Properties

Label 8.0.1327833.1
Degree $8$
Signature $[0, 4]$
Discriminant $3^{4}\cdot 13^{2}\cdot 97$
Root discriminant $5.83$
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, 4, -4, 3, -2, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + x^6 - 2*x^5 + 3*x^4 - 4*x^3 + 4*x^2 - 2*x + 1)
gp: K = bnfinit(x^8 - x^7 + x^6 - 2*x^5 + 3*x^4 - 4*x^3 + 4*x^2 - 2*x + 1, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 3 x^{4} \) \(\mathstrut -\mathstrut 4 x^{3} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(1327833=3^{4}\cdot 13^{2}\cdot 97\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $5.83$
magma: Abs(Discriminant(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(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:  $3$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( 2 a^{7} - 2 a^{6} + a^{5} - 3 a^{4} + 6 a^{3} - 6 a^{2} + 5 a - 1 \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a^{6} - a^{5} - 2 a^{3} + 3 a^{2} - 3 a + 2 \),  \( a^{7} - 2 a^{4} + a^{3} - a^{2} + a + 1 \),  \( a^{7} - a^{6} - 2 a^{4} + 3 a^{3} - 2 a^{2} + 2 a - 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.963083858701 \)
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

\(\Q(\sqrt{-3}) \), 4.0.117.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }$

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$$[\ ]_{2}^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97Data not computed

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_97.2t1.1c1$1$ $ 3 \cdot 97 $ $x^{2} - x + 73$ $C_2$ (as 2T1) $1$ $-1$
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_13_97.2t1.1c1$1$ $ 3 \cdot 13 \cdot 97 $ $x^{2} - x + 946$ $C_2$ (as 2T1) $1$ $-1$
1.13.2t1.1c1$1$ $ 13 $ $x^{2} - x - 3$ $C_2$ (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) $1$ $1$
2.3_13_97.4t3.4c1$2$ $ 3 \cdot 13 \cdot 97 $ $x^{4} - x^{3} - 17 x^{2} + 9 x + 81$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13_97.4t3.3c1$2$ $ 3 \cdot 13 \cdot 97 $ $x^{4} - 2 x^{3} - 14 x^{2} + 15 x + 75$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13.4t3.2c1$2$ $ 3 \cdot 13 $ $x^{4} - x^{3} - x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.3_13_97e2.4t3.2c1$2$ $ 3 \cdot 13 \cdot 97^{2}$ $x^{4} - x^{3} - 169 x^{2} + 121 x + 7657$ $D_{4}$ (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_97.4t3.2c1$2$ $ 3 \cdot 97 $ $x^{4} - x^{3} - 4 x^{2} + x + 7$ $D_{4}$ (as 4T3) $1$ $0$
4.3e3_13e2_97e2.8t29.2c1$4$ $ 3^{3} \cdot 13^{2} \cdot 97^{2}$ $x^{8} - x^{7} + 5 x^{6} + 44 x^{4} - 171 x^{3} + 263 x^{2} - 13 x + 88$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $-2$
4.3e2_13e3_97.8t35.1c1$4$ $ 3^{2} \cdot 13^{3} \cdot 97 $ $x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e2_13_97e3.8t35.1c1$4$ $ 3^{2} \cdot 13 \cdot 97^{3}$ $x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3_13e2_97e2.8t29.2c1$4$ $ 3 \cdot 13^{2} \cdot 97^{2}$ $x^{8} - x^{7} + 5 x^{6} + 44 x^{4} - 171 x^{3} + 263 x^{2} - 13 x + 88$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $2$
4.3e2_13e3_97e3.8t35.1c1$4$ $ 3^{2} \cdot 13^{3} \cdot 97^{3}$ $x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$
4.3e2_13_97.8t35.1c1$4$ $ 3^{2} \cdot 13 \cdot 97 $ $x^{8} - x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 4 x^{3} + 4 x^{2} - 2 x + 1$ $C_2 \wr C_2\wr C_2$ (as 8T35) $1$ $0$

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 *.