# Properties

 Label 6.0.937024.1 Degree $6$ Signature $[0, 3]$ Discriminant $-\,2^{6}\cdot 11^{4}$ Root discriminant $9.89$ Ramified primes $2, 11$ Class number $1$ Class group Trivial Galois group $S_3$ (as 6T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 7*x^4 + 20*x^2 + 4)

gp: K = bnfinit(x^6 - 7*x^4 + 20*x^2 + 4, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 20, 0, -7, 0, 1]);

## Normalizeddefining polynomial

$$x^{6} - 7 x^{4} + 20 x^{2} + 4$$

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: $$-937024=-\,2^{6}\cdot 11^{4}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.89$ 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: $2, 11$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $6$ This field is Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{12} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{24} a^{5} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{5}{12} a - \frac{1}{2}$

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: $$-\frac{1}{12} a^{5} + \frac{3}{4} a^{3} - \frac{13}{6} a$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{1}{6} a^{5} + \frac{1}{4} a^{4} - a^{3} - \frac{9}{4} a^{2} - \frac{1}{6} a - \frac{1}{2}$$,  $$\frac{17}{12} a^{5} + \frac{1}{12} a^{4} - \frac{41}{4} a^{3} - \frac{3}{4} a^{2} + \frac{91}{3} a + \frac{13}{6}$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$28.1184752065$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$S_3$ (as 6T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 6 The 3 conjugacy class representatives for $S_3$ Character table for $S_3$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: 3.1.484.1 $\times$ $$\Q$$ $\times$ $$\Q$$ $\times$ $$\Q$$ Degree 3 sibling: 3.1.484.1

## Multiplicative Galois module structure

 $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A'$

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 1111.6.4.1x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$

## 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) $1$ $-1$
*2 2.484.3t2.a.a$2$ $2^{2} \cdot 11^{2}$ $x^{6} - 7 x^{4} + 20 x^{2} + 4$ $S_3$ (as 6T2) $1$ $0$

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 *.