Properties

Label 8.2.9885738095.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,5\cdot 19\cdot 101^{4}$
Root discriminant $17.76$
Ramified primes $5, 19, 101$
Class number $1$
Class group Trivial
Galois Group $S_4\wr C_2$ (as 8T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, -10, 5, -1, -4, 5, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 5*x^6 - 4*x^5 - x^4 + 5*x^3 - 10*x^2 - 19)
gp: K = bnfinit(x^8 - 2*x^7 + 5*x^6 - 4*x^5 - x^4 + 5*x^3 - 10*x^2 - 19, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 5 x^{6} \) \(\mathstrut -\mathstrut 4 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut 5 x^{3} \) \(\mathstrut -\mathstrut 10 x^{2} \) \(\mathstrut -\mathstrut 19 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 3]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-9885738095=-\,5\cdot 19\cdot 101^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $17.76$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 19, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2015} a^{7} - \frac{836}{2015} a^{6} + \frac{3}{155} a^{5} - \frac{58}{403} a^{4} + \frac{59}{2015} a^{3} - \frac{841}{2015} a^{2} + \frac{164}{2015} a + \frac{244}{2015}$

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:  $4$
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:  \( 126.257684791 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_4\wr C_2$ (as 8T47):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 1152
The 20 conjugacy class representatives for $S_4\wr C_2$
Character table for $S_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{101}) \)

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

Sibling fields

Degree 12 siblings: data not computed
Degree 16 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 siblings: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
101Data not computed

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_19.2t1.1c1$1$ $ 5 \cdot 19 $ $x^{2} - x + 24$ $C_2$ (as 2T1) $1$ $-1$
1.5_19_101.2t1.1c1$1$ $ 5 \cdot 19 \cdot 101 $ $x^{2} - x + 2399$ $C_2$ (as 2T1) $1$ $-1$
* 1.101.2t1.1c1$1$ $ 101 $ $x^{2} - x - 25$ $C_2$ (as 2T1) $1$ $1$
2.5_19_101.4t3.7c1$2$ $ 5 \cdot 19 \cdot 101 $ $x^{4} - x^{3} + 22 x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $0$
4.5e2_19e2_101.6t13.1c1$4$ $ 5^{2} \cdot 19^{2} \cdot 101 $ $x^{6} - 2 x^{5} + 6 x^{4} + 5 x^{3} + 20 x^{2} + 25 x + 25$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5e3_19e3_101e2.12t34.1c1$4$ $ 5^{3} \cdot 19^{3} \cdot 101^{2}$ $x^{6} - 2 x^{5} + 6 x^{4} + 5 x^{3} + 20 x^{2} + 25 x + 25$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.5e2_19e2_101e3.12t34.1c1$4$ $ 5^{2} \cdot 19^{2} \cdot 101^{3}$ $x^{6} - 2 x^{5} + 6 x^{4} + 5 x^{3} + 20 x^{2} + 25 x + 25$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.5_19_101e2.6t13.1c1$4$ $ 5 \cdot 19 \cdot 101^{2}$ $x^{6} - 2 x^{5} + 6 x^{4} + 5 x^{3} + 20 x^{2} + 25 x + 25$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.5e4_19e4_101e3.12t201.1c1$6$ $ 5^{4} \cdot 19^{4} \cdot 101^{3}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.5e5_19e5_101e3.12t202.1c1$6$ $ 5^{5} \cdot 19^{5} \cdot 101^{3}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.5_19_101e3.8t47.1c1$6$ $ 5 \cdot 19 \cdot 101^{3}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.5e2_19e2_101e3.12t200.1c1$6$ $ 5^{2} \cdot 19^{2} \cdot 101^{3}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.5e3_19e3_101e3.16t1294.1c1$9$ $ 5^{3} \cdot 19^{3} \cdot 101^{3}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.5e6_19e6_101e3.18t272.1c1$9$ $ 5^{6} \cdot 19^{6} \cdot 101^{3}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.5e6_19e6_101e6.18t273.1c1$9$ $ 5^{6} \cdot 19^{6} \cdot 101^{6}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.5e3_19e3_101e6.18t274.1c1$9$ $ 5^{3} \cdot 19^{3} \cdot 101^{6}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.5e7_19e7_101e6.36t1944.1c1$12$ $ 5^{7} \cdot 19^{7} \cdot 101^{6}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.5e5_19e5_101e6.24t2821.1c1$12$ $ 5^{5} \cdot 19^{5} \cdot 101^{6}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.5e9_19e9_101e9.36t1758.1c1$18$ $ 5^{9} \cdot 19^{9} \cdot 101^{9}$ $x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} - x^{4} + 5 x^{3} - 10 x^{2} - 19$ $S_4\wr C_2$ (as 8T47) $1$ $0$

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 *.