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Properties

Label 8.2.213311234375.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,5^{6}\cdot 239^{3}$
Root discriminant $26.07$
Ramified primes $5, 239$
Class number $1$
Class group Trivial
Galois group $QD_{16}$ (as 8T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -30, -60, -60, -59, -12, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 - x^6 - 12*x^5 - 59*x^4 - 60*x^3 - 60*x^2 - 30*x - 5)
 
gp: K = bnfinit(x^8 - 3*x^7 - x^6 - 12*x^5 - 59*x^4 - 60*x^3 - 60*x^2 - 30*x - 5, 1)
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} - x^{6} - 12 x^{5} - 59 x^{4} - 60 x^{3} - 60 x^{2} - 30 x - 5 \)

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:  \(-213311234375=-\,5^{6}\cdot 239^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.07$
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:  $5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{8951} a^{7} - \frac{3319}{8951} a^{6} - \frac{3927}{8951} a^{5} - \frac{1785}{8951} a^{4} + \frac{2390}{8951} a^{3} - \frac{3665}{8951} a^{2} - \frac{2378}{8951} a - \frac{413}{8951}$

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:  \( 248.890865982 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 8T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.5975.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
239Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.239.2t1.1c1$1$ $ 239 $ $x^{2} - x + 60$ $C_2$ (as 2T1) $1$ $-1$
1.5_239.2t1.1c1$1$ $ 5 \cdot 239 $ $x^{2} - x + 299$ $C_2$ (as 2T1) $1$ $-1$
* 2.5_239.4t3.1c1$2$ $ 5 \cdot 239 $ $x^{4} - x^{3} + 3 x^{2} + 8 x - 16$ $D_{4}$ (as 4T3) $1$ $0$
* 2.5e2_239.8t8.1c1$2$ $ 5^{2} \cdot 239 $ $x^{8} - 3 x^{7} - x^{6} - 12 x^{5} - 59 x^{4} - 60 x^{3} - 60 x^{2} - 30 x - 5$ $QD_{16}$ (as 8T8) $0$ $0$
* 2.5e2_239.8t8.1c2$2$ $ 5^{2} \cdot 239 $ $x^{8} - 3 x^{7} - x^{6} - 12 x^{5} - 59 x^{4} - 60 x^{3} - 60 x^{2} - 30 x - 5$ $QD_{16}$ (as 8T8) $0$ $0$

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 *.