Properties

Label 8.0.414720000000.3
Degree $8$
Signature $[0, 4]$
Discriminant $2^{16}\cdot 3^{4}\cdot 5^{7}$
Root discriminant $28.33$
Ramified primes $2, 3, 5$
Class number $20$
Class group $[2, 10]$
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![405, 0, 540, 0, 225, 0, 30, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 30*x^6 + 225*x^4 + 540*x^2 + 405)
gp: K = bnfinit(x^8 + 30*x^6 + 225*x^4 + 540*x^2 + 405, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut +\mathstrut 30 x^{6} \) \(\mathstrut +\mathstrut 225 x^{4} \) \(\mathstrut +\mathstrut 540 x^{2} \) \(\mathstrut +\mathstrut 405 \)

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:  \(414720000000=2^{16}\cdot 3^{4}\cdot 5^{7}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $28.33$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 5$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C2 x C10, order $20$

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:  \( \frac{2}{27} a^{6} + \frac{19}{9} a^{4} + \frac{40}{3} a^{2} + 17 \),  \( \frac{5}{27} a^{6} + \frac{47}{9} a^{4} + \frac{97}{3} a^{2} + 43 \),  \( \frac{1}{9} a^{6} + \frac{28}{9} a^{4} + \frac{56}{3} a^{2} + 22 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 14.8224845118 \)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\)

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 R R R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.8.16.35$x^{8} + 6 x^{6} + 4 x^{5} + 10 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_5.2t1.1c1$1$ $ 2^{3} \cdot 5 $ $x^{2} - 10$ $C_2$ (as 2T1) $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.2e2_5.4t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{4} - 5 x^{2} + 5$ $C_4$ (as 4T1) $0$ $1$
1.2e3_5.4t1.1c1$1$ $ 2^{3} \cdot 5 $ $x^{4} - 10 x^{2} + 20$ $C_4$ (as 4T1) $0$ $1$
1.2e3_5.4t1.1c2$1$ $ 2^{3} \cdot 5 $ $x^{4} - 10 x^{2} + 20$ $C_4$ (as 4T1) $0$ $1$
* 1.2e2_5.4t1.1c2$1$ $ 2^{2} \cdot 5 $ $x^{4} - 5 x^{2} + 5$ $C_4$ (as 4T1) $0$ $1$
* 2.2e6_3e2_5e2.8t7.4c1$2$ $ 2^{6} \cdot 3^{2} \cdot 5^{2}$ $x^{8} + 30 x^{6} + 225 x^{4} + 540 x^{2} + 405$ $C_8:C_2$ (as 8T7) $0$ $-2$
* 2.2e6_3e2_5e2.8t7.4c2$2$ $ 2^{6} \cdot 3^{2} \cdot 5^{2}$ $x^{8} + 30 x^{6} + 225 x^{4} + 540 x^{2} + 405$ $C_8:C_2$ (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 *.