Properties

Label 8.0.6472063200625.8
Degree $8$
Signature $[0, 4]$
Discriminant $5^{4}\cdot 11^{4}\cdot 29^{4}$
Root discriminant $39.94$
Ramified primes $5, 11, 29$
Class number $16$
Class group $[4, 4]$
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![8771, -6643, 5673, -1870, 491, -120, 28, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 28*x^6 - 120*x^5 + 491*x^4 - 1870*x^3 + 5673*x^2 - 6643*x + 8771)
gp: K = bnfinit(x^8 - 2*x^7 + 28*x^6 - 120*x^5 + 491*x^4 - 1870*x^3 + 5673*x^2 - 6643*x + 8771, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 28 x^{6} \) \(\mathstrut -\mathstrut 120 x^{5} \) \(\mathstrut +\mathstrut 491 x^{4} \) \(\mathstrut -\mathstrut 1870 x^{3} \) \(\mathstrut +\mathstrut 5673 x^{2} \) \(\mathstrut -\mathstrut 6643 x \) \(\mathstrut +\mathstrut 8771 \)

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:  \(6472063200625=5^{4}\cdot 11^{4}\cdot 29^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $39.94$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2755} a^{6} + \frac{1057}{2755} a^{5} - \frac{17}{95} a^{4} - \frac{18}{551} a^{3} - \frac{337}{2755} a^{2} + \frac{37}{2755} a - \frac{256}{2755}$, $\frac{1}{263799515} a^{7} + \frac{39702}{263799515} a^{6} - \frac{3427379}{37685645} a^{5} - \frac{8034932}{52759903} a^{4} + \frac{1806472}{13884185} a^{3} - \frac{34920068}{263799515} a^{2} - \frac{98328941}{263799515} a + \frac{1708496}{7537129}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C4 x C4, order $16$

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:  \( \frac{2916}{13884185} a^{7} + \frac{39}{95753} a^{6} + \frac{9569}{1983455} a^{5} - \frac{109204}{13884185} a^{4} + \frac{430203}{13884185} a^{3} - \frac{1933299}{13884185} a^{2} + \frac{401339}{2776837} a + \frac{2656036}{1983455} \),  \( \frac{1002}{7537129} a^{7} + \frac{2872}{7537129} a^{6} + \frac{2442}{7537129} a^{5} + \frac{173}{396691} a^{4} - \frac{374086}{7537129} a^{3} + \frac{1794604}{7537129} a^{2} - \frac{2314123}{7537129} a + \frac{42644857}{7537129} \),  \( \frac{84898}{263799515} a^{7} - \frac{15342}{52759903} a^{6} + \frac{301067}{37685645} a^{5} - \frac{3826972}{263799515} a^{4} + \frac{15191864}{263799515} a^{3} - \frac{60770362}{263799515} a^{2} + \frac{13319366}{52759903} a - \frac{19307332}{37685645} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 40.3281811419 \)
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

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.508805.3 x2, 4.0.87725.1 x2

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

Sibling fields

Degree 4 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$