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Properties

Label 8.0.36790308864.1
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 3^{8}\cdot 37^{2}$
Root discriminant $20.93$
Ramified primes $2, 3, 37$
Class number $1$
Class group Trivial
Galois group $((C_2 \times D_4): C_2):C_3$ (as 8T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![55, -136, 116, -10, -17, 22, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 4*x^6 + 22*x^5 - 17*x^4 - 10*x^3 + 116*x^2 - 136*x + 55)
 
gp: K = bnfinit(x^8 - 2*x^7 - 4*x^6 + 22*x^5 - 17*x^4 - 10*x^3 + 116*x^2 - 136*x + 55, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 4 x^{6} + 22 x^{5} - 17 x^{4} - 10 x^{3} + 116 x^{2} - 136 x + 55 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(36790308864=2^{12}\cdot 3^{8}\cdot 37^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.93$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{21} a^{6} + \frac{3}{7} a^{5} + \frac{1}{3} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{5}{21}$, $\frac{1}{10437} a^{7} - \frac{20}{1491} a^{6} - \frac{1679}{3479} a^{5} - \frac{593}{1491} a^{4} - \frac{1214}{10437} a^{3} - \frac{983}{3479} a^{2} + \frac{5}{1491} a - \frac{4966}{10437}$

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:  $3$
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:  \( 179.464154724 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:A_4$ (as 8T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 11 conjugacy class representatives for $((C_2 \times D_4): C_2):C_3$
Character table for $((C_2 \times D_4): C_2):C_3$

Intermediate fields

4.0.5184.1

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

Sibling fields

Degree 8 siblings: 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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.8.12.16$x^{8} + 24 x^{2} + 4$$4$$2$$12$$A_4\times C_2$$[2, 2]^{6}$
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3e2.3t1.1c1$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.3e2.3t1.1c2$1$ $ 3^{2}$ $x^{3} - 3 x - 1$ $C_3$ (as 3T1) $0$ $1$
3.2e6_3e4_37e2.4t4.3c1$3$ $ 2^{6} \cdot 3^{4} \cdot 37^{2}$ $x^{4} - 2 x^{3} - 42 x^{2} + 6 x + 231$ $A_4$ (as 4T4) $1$ $3$
3.2e6_3e4_37e2.4t4.2c1$3$ $ 2^{6} \cdot 3^{4} \cdot 37^{2}$ $x^{4} - 2 x^{3} + 6 x^{2} + 32 x + 182$ $A_4$ (as 4T4) $1$ $-1$
3.3e4_37e2.4t4.1c1$3$ $ 3^{4} \cdot 37^{2}$ $x^{4} - x^{3} + 6 x^{2} + 11 x + 10$ $A_4$ (as 4T4) $1$ $-1$
3.2e6_3e4_37e2.4t4.1c1$3$ $ 2^{6} \cdot 3^{4} \cdot 37^{2}$ $x^{4} - 2 x^{3} + 18 x^{2} + 20 x + 26$ $A_4$ (as 4T4) $1$ $-1$
* 3.2e6_3e4.4t4.1c1$3$ $ 2^{6} \cdot 3^{4}$ $x^{4} - 2 x^{3} + 6 x^{2} - 4 x + 2$ $A_4$ (as 4T4) $1$ $-1$
* 4.2e6_3e4_37e2.8t32.3c1$4$ $ 2^{6} \cdot 3^{4} \cdot 37^{2}$ $x^{8} - 2 x^{7} - 4 x^{6} + 22 x^{5} - 17 x^{4} - 10 x^{3} + 116 x^{2} - 136 x + 55$ $((C_2 \times D_4): C_2):C_3$ (as 8T32) $1$ $0$
4.2e6_3e6_37e2.24t97.3c1$4$ $ 2^{6} \cdot 3^{6} \cdot 37^{2}$ $x^{8} - 2 x^{7} - 4 x^{6} + 22 x^{5} - 17 x^{4} - 10 x^{3} + 116 x^{2} - 136 x + 55$ $((C_2 \times D_4): C_2):C_3$ (as 8T32) $0$ $0$
4.2e6_3e6_37e2.24t97.3c2$4$ $ 2^{6} \cdot 3^{6} \cdot 37^{2}$ $x^{8} - 2 x^{7} - 4 x^{6} + 22 x^{5} - 17 x^{4} - 10 x^{3} + 116 x^{2} - 136 x + 55$ $((C_2 \times D_4): C_2):C_3$ (as 8T32) $0$ $0$

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 *.