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Properties

Label 8.0.20025646144.2
Degree $8$
Signature $[0, 4]$
Discriminant $2^{6}\cdot 7^{4}\cdot 19^{4}$
Root discriminant $19.40$
Ramified primes $2, 7, 19$
Class number $4$
Class group $[4]$
Galois group $((C_2 \times D_4): C_2):C_3$ (as 8T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 7, 0, 3, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 3*x^4 + 7*x^2 + 4)
 
gp: K = bnfinit(x^8 + 3*x^4 + 7*x^2 + 4, 1)
 

Normalized defining polynomial

\( x^{8} + 3 x^{4} + 7 x^{2} + 4 \)

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:  \(20025646144=2^{6}\cdot 7^{4}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.40$
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:  $2, 7, 19$
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 \(a\))

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

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

Class group and class number

$C_{4}$, which has order $4$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( a^{2} + 1 \),  \( \frac{1}{4} a^{7} + \frac{1}{2} a^{6} + \frac{3}{4} a^{3} + \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{3}{2} \),  \( \frac{3}{4} a^{7} + \frac{1}{2} a^{6} - a^{5} + \frac{13}{4} a^{3} + \frac{3}{2} a^{2} + \frac{9}{4} a + \frac{7}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 62.4040738617 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:A_4$ (as 8T32):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 96
The 11 conjugacy class representatives for $((C_2 \times D_4): C_2):C_3$
Character table for $((C_2 \times D_4): C_2):C_3$

Intermediate fields

4.0.17689.1

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

Sibling fields

Degree 8 siblings: data not computed
Degree 24 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.6.4.3$x^{6} + 95 x^{3} + 2888$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.19.3t1.1c1$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
1.19.3t1.1c2$1$ $ 19 $ $x^{3} - x^{2} - 6 x + 7$ $C_3$ (as 3T1) $0$ $1$
3.2e6_19e2.4t4.1c1$3$ $ 2^{6} \cdot 19^{2}$ $x^{4} - 2 x^{3} + 10 x^{2} - 8 x + 2$ $A_4$ (as 4T4) $1$ $-1$
3.2e6_7e2_19e2.4t4.3c1$3$ $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ $x^{4} - 2 x^{3} + 6 x^{2} + 2 x + 15$ $A_4$ (as 4T4) $1$ $-1$
3.2e6_7e2_19e2.4t4.1c1$3$ $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ $x^{4} - 30 x^{2} - 56 x + 1$ $A_4$ (as 4T4) $1$ $3$
3.2e6_7e2_19e2.4t4.6c1$3$ $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ $x^{4} - 2 x^{3} + 14 x^{2} - 6 x + 107$ $A_4$ (as 4T4) $1$ $-1$
* 3.7e2_19e2.4t4.1c1$3$ $ 7^{2} \cdot 19^{2}$ $x^{4} + 3 x^{2} - 7 x + 4$ $A_4$ (as 4T4) $1$ $-1$
* 4.2e6_7e2_19e2.8t32.3c1$4$ $ 2^{6} \cdot 7^{2} \cdot 19^{2}$ $x^{8} + 3 x^{4} + 7 x^{2} + 4$ $((C_2 \times D_4): C_2):C_3$ (as 8T32) $1$ $0$
4.2e6_7e2_19e3.24t97.3c1$4$ $ 2^{6} \cdot 7^{2} \cdot 19^{3}$ $x^{8} + 3 x^{4} + 7 x^{2} + 4$ $((C_2 \times D_4): C_2):C_3$ (as 8T32) $0$ $0$
4.2e6_7e2_19e3.24t97.3c2$4$ $ 2^{6} \cdot 7^{2} \cdot 19^{3}$ $x^{8} + 3 x^{4} + 7 x^{2} + 4$ $((C_2 \times D_4): C_2):C_3$ (as 8T32) $0$ $0$

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 *.