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)

Related objects

Show commands for: SageMath / Pari/GP / Magma

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)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -20, 16, -8, 9, -8, 3, -1, 1]);

Normalizeddefining polynomial

$$x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$$

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: $$307827025=5^{2}\cdot 11^{4}\cdot 29^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $11.51$ 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: $5, 11, 29$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

Integral basis (with respect to field generator $$a$$)

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

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: 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 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$12.1412815954$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 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

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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ 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.

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$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2} 5.2.0.1x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 1111.8.4.1x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$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$$[\ ]_{2}^{2} 29.4.0.1x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.1595.2t1.a.a$1$ $5 \cdot 11 \cdot 29$ $x^{2} - x + 399$ $C_2$ (as 2T1) $1$ $-1$
1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.319.2t1.a.a$1$ $11 \cdot 29$ $x^{2} - x + 80$ $C_2$ (as 2T1) $1$ $-1$
1.29.2t1.a.a$1$ $29$ $x^{2} - x - 7$ $C_2$ (as 2T1) $1$ $1$
1.55.2t1.a.a$1$ $5 \cdot 11$ $x^{2} - x + 14$ $C_2$ (as 2T1) $1$ $-1$
1.145.2t1.a.a$1$ $5 \cdot 29$ $x^{2} - x - 36$ $C_2$ (as 2T1) $1$ $1$
* 1.11.2t1.a.a$1$ $11$ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 2.55.4t3.b.a$2$ $5 \cdot 11$ $x^{4} - x^{3} + x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $0$
2.17545.4t3.b.a$2$ $5 \cdot 11^{2} \cdot 29$ $x^{4} - x^{3} + 30 x^{2} - 8 x + 229$ $D_{4}$ (as 4T3) $1$ $-2$
2.145.4t3.b.a$2$ $5 \cdot 29$ $x^{4} - x^{3} - 3 x^{2} + x + 1$ $D_{4}$ (as 4T3) $1$ $2$
2.1595.4t3.d.a$2$ $5 \cdot 11 \cdot 29$ $x^{4} - x^{3} + 3 x^{2} - 12 x - 16$ $D_{4}$ (as 4T3) $1$ $0$
2.46255.4t3.a.a$2$ $5 \cdot 11 \cdot 29^{2}$ $x^{4} - x^{3} - 20 x^{2} + 50 x + 267$ $D_{4}$ (as 4T3) $1$ $0$
2.1595.4t3.c.a$2$ $5 \cdot 11 \cdot 29$ $x^{4} - 2 x^{3} + x^{2} - 20$ $D_{4}$ (as 4T3) $1$ $0$
4.12720125.8t29.c.a$4$ $5^{3} \cdot 11^{2} \cdot 29^{2}$ $x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) $1$ $0$
* 4.508805.8t29.c.a$4$ $5 \cdot 11^{2} \cdot 29^{2}$ $x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ $(((C_4 \times C_2): C_2):C_2):C_2$ (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 *.