# Properties

 Label 6.0.2685619.1 Degree $6$ Signature $[0, 3]$ Discriminant $-\,139^{3}$ Root discriminant $11.79$ Ramified prime $139$ Class number $1$ Class group Trivial Galois Group $S_3$ (as 6T2)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, -18, 37, -12, 3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 3*x^4 - 12*x^3 + 37*x^2 - 18*x + 36)
gp: K = bnfinit(x^6 + 3*x^4 - 12*x^3 + 37*x^2 - 18*x + 36, 1)

## Normalizeddefining polynomial

$$x^{6}$$ $$\mathstrut +\mathstrut 3 x^{4}$$ $$\mathstrut -\mathstrut 12 x^{3}$$ $$\mathstrut +\mathstrut 37 x^{2}$$ $$\mathstrut -\mathstrut 18 x$$ $$\mathstrut +\mathstrut 36$$

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: $$-2685619=-\,139^{3}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $11.79$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $139$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{564} a^{5} + \frac{11}{94} a^{4} - \frac{1}{47} a^{3} + \frac{61}{188} a^{2} - \frac{38}{141} a - \frac{15}{47}$

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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental units: $$\frac{35}{564} a^{5} + \frac{9}{94} a^{4} + \frac{12}{47} a^{3} - \frac{121}{188} a^{2} + \frac{80}{141} a - \frac{55}{47}$$,  $$\frac{7}{188} a^{5} - \frac{2}{47} a^{4} + \frac{5}{94} a^{3} - \frac{129}{188} a^{2} + \frac{63}{47} a - \frac{127}{47}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$8.30224307268$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$S_3$ (as 6T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 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: 3.1.139.1 $\times$ $$\Q$$ $\times$ $$\Q$$ $\times$ $$\Q$$ Degree 3 sibling: 3.1.139.1

## 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$139$139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2} 139.2.1.2x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.1.2$x^{2} + 556$$2$$1$$1$$C_2$$[\ ]_{2}$