Properties

Label 8.8.333621760000.1
Degree $8$
Signature $[8, 0]$
Discriminant $2^{12}\cdot 5^{4}\cdot 19^{4}$
Root discriminant $27.57$
Ramified primes $2, 5, 19$
Class number $1$
Class group Trivial
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![-1899, -3402, -1438, 562, 458, 2, -38, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 38*x^6 + 2*x^5 + 458*x^4 + 562*x^3 - 1438*x^2 - 3402*x - 1899)
gp: K = bnfinit(x^8 - 2*x^7 - 38*x^6 + 2*x^5 + 458*x^4 + 562*x^3 - 1438*x^2 - 3402*x - 1899, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut -\mathstrut 38 x^{6} \) \(\mathstrut +\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 458 x^{4} \) \(\mathstrut +\mathstrut 562 x^{3} \) \(\mathstrut -\mathstrut 1438 x^{2} \) \(\mathstrut -\mathstrut 3402 x \) \(\mathstrut -\mathstrut 1899 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[8, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(333621760000=2^{12}\cdot 5^{4}\cdot 19^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $27.57$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{114} a^{6} - \frac{3}{38} a^{5} + \frac{4}{57} a^{4} + \frac{2}{57} a^{3} - \frac{29}{114} a^{2} + \frac{13}{114} a - \frac{8}{19}$, $\frac{1}{114} a^{7} + \frac{1}{38} a^{5} - \frac{1}{6} a^{4} + \frac{7}{114} a^{3} + \frac{28}{57} a^{2} - \frac{7}{114} a - \frac{11}{38}$

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:  $7$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1758.1037263 \)
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

\(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{19})\), 4.4.115520.1 x2, 4.4.7600.1 x2

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

Sibling fields

Degree 4 siblings: 4.4.115520.1, 4.4.7600.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.2e2_19.2t1.1c1$1$ $ 2^{2} \cdot 19 $ $x^{2} - 19$ $C_2$ (as 2T1) $1$ $1$
* 1.2e2_5_19.2t1.1c1$1$ $ 2^{2} \cdot 5 \cdot 19 $ $x^{2} - 95$ $C_2$ (as 2T1) $1$ $1$
*2 2.2e4_5_19.4t3.7c1$2$ $ 2^{4} \cdot 5 \cdot 19 $ $x^{8} - 2 x^{7} - 38 x^{6} + 2 x^{5} + 458 x^{4} + 562 x^{3} - 1438 x^{2} - 3402 x - 1899$ $D_4$ (as 8T4) $1$ $2$

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 *.