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Properties

Label 8.2.7348698545643.4
Degree $8$
Signature $[2, 3]$
Discriminant $-\,3^{7}\cdot 7^{6}\cdot 13^{4}$
Root discriminant $40.58$
Ramified primes $3, 7, 13$
Class number $12$
Class group $[2, 6]$
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![-16253, -5234, 4159, 1195, -308, -68, 1, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + x^6 - 68*x^5 - 308*x^4 + 1195*x^3 + 4159*x^2 - 5234*x - 16253)
 
gp: K = bnfinit(x^8 - 2*x^7 + x^6 - 68*x^5 - 308*x^4 + 1195*x^3 + 4159*x^2 - 5234*x - 16253, 1)
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + x^{6} - 68 x^{5} - 308 x^{4} + 1195 x^{3} + 4159 x^{2} - 5234 x - 16253 \)

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:  \(-7348698545643=-\,3^{7}\cdot 7^{6}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.58$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{26511200185} a^{7} - \frac{2403840863}{26511200185} a^{6} + \frac{4855256716}{26511200185} a^{5} + \frac{6977304084}{26511200185} a^{4} + \frac{3545558021}{26511200185} a^{3} + \frac{12366738493}{26511200185} a^{2} + \frac{2069365021}{26511200185} a - \frac{9245567244}{26511200185}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$

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:  \( 241.228196926 \)
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

\(\Q(\sqrt{21}) \), 4.2.1323.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.8.7.2$x^{8} - 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_7.2t1.1c1$1$ $ 3 \cdot 7 $ $x^{2} - x - 5$ $C_2$ (as 2T1) $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.1c1$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 2.3e2_7.4t3.1c1$2$ $ 3^{2} \cdot 7 $ $x^{4} - x^{3} - 3 x^{2} - x + 1$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3e2_7e2_13e2.8t8.1c1$2$ $ 3^{2} \cdot 7^{2} \cdot 13^{2}$ $x^{8} - 2 x^{7} + x^{6} - 68 x^{5} - 308 x^{4} + 1195 x^{3} + 4159 x^{2} - 5234 x - 16253$ $QD_{16}$ (as 8T8) $0$ $0$
* 2.3e2_7e2_13e2.8t8.1c2$2$ $ 3^{2} \cdot 7^{2} \cdot 13^{2}$ $x^{8} - 2 x^{7} + x^{6} - 68 x^{5} - 308 x^{4} + 1195 x^{3} + 4159 x^{2} - 5234 x - 16253$ $QD_{16}$ (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 *.