Properties

Label 8.0.9150625.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{4}\cdot 11^{4}$
Root discriminant $7.42$
Ramified primes $5, 11$
Class number $1$
Class group Trivial
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 - 3*x^7 + 9*x^6 - 13*x^5 + 18*x^4 - 11*x^3 + 11*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^8 - 3*x^7 + 9*x^6 - 13*x^5 + 18*x^4 - 11*x^3 + 11*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 11, -11, 18, -13, 9, -3, 1]);
 

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 9 x^{6} - 13 x^{5} + 18 x^{4} - 11 x^{3} + 11 x^{2} - 4 x + 1 \)

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:  \(9150625=5^{4}\cdot 11^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $7.42$
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:  $5, 11$
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 \(a\))

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

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

Class group and class number

Trivial group, which has order $1$

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:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{1}{3} a^{7} - \frac{5}{6} a^{6} + \frac{7}{3} a^{5} - \frac{7}{3} a^{4} + \frac{17}{6} a^{3} + \frac{1}{6} a^{2} + \frac{11}{6} a + \frac{1}{6} \),  \( \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{4}{3} a^{4} - \frac{3}{2} a^{3} + \frac{17}{6} a^{2} - \frac{7}{6} a + \frac{7}{6} \),  \( \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{4}{3} a^{5} - \frac{11}{6} a^{4} + \frac{5}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1.13294889772 \)
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

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 4.0.605.1 x2, 4.2.275.1 x2

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

Sibling fields

Degree 4 siblings: 4.2.275.1, 4.0.605.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.11.2t1.a.a$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 1.55.2t1.a.a$1$ $ 5 \cdot 11 $ $x^{2} - x + 14$ $C_2$ (as 2T1) $1$ $-1$
*2 2.55.4t3.a.a$2$ $ 5 \cdot 11 $ $x^{8} - 3 x^{7} + 9 x^{6} - 13 x^{5} + 18 x^{4} - 11 x^{3} + 11 x^{2} - 4 x + 1$ $D_4$ (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 *.