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Properties

Label 8.0.1050340608.4
Degree $8$
Signature $[0, 4]$
Discriminant $2^{8}\cdot 3^{4}\cdot 37^{3}$
Root discriminant $13.42$
Ramified primes $2, 3, 37$
Class number $2$
Class group $[2]$
Galois group $D_{8}$ (as 8T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37, 0, 17, 0, 18, 0, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 8*x^6 + 18*x^4 + 17*x^2 + 37)
 
gp: K = bnfinit(x^8 + 8*x^6 + 18*x^4 + 17*x^2 + 37, 1)
 

Normalized defining polynomial

\( x^{8} + 8 x^{6} + 18 x^{4} + 17 x^{2} + 37 \)

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:  \(1050340608=2^{8}\cdot 3^{4}\cdot 37^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.42$
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:  $2, 3, 37$
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 \(a\))

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

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:  \( \frac{2}{9} a^{6} + a^{4} + a^{2} + \frac{25}{9} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{9} a^{6} + \frac{2}{3} a^{4} + \frac{2}{3} a^{2} - \frac{4}{9} \),  \( \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} + \frac{2}{3} a^{4} + \frac{1}{3} a^{3} + \frac{5}{3} a^{2} + \frac{29}{9} a + \frac{32}{9} \),  \( \frac{5}{9} a^{7} - \frac{4}{9} a^{6} + \frac{7}{3} a^{5} - \frac{5}{3} a^{4} + \frac{4}{3} a^{3} - \frac{5}{3} a^{2} + \frac{43}{9} a - \frac{38}{9} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 66.5492709743 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_8$ (as 8T6):

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 $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.333.1

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 sibling: 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 R R ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$3$3.2.1.2$x^{2} + 3$$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$$[\ ]_{2}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.37.2t1.1c1$1$ $ 37 $ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_37.2t1.1c1$1$ $ 3 \cdot 37 $ $x^{2} - x + 28$ $C_2$ (as 2T1) $1$ $-1$
* 2.3_37.4t3.1c1$2$ $ 3 \cdot 37 $ $x^{4} + 5 x^{2} - 3$ $D_{4}$ (as 4T3) $1$ $0$
* 2.2e4_3_37.8t6.2c1$2$ $ 2^{4} \cdot 3 \cdot 37 $ $x^{8} + 8 x^{6} + 18 x^{4} + 17 x^{2} + 37$ $D_{8}$ (as 8T6) $1$ $0$
* 2.2e4_3_37.8t6.2c2$2$ $ 2^{4} \cdot 3 \cdot 37 $ $x^{8} + 8 x^{6} + 18 x^{4} + 17 x^{2} + 37$ $D_{8}$ (as 8T6) $1$ $0$

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 *.