Properties

 Label 8.0.741610573.1 Degree $8$ Signature $[0, 4]$ Discriminant $11^{4}\cdot 37^{3}$ Root discriminant $12.85$ Ramified primes $11, 37$ Class number $1$ Class group Trivial Galois Group $D_{8}$ (as 8T6)

Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 20, -15, 9, -1, -1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - x^6 - x^5 + 9*x^4 - 15*x^3 + 20*x^2 - 8*x + 1)
gp: K = bnfinit(x^8 - x^7 - x^6 - x^5 + 9*x^4 - 15*x^3 + 20*x^2 - 8*x + 1, 1)

Normalizeddefining polynomial

$$x^{8}$$ $$\mathstrut -\mathstrut x^{7}$$ $$\mathstrut -\mathstrut x^{6}$$ $$\mathstrut -\mathstrut x^{5}$$ $$\mathstrut +\mathstrut 9 x^{4}$$ $$\mathstrut -\mathstrut 15 x^{3}$$ $$\mathstrut +\mathstrut 20 x^{2}$$ $$\mathstrut -\mathstrut 8 x$$ $$\mathstrut +\mathstrut 1$$

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: $$741610573=11^{4}\cdot 37^{3}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $12.85$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $11, 37$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} + \frac{3}{11} a^{5} - \frac{4}{11} a^{4} + \frac{4}{11} a^{3} - \frac{3}{11} a^{2} + \frac{1}{11} a + \frac{3}{11}$, $\frac{1}{11} a^{7} - \frac{2}{11} a^{5} + \frac{5}{11} a^{4} - \frac{4}{11} a^{3} - \frac{1}{11} a^{2} + \frac{2}{11}$

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

Class group and class number

Trivial Abelian group, 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: $$\frac{10}{11} a^{7} - \frac{5}{11} a^{6} - \frac{13}{11} a^{5} - \frac{18}{11} a^{4} + \frac{83}{11} a^{3} - \frac{105}{11} a^{2} + \frac{149}{11} a - \frac{17}{11}$$,  $$a$$,  $$\frac{7}{11} a^{7} - \frac{6}{11} a^{6} - \frac{10}{11} a^{5} - \frac{7}{11} a^{4} + \frac{69}{11} a^{3} - 8 a^{2} + \frac{104}{11} a - \frac{26}{11}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$15.9941542144$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

Galois group

$D_8$ (as 8T6):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A solvable group of order 16 The 7 conjugacy class representatives for $D_{8}$ Character table for $D_{8}$

Intermediate fields

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

Sibling fields

 Galois closure: data not computed Degree 8 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.2x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.1.2x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$37$$\Q_{37}$$x + 2$$1$$1$$0Trivial[\ ] \Q_{37}$$x + 2$$1$$1$$0Trivial[\ ] 37.2.1.1x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$
37.2.1.1$x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2} 37.2.1.1x^{2} - 37$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.11.2t1.1c1$1$ $11$ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
1.37.2t1.1c1$1$ $37$ $x^{2} - x - 9$ $C_2$ (as 2T1) $1$ $1$
1.11_37.2t1.1c1$1$ $11 \cdot 37$ $x^{2} - x + 102$ $C_2$ (as 2T1) $1$ $-1$
* 2.11_37.4t3.2c1$2$ $11 \cdot 37$ $x^{4} - x^{3} - 4 x + 5$ $D_{4}$ (as 4T3) $1$ $0$
* 2.11_37.8t6.2c1$2$ $11 \cdot 37$ $x^{8} - x^{7} - x^{6} - x^{5} + 9 x^{4} - 15 x^{3} + 20 x^{2} - 8 x + 1$ $D_{8}$ (as 8T6) $1$ $0$
* 2.11_37.8t6.2c2$2$ $11 \cdot 37$ $x^{8} - x^{7} - x^{6} - x^{5} + 9 x^{4} - 15 x^{3} + 20 x^{2} - 8 x + 1$ $D_{8}$ (as 8T6) $1$ $0$

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 *.