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Properties

Label 8.0.9907677.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{4}\cdot 13\cdot 97^{2}$
Root discriminant $7.49$
Ramified primes $3, 13, 97$
Class number $1$
Class group Trivial
Galois group $C_2 \wr C_2\wr C_2$ (as 8T35)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 3, 1, -1, -2, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 2*x^5 - x^4 + x^3 + 3*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^8 - x^7 - 2*x^5 - x^4 + x^3 + 3*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{8} - x^{7} - 2 x^{5} - x^{4} + x^{3} + 3 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $8$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9907677=3^{4}\cdot 13\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $7.49$
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:  $3, 13, 97$
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 \(a\))

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

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:  $3$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{1}{2} a^{7} + a^{5} + \frac{3}{2} a^{3} - a^{2} - \frac{3}{2} a - \frac{1}{2} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{1}{2} a^{7} - a^{6} - \frac{1}{2} a^{3} + 2 a^{2} + \frac{1}{2} a - \frac{1}{2} \),  \( a^{7} - a^{6} - 2 a^{4} - a^{3} + 2 a^{2} + 2 a + 2 \),  \( \frac{1}{2} a^{7} - a^{6} + a^{5} - 2 a^{4} + \frac{3}{2} a^{3} + \frac{3}{2} a + \frac{1}{2} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4.30215121927 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4^2.C_2$ (as 8T35):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$
Character table for $C_2 \wr C_2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.873.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$97$97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.97.2t1.1c1$1$ $ 97 $ $x^{2} - x - 24$ $C_2$ (as 2T1) $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.3_97.2t1.1c1$1$ $ 3 \cdot 97 $ $x^{2} - x + 73$ $C_2$ (as 2T1) $1$ $-1$
1.3_13_97.2t1.1c1$1$ $ 3 \cdot 13 \cdot 97 $ $x^{2} - x + 946$ $C_2$ (as 2T1) $1$ $-1$
1.3_13.2t1.1c1$1$ $ 3 \cdot 13 $ $x^{2} - x + 10$ $C_2$ (as 2T1) $1$ $-1$
1.13_97.2t1.1c1$1$ $ 13 \cdot 97 $ $x^{2} - x - 315$ $C_2$ (as 2T1) $1$ $1$
1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
2.3_13_97.4t3.2c1$2$ $ 3 \cdot 13 \cdot 97 $ $x^{4} - 31 x^{2} - 75$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13_97e2.4t3.1c1$2$ $ 3 \cdot 13 \cdot 97^{2}$ $x^{4} - x^{3} + 23 x^{2} - 169 x - 1703$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13.4t3.1c1$2$ $ 3 \cdot 13 $ $x^{4} - x^{3} - x^{2} - x + 1$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3_97.4t3.2c1$2$ $ 3 \cdot 97 $ $x^{4} - x^{3} - 4 x^{2} + x + 7$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13e2_97.4t3.2c1$2$ $ 3 \cdot 13^{2} \cdot 97 $ $x^{4} - x^{3} - 61 x^{2} + 16 x + 1036$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13_97.4t3.1c1$2$ $ 3 \cdot 13 \cdot 97 $ $x^{4} + 17 x^{2} - 243$ $D_{4}$ (as 4T3) $1$ $0$
* 4.3e2_13_97.8t35.2c1$4$ $ 3^{2} \cdot 13 \cdot 97 $ $x^{8} - x^{7} - 2 x^{5} - x^{4} + x^{3} + 3 x^{2} + 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e2_13e3_97.8t35.2c1$4$ $ 3^{2} \cdot 13^{3} \cdot 97 $ $x^{8} - x^{7} - 2 x^{5} - x^{4} + x^{3} + 3 x^{2} + 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3_13e2_97e2.8t29.1c1$4$ $ 3 \cdot 13^{2} \cdot 97^{2}$ $x^{8} - x^{7} - 8 x^{6} + x^{5} + 14 x^{4} + 22 x^{3} + 20 x^{2} - 35 x - 22$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $2$
4.3e2_13e3_97e3.8t35.2c1$4$ $ 3^{2} \cdot 13^{3} \cdot 97^{3}$ $x^{8} - x^{7} - 2 x^{5} - x^{4} + x^{3} + 3 x^{2} + 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e2_13_97e3.8t35.2c1$4$ $ 3^{2} \cdot 13 \cdot 97^{3}$ $x^{8} - x^{7} - 2 x^{5} - x^{4} + x^{3} + 3 x^{2} + 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e3_13e2_97e2.8t29.1c1$4$ $ 3^{3} \cdot 13^{2} \cdot 97^{2}$ $x^{8} - x^{7} - 8 x^{6} + x^{5} + 14 x^{4} + 22 x^{3} + 20 x^{2} - 35 x - 22$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $-2$

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 *.