# Properties

 Label 6.0.16807.1 Degree $6$ Signature $[0, 3]$ Discriminant $-\,7^{5}$ Root discriminant $5.06$ Ramified prime $7$ Class number $1$ Class group Trivial Galois group $C_6$ (as 6T1)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)

gp: K = bnfinit(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

## Normalizeddefining polynomial

$$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $6$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 3]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$-16807=-\,7^{5}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $5.06$ 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: $7$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$7$$ Dirichlet character group: $\lbrace$$\chi_{7}(1,·), \chi_{7}(2,·), \chi_{7}(3,·), \chi_{7}(4,·), \chi_{7}(5,·), \chi_{7}(6,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$

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: $2$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$a$$ (order $14$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: $$a - 1$$,  $$a^{4} - a$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$2.10181872849$$ magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 6 The 6 conjugacy class representatives for $C_6$ Character table for $C_6$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: $$\Q(\zeta_{7})^+$$ $\times$ $$\Q(\sqrt{-7})$$ $\times$ $$\Q$$

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.7.2t1.1c1$1$ $7$ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.6t1.1c1$1$ $7$ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c2$1$ $7$ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.6t1.1c2$1$ $7$ $x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_6$ (as 6T1) $0$ $-1$

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 *.