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Properties

Label 8.0.10093618089.2
Degree $8$
Signature $[0, 4]$
Discriminant $3^{6}\cdot 61^{4}$
Root discriminant $17.80$
Ramified primes $3, 61$
Class number $2$
Class group $[2]$
Galois group $C_2^3 : C_4 $ (as 8T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![129, -240, 228, -51, -32, 20, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 6*x^6 + 20*x^5 - 32*x^4 - 51*x^3 + 228*x^2 - 240*x + 129)
 
gp: K = bnfinit(x^8 - x^7 - 6*x^6 + 20*x^5 - 32*x^4 - 51*x^3 + 228*x^2 - 240*x + 129, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 6 x^{6} + 20 x^{5} - 32 x^{4} - 51 x^{3} + 228 x^{2} - 240 x + 129 \)

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:  \(10093618089=3^{6}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.80$
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:  $3, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{197080} a^{7} + \frac{222}{24635} a^{6} + \frac{1333}{98540} a^{5} + \frac{3791}{98540} a^{4} + \frac{35871}{98540} a^{3} - \frac{25277}{197080} a^{2} + \frac{17239}{197080} a + \frac{86063}{197080}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{271}{15160} a^{7} + \frac{17}{7580} a^{6} + \frac{351}{3790} a^{5} - \frac{1083}{3790} a^{4} + \frac{551}{1895} a^{3} + \frac{16697}{15160} a^{2} - \frac{40389}{15160} a + \frac{34697}{15160} \) (order $6$)
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:  \( 131.269561521 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 8T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3 : C_4 $
Character table for $C_2^3 : C_4 $

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.549.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
61Data 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_61.2t1.1c1$1$ $ 3 \cdot 61 $ $x^{2} - x + 46$ $C_2$ (as 2T1) $1$ $-1$
1.61.2t1.1c1$1$ $ 61 $ $x^{2} - x - 15$ $C_2$ (as 2T1) $1$ $1$
1.3_61.4t1.1c1$1$ $ 3 \cdot 61 $ $x^{4} - x^{3} - 53 x^{2} + 141 x + 117$ $C_4$ (as 4T1) $0$ $1$
1.61.4t1.1c1$1$ $ 61 $ $x^{4} - x^{3} + 8 x^{2} - 42 x + 117$ $C_4$ (as 4T1) $0$ $-1$
1.61.4t1.1c2$1$ $ 61 $ $x^{4} - x^{3} + 8 x^{2} - 42 x + 117$ $C_4$ (as 4T1) $0$ $-1$
1.3_61.4t1.1c2$1$ $ 3 \cdot 61 $ $x^{4} - x^{3} - 53 x^{2} + 141 x + 117$ $C_4$ (as 4T1) $0$ $1$
2.3_61e2.4t3.1c1$2$ $ 3 \cdot 61^{2}$ $x^{4} - 2 x^{3} - 29 x^{2} + 30 x - 19$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3_61.4t3.2c1$2$ $ 3 \cdot 61 $ $x^{4} - 2 x^{3} - 2 x^{2} + 3 x + 3$ $D_{4}$ (as 4T3) $1$ $0$
* 4.3e4_61e3.8t21.1c1$4$ $ 3^{4} \cdot 61^{3}$ $x^{8} - x^{7} - 6 x^{6} + 20 x^{5} - 32 x^{4} - 51 x^{3} + 228 x^{2} - 240 x + 129$ $C_2^3 : C_4 $ (as 8T19) $1$ $0$

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 *.