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Properties

Label 8.2.67381875.1
Degree $8$
Signature $[2, 3]$
Discriminant $-\,3^{4}\cdot 5^{4}\cdot 11^{3}$
Root discriminant $9.52$
Ramified primes $3, 5, 11$
Class number $1$
Class group Trivial
Galois group $(C_4^2 : C_2):C_2$ (as 8T26)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 0, -8, 1, 9, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 5*x^6 + 9*x^5 + x^4 - 8*x^3 + 3*x + 1)
 
gp: K = bnfinit(x^8 - x^7 - 5*x^6 + 9*x^5 + x^4 - 8*x^3 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 5 x^{6} + 9 x^{5} + x^{4} - 8 x^{3} + 3 x + 1 \)

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:  \(-67381875=-\,3^{4}\cdot 5^{4}\cdot 11^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $9.52$
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:  $3, 5, 11$
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 \(a\))

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

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

Class group and class number

Trivial group, which has order $1$

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:  \( a^{6} - a^{5} - 5 a^{4} + 8 a^{3} - 4 a - 1 \),  \( a^{7} - a^{6} - 4 a^{5} + 8 a^{4} - 4 a^{3} - 1 \),  \( a^{6} - 4 a^{4} + 4 a^{3} \),  \( a^{7} - a^{6} - 4 a^{5} + 8 a^{4} - 4 a^{3} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5.48804517253 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4:D_4$ (as 8T26):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$
Character table for $(C_4^2 : C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.2475.1

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 16 siblings: data not computed
Degree 32 siblings: 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 R ${\href{/LocalNumberField/7.8.0.1}{8} }$ R ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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

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$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{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_5.2t1.1c1$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
1.5_11.2t1.1c1$1$ $ 5 \cdot 11 $ $x^{2} - x + 14$ $C_2$ (as 2T1) $1$ $-1$
1.3_11.2t1.1c1$1$ $ 3 \cdot 11 $ $x^{2} - x - 8$ $C_2$ (as 2T1) $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3_5_11.2t1.1c1$1$ $ 3 \cdot 5 \cdot 11 $ $x^{2} - x - 41$ $C_2$ (as 2T1) $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 2.3e2_5_11.4t3.1c1$2$ $ 3^{2} \cdot 5 \cdot 11 $ $x^{4} - x^{3} - x^{2} - 5 x - 5$ $D_{4}$ (as 4T3) $1$ $0$
2.3_11e2.4t3.1c1$2$ $ 3 \cdot 11^{2}$ $x^{4} - 2 x^{3} - 4 x^{2} + 5 x - 2$ $D_{4}$ (as 4T3) $1$ $0$
2.3_5_11e2.4t3.4c1$2$ $ 3 \cdot 5 \cdot 11^{2}$ $x^{4} - x^{3} - x^{2} - 9 x + 15$ $D_{4}$ (as 4T3) $1$ $0$
2.3_5e2_11e2.4t3.1c1$2$ $ 3 \cdot 5^{2} \cdot 11^{2}$ $x^{4} - x^{3} + 10 x^{2} + 57 x - 51$ $D_{4}$ (as 4T3) $1$ $0$
2.3_5_11e2.4t3.3c1$2$ $ 3 \cdot 5 \cdot 11^{2}$ $x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 9$ $D_{4}$ (as 4T3) $1$ $0$
2.5_11.4t3.1c1$2$ $ 5 \cdot 11 $ $x^{4} - x^{3} + 2 x - 1$ $D_{4}$ (as 4T3) $1$ $0$
* 4.3e2_5e2_11e2.8t26.4c1$4$ $ 3^{2} \cdot 5^{2} \cdot 11^{2}$ $x^{8} - x^{7} - 5 x^{6} + 9 x^{5} + x^{4} - 8 x^{3} + 3 x + 1$ $(C_4^2 : C_2):C_2$ (as 8T26) $1$ $0$
4.3e2_5e2_11e4.8t26.4c1$4$ $ 3^{2} \cdot 5^{2} \cdot 11^{4}$ $x^{8} - x^{7} - 5 x^{6} + 9 x^{5} + x^{4} - 8 x^{3} + 3 x + 1$ $(C_4^2 : C_2):C_2$ (as 8T26) $1$ $0$

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 *.