Properties

Label 8.0.796594176.14
Degree $8$
Signature $[0, 4]$
Discriminant $2^{12}\cdot 3^{4}\cdot 7^{4}$
Root discriminant $12.96$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $D_4$ (as 8T4)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 4*x^6 + 12*x^5 - 21*x^4 + 36*x^3 + 36*x^2 - 54*x + 81)
 
gp: K = bnfinit(x^8 - 2*x^7 + 4*x^6 + 12*x^5 - 21*x^4 + 36*x^3 + 36*x^2 - 54*x + 81, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -54, 36, 36, -21, 12, 4, -2, 1]);
 

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 4 x^{6} + 12 x^{5} - 21 x^{4} + 36 x^{3} + 36 x^{2} - 54 x + 81 \)

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

Invariants

Degree:  $8$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(796594176=2^{12}\cdot 3^{4}\cdot 7^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.96$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{6} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a$

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:  $3$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{2}{27} a^{7} + \frac{4}{27} a^{6} - \frac{8}{27} a^{5} - \frac{5}{9} a^{4} + \frac{8}{9} a^{3} - \frac{4}{3} a^{2} - \frac{2}{3} a + 1 \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{1}{27} a^{7} + \frac{1}{27} a^{6} - \frac{2}{27} a^{5} + \frac{5}{9} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} + \frac{4}{3} a \),  \( \frac{1}{27} a^{7} + \frac{4}{27} a^{6} + \frac{1}{27} a^{5} + \frac{2}{3} a^{4} + \frac{20}{9} a^{3} + \frac{2}{3} a^{2} + \frac{4}{3} a + 4 \),  \( \frac{2}{27} a^{7} - \frac{4}{27} a^{6} + \frac{5}{27} a^{5} + \frac{4}{9} a^{4} - \frac{4}{3} a^{3} + a^{2} + \frac{1}{3} a - 3 \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 63.8564851446 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$D_4$ (as 8T4):

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 $D_4$
Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{7})\), 4.2.9408.2 x2, 4.0.1008.1 x2

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

Sibling fields

Degree 4 siblings: 4.0.1008.1, 4.2.9408.2

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 ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.28.2t1.a.a$1$ $ 2^{2} \cdot 7 $ $x^{2} - 7$ $C_2$ (as 2T1) $1$ $1$
* 1.84.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 7 $ $x^{2} + 21$ $C_2$ (as 2T1) $1$ $-1$
*2 2.336.4t3.f.a$2$ $ 2^{4} \cdot 3 \cdot 7 $ $x^{8} - 2 x^{7} + 4 x^{6} + 12 x^{5} - 21 x^{4} + 36 x^{3} + 36 x^{2} - 54 x + 81$ $D_4$ (as 8T4) $1$ $0$

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 *.