Properties

Label 8.0.54731953125.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{6}\cdot 5^{7}\cdot 31^{2}$
Root discriminant $21.99$
Ramified primes $3, 5, 31$
Class number $8$
Class group $[2, 2, 2]$
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![331, -359, 422, -182, 130, -28, 17, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 17*x^6 - 28*x^5 + 130*x^4 - 182*x^3 + 422*x^2 - 359*x + 331)
gp: K = bnfinit(x^8 - x^7 + 17*x^6 - 28*x^5 + 130*x^4 - 182*x^3 + 422*x^2 - 359*x + 331, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut 17 x^{6} \) \(\mathstrut -\mathstrut 28 x^{5} \) \(\mathstrut +\mathstrut 130 x^{4} \) \(\mathstrut -\mathstrut 182 x^{3} \) \(\mathstrut +\mathstrut 422 x^{2} \) \(\mathstrut -\mathstrut 359 x \) \(\mathstrut +\mathstrut 331 \)

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:  \(54731953125=3^{6}\cdot 5^{7}\cdot 31^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $21.99$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1972801} a^{7} - \frac{253170}{1972801} a^{6} + \frac{464058}{1972801} a^{5} - \frac{854678}{1972801} a^{4} - \frac{811769}{1972801} a^{3} + \frac{174405}{1972801} a^{2} - \frac{679842}{1972801} a - \frac{131505}{1972801}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C2 x C2 x C2, order $8$

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:  \( \frac{221}{32341} a^{7} - \frac{640}{32341} a^{6} + \frac{3507}{32341} a^{5} - \frac{12398}{32341} a^{4} + \frac{26919}{32341} a^{3} - \frac{71649}{32341} a^{2} + \frac{75886}{32341} a - \frac{85069}{32341} \),  \( \frac{6530}{1972801} a^{7} + \frac{7138}{1972801} a^{6} + \frac{76404}{1972801} a^{5} + \frac{6689}{1972801} a^{4} + \frac{64717}{1972801} a^{3} + \frac{558473}{1972801} a^{2} - \frac{566010}{1972801} a + \frac{1413586}{1972801} \),  \( \frac{13247}{1972801} a^{7} + \frac{18710}{1972801} a^{6} + \frac{128410}{1972801} a^{5} - \frac{14527}{1972801} a^{4} + \frac{234308}{1972801} a^{3} + \frac{193064}{1972801} a^{2} - \frac{30409}{1972801} a - \frac{2036253}{1972801} \)
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 ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$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}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3_5_31.2t1.1c1$1$ $ 3 \cdot 5 \cdot 31 $ $x^{2} - x - 116$ $C_2$ (as 2T1) $1$ $1$
1.3_31.2t1.1c1$1$ $ 3 \cdot 31 $ $x^{2} - x - 23$ $C_2$ (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_31.4t1.1c1$1$ $ 5 \cdot 31 $ $x^{4} - x^{3} - 39 x^{2} + 39 x + 281$ $C_4$ (as 4T1) $0$ $1$
1.5_31.4t1.1c2$1$ $ 5 \cdot 31 $ $x^{4} - x^{3} - 39 x^{2} + 39 x + 281$ $C_4$ (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_31.8t7.2c1$2$ $ 3^{2} \cdot 5^{2} \cdot 31 $ $x^{8} - x^{7} + 17 x^{6} - 28 x^{5} + 130 x^{4} - 182 x^{3} + 422 x^{2} - 359 x + 331$ $C_8:C_2$ (as 8T7) $0$ $-2$
* 2.3e2_5e2_31.8t7.2c2$2$ $ 3^{2} \cdot 5^{2} \cdot 31 $ $x^{8} - x^{7} + 17 x^{6} - 28 x^{5} + 130 x^{4} - 182 x^{3} + 422 x^{2} - 359 x + 331$ $C_8:C_2$ (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 *.