# Properties

 Label 6.0.10816.1 Degree $6$ Signature $[0, 3]$ Discriminant $-10816$ Root discriminant $4.70$ Ramified primes $2, 13$ Class number $1$ Class group trivial Galois group $S_3\times C_3$ (as 6T5)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 0, -2, -1, 0, 1]);

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-10816$$$$\medspace = -\,2^{6}\cdot 13^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $4.70$ 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: $2, 13$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $3$ This field is not Galois over $\Q$. This is not a CM field.

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

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

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: $2$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-a^{5} + a^{4} + a^{3} + 2 a^{2} - 2 a - 2$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$2 a^{5} - a^{4} - a^{3} - 3 a^{2} + a + 2$$,  $$2 a^{5} - a^{4} - a^{3} - 3 a^{2} + 2 a + 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.434147804699$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.434147804699 \cdot 1}{4\sqrt{10816}}\approx 0.2588712875508$

## Galois group

$C_3\times S_3$ (as 6T5):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 18 The 9 conjugacy class representatives for $S_3\times C_3$ Character table for $S_3\times C_3$

## Intermediate fields

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

## Sibling algebras

 Galois closure: 18.0.6107453226347659264.1 Twin sextic algebra: 3.3.169.1 $\times$ 3.1.676.1 Degree 9 sibling: 9.3.308915776.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3} 1313.3.0.1x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.13.3t1.a.a$1$ $13$ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.52.6t1.b.a$1$ $2^{2} \cdot 13$ $x^{6} + 9 x^{4} + 14 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.52.6t1.b.b$1$ $2^{2} \cdot 13$ $x^{6} + 9 x^{4} + 14 x^{2} + 1$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.a.b$1$ $13$ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
2.676.3t2.b.a$2$ $2^{2} \cdot 13^{2}$ $x^{3} - x^{2} - 4 x + 12$ $S_3$ (as 3T2) $1$ $0$
* 2.52.6t5.b.a$2$ $2^{2} \cdot 13$ $x^{6} - x^{4} - 2 x^{3} + 2 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.52.6t5.b.b$2$ $2^{2} \cdot 13$ $x^{6} - x^{4} - 2 x^{3} + 2 x + 1$ $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.