Properties

Label 8.8.1438916737499136.2
Degree $8$
Signature $[8, 0]$
Discriminant $1.439\times 10^{15}$
Root discriminant $78.48$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $Q_8$ (as 8T5)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 84*x^6 + 1260*x^4 - 5292*x^2 + 441)
 
gp: K = bnfinit(x^8 - 84*x^6 + 1260*x^4 - 5292*x^2 + 441, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![441, 0, -5292, 0, 1260, 0, -84, 0, 1]);
 

\( x^{8} - 84 x^{6} + 1260 x^{4} - 5292 x^{2} + 441 \)

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[8, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1438916737499136\)\(\medspace = 2^{24}\cdot 3^{6}\cdot 7^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $78.48$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $8$
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{21} a^{4}$, $\frac{1}{21} a^{5}$, $\frac{1}{3003} a^{6} - \frac{2}{429} a^{4} + \frac{61}{143} a^{2} + \frac{14}{143}$, $\frac{1}{3003} a^{7} - \frac{2}{429} a^{5} + \frac{61}{143} a^{3} + \frac{14}{143} a$

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

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 118750.668157 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{8}\cdot(2\pi)^{0}\cdot 118750.668157 \cdot 2}{2\sqrt{1438916737499136}}\approx 0.801416343787$

Galois group

$Q_8$ (as 8T5):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 8
The 5 conjugacy class representatives for $Q_8$
Character table for $Q_8$

Intermediate fields

\(\Q(\sqrt{42}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{14})\)

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

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.24.8$x^{8} + 8 x^{7} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 22$$8$$1$$24$$Q_8$$[2, 3, 4]$
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$7$7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.56.2t1.a.a$1$ $ 2^{3} \cdot 7 $ $x^{2} - 14$ $C_2$ (as 2T1) $1$ $1$
* 1.168.2t1.a.a$1$ $ 2^{3} \cdot 3 \cdot 7 $ $x^{2} - 42$ $C_2$ (as 2T1) $1$ $1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
*2 2.112896.8t5.b.a$2$ $ 2^{8} \cdot 3^{2} \cdot 7^{2}$ $x^{8} - 84 x^{6} + 1260 x^{4} - 5292 x^{2} + 441$ $Q_8$ (as 8T5) $-1$ $2$

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 *.