Properties

Label 8.0.1578125.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{6}\cdot 101$
Root discriminant $5.95$
Ramified primes $5, 101$
Class number $1$
Class group Trivial
Galois Group $((C_8 : C_2):C_2):C_2$ (as 8T27)

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Show commands for: Magma / SageMath / Pari/GP

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

Normalized defining polynomial

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

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:  \(1578125=5^{6}\cdot 101\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $5.95$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 101$
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 \(a\))

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

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:  \( a^{7} - 3 a^{6} + 3 a^{5} + a^{4} - 3 a^{3} + a + 1 \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a \),  \( a^{7} - 2 a^{6} + a^{5} + 2 a^{4} - 2 a^{3} - a^{2} + a \),  \( a^{6} - 2 a^{5} + a^{4} + a^{3} - a^{2} - a - 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1.81195906066 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\wr C_4$ (as 8T27):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 64
The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$
Character table for $((C_8 : C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5_101.2t1.1c1$1$ $ 5 \cdot 101 $ $x^{2} - x - 126$ $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.101.2t1.1c1$1$ $ 101 $ $x^{2} - x - 25$ $C_2$ (as 2T1) $1$ $1$
1.5_101.4t1.3c1$1$ $ 5 \cdot 101 $ $x^{4} - x^{3} + 126 x^{2} - 126 x + 3251$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.1c1$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
1.5_101.4t1.3c2$1$ $ 5 \cdot 101 $ $x^{4} - x^{3} + 126 x^{2} - 126 x + 3251$ $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.1c2$1$ $ 5 $ $x^{4} - x^{3} + x^{2} - x + 1$ $C_4$ (as 4T1) $0$ $-1$
2.5e2_101.4t3.2c1$2$ $ 5^{2} \cdot 101 $ $x^{4} - x^{3} + 11 x^{2} - 6 x + 36$ $D_{4}$ (as 4T3) $1$ $-2$
2.5_101.4t3.1c1$2$ $ 5 \cdot 101 $ $x^{4} - 11 x^{2} + 5$ $D_{4}$ (as 4T3) $1$ $2$
* 4.5e3_101.8t27.1c1$4$ $ 5^{3} \cdot 101 $ $x^{8} - 2 x^{7} + 3 x^{5} - x^{4} - 3 x^{3} + 2 x + 1$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $1$ $0$
4.5e3_101e3.8t27.1c1$4$ $ 5^{3} \cdot 101^{3}$ $x^{8} - 2 x^{7} + 3 x^{5} - x^{4} - 3 x^{3} + 2 x + 1$ $((C_8 : C_2):C_2):C_2$ (as 8T27) $1$ $0$
4.5e3_101e2.8t21.1c1$4$ $ 5^{3} \cdot 101^{2}$ $x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 31 x^{4} - 50 x^{3} + 47 x^{2} - 23 x + 6$ $C_2^3 : C_4 $ (as 8T19) $1$ $0$

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 *.