# Properties

 Label 6.0.184832.1 Degree $6$ Signature $[0, 3]$ Discriminant $-184832$ Root discriminant $$7.55$$ Ramified primes see page Class number $1$ Class group trivial Galois group $S_3\times C_3$ (as 6T5)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -4, 0, 2, 1, -2, 1]);

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

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: $$-184832$$ -184832 $$\medspace = -\,2^{9}\cdot 19^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$7.55$$ 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: $$2$$, $$19$$ 2, 19 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $3$ This field is not Galois over $\Q$. This is not a CM field.

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

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: Not computed Index: $1$ Inessential primes: None

## 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: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{2}{3}a^{5}-a^{4}-\frac{1}{3}a^{3}+\frac{5}{3}a^{2}+\frac{4}{3}a-2$, $\frac{1}{3}a^{5}-\frac{2}{3}a^{3}+\frac{1}{3}a^{2}+\frac{2}{3}a$ 2/3*a^5 - a^4 - 1/3*a^3 + 5/3*a^2 + 4/3*a - 2, 1/3*a^5 - 2/3*a^3 + 1/3*a^2 + 2/3*a sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$2.37416675486$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 2.37416675486 \cdot 1}{2\sqrt{184832}}\approx 0.684908013123$

## 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: 18.0.297066099785564011102208.1 Twin sextic algebra: 3.1.2888.1 $\times$ 3.3.361.1 Degree 9 sibling: 9.3.24087491072.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ ${\href{/padicField/5.6.0.1}{6} }$ ${\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.3.0.1}{3} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}$ R ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.6.0.1}{6} }$ ${\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.2.0.1}{2} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.6.0.1}{6} }$ ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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
$$2$$ 2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3} $$19$$ 19.3.2.1x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.8.2t1.b.a$1$ $2^{3}$ $$\Q(\sqrt{-2})$$ $C_2$ (as 2T1) $1$ $-1$
1.19.3t1.a.a$1$ $19$ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.152.6t1.c.a$1$ $2^{3} \cdot 19$ 6.0.66724352.1 $C_6$ (as 6T1) $0$ $-1$
1.152.6t1.c.b$1$ $2^{3} \cdot 19$ 6.0.66724352.1 $C_6$ (as 6T1) $0$ $-1$
1.19.3t1.a.b$1$ $19$ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
2.2888.3t2.b.a$2$ $2^{3} \cdot 19^{2}$ 3.1.2888.1 $S_3$ (as 3T2) $1$ $0$
* 2.152.6t5.a.a$2$ $2^{3} \cdot 19$ 6.0.184832.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.152.6t5.a.b$2$ $2^{3} \cdot 19$ 6.0.184832.1 $S_3\times C_3$ (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 *.