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Properties

Label 8.4.53216631070729.2
Degree $8$
Signature $[4, 2]$
Discriminant $7^{4}\cdot 53^{6}$
Root discriminant $51.97$
Ramified primes $7, 53$
Class number $1$
Class group Trivial
Galois group $V_4^2:S_3$ (as 8T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-59, -561, -158, 167, -67, 47, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 16*x^6 + 47*x^5 - 67*x^4 + 167*x^3 - 158*x^2 - 561*x - 59)
 
gp: K = bnfinit(x^8 - x^7 - 16*x^6 + 47*x^5 - 67*x^4 + 167*x^3 - 158*x^2 - 561*x - 59, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 16 x^{6} + 47 x^{5} - 67 x^{4} + 167 x^{3} - 158 x^{2} - 561 x - 59 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(53216631070729=7^{4}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.97$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $7, 53$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{35} a^{6} - \frac{13}{35} a^{5} - \frac{4}{35} a^{4} + \frac{1}{5} a^{3} + \frac{1}{7} a^{2} + \frac{9}{35} a - \frac{6}{35}$, $\frac{1}{1723645} a^{7} + \frac{1201}{246235} a^{6} - \frac{574369}{1723645} a^{5} - \frac{576908}{1723645} a^{4} + \frac{66942}{344729} a^{3} + \frac{563749}{1723645} a^{2} + \frac{470154}{1723645} a + \frac{117709}{344729}$

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:  $5$
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:  \( 12593.037021 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:S_4$ (as 8T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 10 conjugacy class representatives for $V_4^2:S_3$
Character table for $V_4^2:S_3$

Intermediate fields

\(\Q(\sqrt{53}) \)

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 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.53.2t1.1c1$1$ $ 53 $ $x^{2} - x - 13$ $C_2$ (as 2T1) $1$ $1$
2.7e2_53.3t2.1c1$2$ $ 7^{2} \cdot 53 $ $x^{3} - x^{2} - 9 x + 8$ $S_3$ (as 3T2) $1$ $2$
3.7e2_53e2.6t8.2c1$3$ $ 7^{2} \cdot 53^{2}$ $x^{4} - x^{3} + 7 x^{2} - 63 x + 259$ $S_4$ (as 4T5) $1$ $-1$
3.7e2_53.4t5.1c1$3$ $ 7^{2} \cdot 53 $ $x^{4} + 4 x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.7e2_53e2.6t8.3c1$3$ $ 7^{2} \cdot 53^{2}$ $x^{4} + 53 x^{2} - 106 x + 848$ $S_4$ (as 4T5) $1$ $-1$
3.7e2_53e3.4t5.1c1$3$ $ 7^{2} \cdot 53^{3}$ $x^{4} - x^{3} + 7 x^{2} - 63 x + 259$ $S_4$ (as 4T5) $1$ $-1$
3.7e2_53e2.6t8.1c1$3$ $ 7^{2} \cdot 53^{2}$ $x^{4} + 4 x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
3.7e2_53e3.4t5.2c1$3$ $ 7^{2} \cdot 53^{3}$ $x^{4} + 53 x^{2} - 106 x + 848$ $S_4$ (as 4T5) $1$ $-1$
* 6.7e4_53e5.8t34.1c1$6$ $ 7^{4} \cdot 53^{5}$ $x^{8} - x^{7} - 16 x^{6} + 47 x^{5} - 67 x^{4} + 167 x^{3} - 158 x^{2} - 561 x - 59$ $V_4^2:S_3$ (as 8T34) $1$ $2$

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 *.