Properties

Label 8.0.307827025.1
Degree $8$
Signature $[0, 4]$
Discriminant $5^{2}\cdot 11^{4}\cdot 29^{2}$
Root discriminant $11.51$
Ramified primes $5, 11, 29$
Class number $1$
Class group Trivial
Galois Group $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -20, 16, -8, 9, -8, 3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 3*x^6 - 8*x^5 + 9*x^4 - 8*x^3 + 16*x^2 - 20*x + 9)
gp: K = bnfinit(x^8 - x^7 + 3*x^6 - 8*x^5 + 9*x^4 - 8*x^3 + 16*x^2 - 20*x + 9, 1)

Normalized defining polynomial

\(x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut 3 x^{6} \) \(\mathstrut -\mathstrut 8 x^{5} \) \(\mathstrut +\mathstrut 9 x^{4} \) \(\mathstrut -\mathstrut 8 x^{3} \) \(\mathstrut +\mathstrut 16 x^{2} \) \(\mathstrut -\mathstrut 20 x \) \(\mathstrut +\mathstrut 9 \)

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:  \(307827025=5^{2}\cdot 11^{4}\cdot 29^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.51$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{95} a^{7} - \frac{32}{95} a^{6} + \frac{9}{19} a^{5} + \frac{22}{95} a^{4} - \frac{8}{95} a^{3} - \frac{9}{19} a^{2} - \frac{14}{95} a + \frac{34}{95}$

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 12.1412815954 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\wr C_2^2$ (as 8T29):

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 $(((C_4 \times C_2): C_2):C_2):C_2$
Character table for $(((C_4 \times C_2): C_2):C_2):C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.605.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5_11_29.2t1.1c1$1$ $ 5 \cdot 11 \cdot 29 $ $x^{2} - x + 399$ $C_2$ (as 2T1) $1$ $-1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.11_29.2t1.1c1$1$ $ 11 \cdot 29 $ $x^{2} - x + 80$ $C_2$ (as 2T1) $1$ $-1$
1.29.2t1.1c1$1$ $ 29 $ $x^{2} - x - 7$ $C_2$ (as 2T1) $1$ $1$
1.5_11.2t1.1c1$1$ $ 5 \cdot 11 $ $x^{2} - x + 14$ $C_2$ (as 2T1) $1$ $-1$
1.5_29.2t1.1c1$1$ $ 5 \cdot 29 $ $x^{2} - x - 36$ $C_2$ (as 2T1) $1$ $1$
1.11.2t1.1c1$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
2.5_11.4t3.2c1$2$ $ 5 \cdot 11 $ $x^{4} - x^{3} + x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.5_11e2_29.4t3.2c1$2$ $ 5 \cdot 11^{2} \cdot 29 $ $x^{4} - x^{3} + 30 x^{2} - 8 x + 229$ $D_{4}$ (as 4T3) $1$ $-2$
2.5_29.4t3.2c1$2$ $ 5 \cdot 29 $ $x^{4} - x^{3} - 3 x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $2$
2.5_11_29.4t3.2c1$2$ $ 5 \cdot 11 \cdot 29 $ $x^{4} - x^{3} + 3 x^{2} - 12 x - 16$ $D_{4}$ (as 4T3) $1$ $0$
2.5_11_29e2.4t3.2c1$2$ $ 5 \cdot 11 \cdot 29^{2}$ $x^{4} - x^{3} - 20 x^{2} + 50 x + 267$ $D_{4}$ (as 4T3) $1$ $0$
2.5_11_29.4t3.1c1$2$ $ 5 \cdot 11 \cdot 29 $ $x^{4} - 2 x^{3} + x^{2} - 20$ $D_{4}$ (as 4T3) $1$ $0$
4.5e3_11e2_29e2.8t29.3c1$4$ $ 5^{3} \cdot 11^{2} \cdot 29^{2}$ $x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $0$
4.5_11e2_29e2.8t29.3c1$4$ $ 5 \cdot 11^{2} \cdot 29^{2}$ $x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $0$

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 *.