Properties

Label 8.2.1568381796352.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{16}\cdot 7\cdot 43^{4}$
Root discriminant $33.45$
Ramified primes $2, 7, 43$
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![-85, -152, -150, -92, -72, -20, -8, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 8*x^6 - 20*x^5 - 72*x^4 - 92*x^3 - 150*x^2 - 152*x - 85)
gp: K = bnfinit(x^8 - 8*x^6 - 20*x^5 - 72*x^4 - 92*x^3 - 150*x^2 - 152*x - 85, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 8 x^{6} \) \(\mathstrut -\mathstrut 20 x^{5} \) \(\mathstrut -\mathstrut 72 x^{4} \) \(\mathstrut -\mathstrut 92 x^{3} \) \(\mathstrut -\mathstrut 150 x^{2} \) \(\mathstrut -\mathstrut 152 x \) \(\mathstrut -\mathstrut 85 \)

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:  \(-1568381796352=-\,2^{16}\cdot 7\cdot 43^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $33.45$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{45815} a^{7} - \frac{271}{45815} a^{6} - \frac{18197}{45815} a^{5} - \frac{2379}{6545} a^{4} - \frac{22794}{45815} a^{3} - \frac{7943}{45815} a^{2} - \frac{82}{4165} a + \frac{179}{539}$

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:  \( 2901.60195823 \)
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

\(\Q(\sqrt{86}) \)

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.8.16.59$x^{8} + 6 x^{6} + 4$$8$$1$$16$$S_4\times C_2$$[4/3, 4/3, 3]_{3}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.2e3_7_43.2t1.2c1$1$ $ 2^{3} \cdot 7 \cdot 43 $ $x^{2} + 602$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e3_43.2t1.1c1$1$ $ 2^{3} \cdot 43 $ $x^{2} - 86$ $C_2$ (as 2T1) $1$ $1$
2.2e3_7_43.4t3.4c1$2$ $ 2^{3} \cdot 7 \cdot 43 $ $x^{4} - x^{3} - 8 x^{2} + 5 x + 25$ $D_{4}$ (as 4T3) $1$ $0$
4.2e9_7e2_43e3.12t36.1c1$4$ $ 2^{9} \cdot 7^{2} \cdot 43^{3}$ $x^{6} - 3 x^{5} + 2 x^{4} + 2 x^{2} - 4 x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e8_7e3_43e2.12t34.1c1$4$ $ 2^{8} \cdot 7^{3} \cdot 43^{2}$ $x^{6} - 3 x^{5} + 2 x^{4} + 2 x^{2} - 4 x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.2e5_7e2_43.6t13.2c1$4$ $ 2^{5} \cdot 7^{2} \cdot 43 $ $x^{6} - 3 x^{5} + 2 x^{4} + 2 x^{2} - 4 x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e8_7_43e2.6t13.2c1$4$ $ 2^{8} \cdot 7 \cdot 43^{2}$ $x^{6} - 3 x^{5} + 2 x^{4} + 2 x^{2} - 4 x + 4$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.2e13_7e4_43e3.12t201.1c1$6$ $ 2^{13} \cdot 7^{4} \cdot 43^{3}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.2e13_7e5_43e3.12t202.1c1$6$ $ 2^{13} \cdot 7^{5} \cdot 43^{3}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.2e13_7_43e3.8t47.1c1$6$ $ 2^{13} \cdot 7 \cdot 43^{3}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.2e13_7e2_43e3.12t200.1c1$6$ $ 2^{13} \cdot 7^{2} \cdot 43^{3}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.2e15_7e3_43e3.16t1294.1c1$9$ $ 2^{15} \cdot 7^{3} \cdot 43^{3}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.2e15_7e6_43e3.18t272.1c1$9$ $ 2^{15} \cdot 7^{6} \cdot 43^{3}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.2e22_7e6_43e6.18t273.1c1$9$ $ 2^{22} \cdot 7^{6} \cdot 43^{6}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.2e22_7e3_43e6.18t274.1c1$9$ $ 2^{22} \cdot 7^{3} \cdot 43^{6}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.2e26_7e7_43e6.36t1944.1c1$12$ $ 2^{26} \cdot 7^{7} \cdot 43^{6}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.2e26_7e5_43e6.24t2821.1c1$12$ $ 2^{26} \cdot 7^{5} \cdot 43^{6}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.2e37_7e9_43e9.36t1758.1c1$18$ $ 2^{37} \cdot 7^{9} \cdot 43^{9}$ $x^{8} - 8 x^{6} - 20 x^{5} - 72 x^{4} - 92 x^{3} - 150 x^{2} - 152 x - 85$ $S_4\wr C_2$ (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 *.