \(\mathstrut +\mathstrut 46 x^{3} \) \(\mathstrut +\mathstrut 133 x^{2} \) \(\mathstrut +\mathstrut 73 x \) \(\mathstrut +\mathstrut 211 \)

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:  \(20560078125=3^{6}\cdot 5^{7}\cdot 19^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $19.46$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{111693} a^{7} - \frac{16016}{111693} a^{6} - \frac{1374}{37231} a^{5} - \frac{854}{111693} a^{4} + \frac{49465}{111693} a^{3} - \frac{14313}{37231} a^{2} + \frac{43171}{111693} a + \frac{39349}{111693}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C2 x C2, order $4$

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:  \( \frac{9}{1201} a^{7} - \frac{24}{1201} a^{6} + \frac{133}{1201} a^{5} - \frac{239}{3603} a^{4} + \frac{1244}{3603} a^{3} + \frac{271}{1201} a^{2} + \frac{647}{3603} a + \frac{739}{3603} \),  \( \frac{568}{37231} a^{7} - \frac{941}{111693} a^{6} + \frac{4257}{37231} a^{5} + \frac{34024}{111693} a^{4} + \frac{23946}{37231} a^{3} + \frac{34184}{37231} a^{2} + \frac{143852}{111693} a + \frac{221051}{111693} \),  \( \frac{137}{111693} a^{7} + \frac{2437}{111693} a^{6} - \frac{2083}{37231} a^{5} + \frac{10642}{37231} a^{4} + \frac{221}{37231} a^{3} + \frac{12362}{37231} a^{2} + \frac{69160}{111693} a - \frac{82144}{111693} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 9.32364155459 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$OD_{16}$ (as 8T7):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 16
The 10 conjugacy class representatives for $C_8:C_2$
Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\)

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

Sibling fields

Galois closure: 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ R ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$3.8.6.3$x^{8} - 3 x^{4} + 18$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$19$19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_5_19.2t1.1c1$1$ $ 3 \cdot 5 \cdot 19 $ $x^{2} - x - 71$ $C_2$ (as 2T1) $1$ $1$
1.3_19.2t1.1c1$1$ $ 3 \cdot 19 $ $x^{2} - x - 14$ $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.3_5.4t1.1c1$1$ $ 3 \cdot 5 $ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
1.5_19.4t1.1c1$1$ $ 5 \cdot 19 $ $x^{4} - x^{3} - 24 x^{2} + 24 x + 101$ $C_4$ (as 4T1) $0$ $1$
1.5_19.4t1.1c2$1$ $ 5 \cdot 19 $ $x^{4} - x^{3} - 24 x^{2} + 24 x + 101$ $C_4$ (as 4T1) $0$ $1$
* 1.3_5.4t1.1c2$1$ $ 3 \cdot 5 $ $x^{4} - x^{3} - 4 x^{2} + 4 x + 1$ $C_4$ (as 4T1) $0$ $1$
* 2.3e2_5e2_19.8t7.2c1$2$ $ 3^{2} \cdot 5^{2} \cdot 19 $ $x^{8} - 2 x^{7} + 13 x^{6} + x^{5} + 55 x^{4} + 46 x^{3} + 133 x^{2} + 73 x + 211$ $C_8:C_2$ (as 8T7) $0$ $-2$
* 2.3e2_5e2_19.8t7.2c2$2$ $ 3^{2} \cdot 5^{2} \cdot 19 $ $x^{8} - 2 x^{7} + 13 x^{6} + x^{5} + 55 x^{4} + 46 x^{3} + 133 x^{2} + 73 x + 211$ $C_8:C_2$ (as 8T7) $0$ $-2$

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 *.