Properties

Label 8.2.2461736148492.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,2^{2}\cdot 3\cdot 673^{4}$
Root discriminant $35.39$
Ramified primes $2, 3, 673$
Class number $1$
Class group Trivial
Galois Group $S_4\wr C_2$ (as 8T47)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![432, 0, -196, -98, 65, 16, -4, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 - 4*x^6 + 16*x^5 + 65*x^4 - 98*x^3 - 196*x^2 + 432)
gp: K = bnfinit(x^8 - 4*x^7 - 4*x^6 + 16*x^5 + 65*x^4 - 98*x^3 - 196*x^2 + 432, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut 4 x^{7} \) \(\mathstrut -\mathstrut 4 x^{6} \) \(\mathstrut +\mathstrut 16 x^{5} \) \(\mathstrut +\mathstrut 65 x^{4} \) \(\mathstrut -\mathstrut 98 x^{3} \) \(\mathstrut -\mathstrut 196 x^{2} \) \(\mathstrut +\mathstrut 432 \)

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:  \(-2461736148492=-\,2^{2}\cdot 3\cdot 673^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $35.39$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{20424} a^{7} - \frac{595}{5106} a^{6} - \frac{643}{5106} a^{5} + \frac{539}{2553} a^{4} + \frac{7601}{20424} a^{3} - \frac{2629}{10212} a^{2} - \frac{1669}{5106} a - \frac{303}{851}$

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:  \( 15148.187659 \)
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

\(\Q(\sqrt{673}) \)

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
673Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_673.2t1.1c1$1$ $ 3 \cdot 673 $ $x^{2} - x + 505$ $C_2$ (as 2T1) $1$ $-1$
* 1.673.2t1.1c1$1$ $ 673 $ $x^{2} - x - 168$ $C_2$ (as 2T1) $1$ $1$
2.3_673.4t3.2c1$2$ $ 3 \cdot 673 $ $x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 39$ $D_{4}$ (as 4T3) $1$ $0$
4.2e4_3e2_673e3.12t34.1c1$4$ $ 2^{4} \cdot 3^{2} \cdot 673^{3}$ $x^{6} - x^{5} + 6 x^{4} - 3 x^{3} + 8 x^{2} - 5 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e2_3e3_673e2.12t34.1c1$4$ $ 2^{2} \cdot 3^{3} \cdot 673^{2}$ $x^{6} - x^{5} + 6 x^{4} - 3 x^{3} + 8 x^{2} - 5 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $-2$
4.2e4_3e2_673.6t13.1c1$4$ $ 2^{4} \cdot 3^{2} \cdot 673 $ $x^{6} - x^{5} + 6 x^{4} - 3 x^{3} + 8 x^{2} - 5 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.2e2_3_673e2.6t13.1c1$4$ $ 2^{2} \cdot 3 \cdot 673^{2}$ $x^{6} - x^{5} + 6 x^{4} - 3 x^{3} + 8 x^{2} - 5 x + 1$ $C_3^2:D_4$ (as 6T13) $1$ $2$
6.2e2_3e4_673e3.12t201.1c1$6$ $ 2^{2} \cdot 3^{4} \cdot 673^{3}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $2$
6.2e2_3e5_673e3.12t202.1c1$6$ $ 2^{2} \cdot 3^{5} \cdot 673^{3}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $0$
* 6.2e2_3_673e3.8t47.1c1$6$ $ 2^{2} \cdot 3 \cdot 673^{3}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $0$
6.2e2_3e2_673e3.12t200.1c1$6$ $ 2^{2} \cdot 3^{2} \cdot 673^{3}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
9.2e6_3e3_673e3.16t1294.1c1$9$ $ 2^{6} \cdot 3^{3} \cdot 673^{3}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
9.2e6_3e6_673e3.18t272.1c1$9$ $ 2^{6} \cdot 3^{6} \cdot 673^{3}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.2e6_3e6_673e6.18t273.1c1$9$ $ 2^{6} \cdot 3^{6} \cdot 673^{6}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $1$
9.2e6_3e3_673e6.18t274.1c1$9$ $ 2^{6} \cdot 3^{3} \cdot 673^{6}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $-1$
12.2e10_3e7_673e6.36t1944.1c1$12$ $ 2^{10} \cdot 3^{7} \cdot 673^{6}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $-2$
12.2e10_3e5_673e6.24t2821.1c1$12$ $ 2^{10} \cdot 3^{5} \cdot 673^{6}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (as 8T47) $1$ $2$
18.2e12_3e9_673e9.36t1758.1c1$18$ $ 2^{12} \cdot 3^{9} \cdot 673^{9}$ $x^{8} - 4 x^{7} - 4 x^{6} + 16 x^{5} + 65 x^{4} - 98 x^{3} - 196 x^{2} + 432$ $S_4\wr C_2$ (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 *.