Properties

Label 8.0.484000000.9
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 5^{6}\cdot 11^{2}$
Root discriminant $12.18$
Ramified primes $2, 5, 11$
Class number $2$
Class group $[2]$
Galois Group $C_2^2:C_4$ (as 8T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![121, 0, 66, 0, 16, 0, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + x^6 + 16*x^4 + 66*x^2 + 121)
gp: K = bnfinit(x^8 + x^6 + 16*x^4 + 66*x^2 + 121, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut +\mathstrut 16 x^{4} \) \(\mathstrut +\mathstrut 66 x^{2} \) \(\mathstrut +\mathstrut 121 \)

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:  \(484000000=2^{8}\cdot 5^{6}\cdot 11^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.18$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{1991} a^{6} + \frac{848}{1991} a^{4} - \frac{479}{1991} a^{2} + \frac{47}{181}$, $\frac{1}{1991} a^{7} + \frac{848}{1991} a^{5} - \frac{479}{1991} a^{3} + \frac{47}{181} a$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{28}{1991} a^{6} - \frac{148}{1991} a^{4} + \frac{525}{1991} a^{2} + \frac{49}{181} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{7}{1991} a^{6} - \frac{37}{1991} a^{4} + \frac{629}{1991} a^{2} + \frac{148}{181} \),  \( \frac{3}{181} a^{7} - \frac{129}{1991} a^{6} + \frac{10}{181} a^{5} + \frac{113}{1991} a^{4} + \frac{11}{181} a^{3} - \frac{1921}{1991} a^{2} + \frac{284}{181} a - \frac{452}{181} \),  \( \frac{61}{1991} a^{7} + \frac{14}{1991} a^{6} - \frac{38}{1991} a^{5} - \frac{74}{1991} a^{4} + \frac{646}{1991} a^{3} + \frac{1258}{1991} a^{2} + \frac{152}{181} a - \frac{66}{181} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 26.3209270812 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2^2:C_4$ (as 8T10):

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 $C_2^2:C_4$
Character table for $C_2^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.4400.1, \(\Q(\zeta_{5})\), 4.0.22000.1

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

Sibling fields

Galois closure: data not computed
Degree 8 sibling: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$