Properties

Label 8.2.6377879282887.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,7\cdot 977^{4}$
Root discriminant $39.86$
Ramified primes $7, 977$
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![28, 0, -33, -33, 34, 2, -1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - x^6 + 2*x^5 + 34*x^4 - 33*x^3 - 33*x^2 + 28)
gp: K = bnfinit(x^8 - 2*x^7 - x^6 + 2*x^5 + 34*x^4 - 33*x^3 - 33*x^2 + 28, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 2 x^{7} \) \(\mathstrut -\mathstrut x^{6} \) \(\mathstrut +\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 34 x^{4} \) \(\mathstrut -\mathstrut 33 x^{3} \) \(\mathstrut -\mathstrut 33 x^{2} \) \(\mathstrut +\mathstrut 28 \)

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:  \(-6377879282887=-\,7\cdot 977^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $39.86$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 977$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{6856} a^{7} - \frac{37}{1714} a^{6} + \frac{1039}{6856} a^{5} - \frac{215}{1714} a^{4} + \frac{1093}{3428} a^{3} + \frac{3043}{6856} a^{2} + \frac{1329}{6856} a - \frac{1033}{3428}$

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:  \( 6316.5578146 \)
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

\(\Q(\sqrt{977}) \)

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 ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
977Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
1.7_977.2t1.1c1$1$ $ 7 \cdot 977 $ $x^{2} - x + 1710$ $C_2$ (as 2T1) $1$ $-1$
* 1.977.2t1.1c1$1$ $ 977 $ $x^{2} - x - 244$ $C_2$ (as 2T1) $1$ $1$
2.7_977.4t3.2c1$2$ $ 7 \cdot 977 $ $x^{4} - 2 x^{3} + 13 x^{2} - 12 x + 64$ $D_{4}$ (as 4T3) $1$ $0$
4.7e2_977.6t13.1c1$4$ $ 7^{2} \cdot 977 $ $x^{6} - 3 x^{5} + 5 x^{4} - 3 x^{3} - x^{2} + x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.7e3_977e2.12t34.1c1$4$ $ 7^{3} \cdot 977^{2}$ $x^{6} - 3 x^{5} + 5 x^{4} - 3 x^{3} - x^{2} + x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.7e2_977e3.12t36.1c1$4$ $ 7^{2} \cdot 977^{3}$ $x^{6} - 3 x^{5} + 5 x^{4} - 3 x^{3} - x^{2} + x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.7_977e2.6t13.1c1$4$ $ 7 \cdot 977^{2}$ $x^{6} - 3 x^{5} + 5 x^{4} - 3 x^{3} - x^{2} + x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.7e4_977e3.12t201.1c1$6$ $ 7^{4} \cdot 977^{3}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.7e5_977e3.12t202.1c1$6$ $ 7^{5} \cdot 977^{3}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.7_977e3.8t47.1c1$6$ $ 7 \cdot 977^{3}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.7e2_977e3.12t200.1c1$6$ $ 7^{2} \cdot 977^{3}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.7e3_977e3.16t1294.1c1$9$ $ 7^{3} \cdot 977^{3}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.7e6_977e3.18t272.1c1$9$ $ 7^{6} \cdot 977^{3}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.7e6_977e6.18t273.1c1$9$ $ 7^{6} \cdot 977^{6}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.7e3_977e6.18t274.1c1$9$ $ 7^{3} \cdot 977^{6}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.7e7_977e6.36t1944.1c1$12$ $ 7^{7} \cdot 977^{6}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.7e5_977e6.24t2821.1c1$12$ $ 7^{5} \cdot 977^{6}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.7e9_977e9.36t1758.1c1$18$ $ 7^{9} \cdot 977^{9}$ $x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 34 x^{4} - 33 x^{3} - 33 x^{2} + 28$ $S_4\wr C_2$ (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 *.