# Properties

 Label 6.0.30808.1 Degree $6$ Signature $[0, 3]$ Discriminant $-30808$ Root discriminant $5.60$ Ramified primes $2, 3851$ Class number $1$ Class group trivial Galois group $S_6$ (as 6T16)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

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

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

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: $$-30808$$$$\medspace = -\,2^{3}\cdot 3851$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.60$ 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: $2, 3851$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ 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: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{5} - a^{3} - a^{2} + a + 1$$,  $$a^{5} - a^{3} + a$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.643679055537$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.643679055537 \cdot 1}{2\sqrt{30808}}\approx 0.4548280438136$

## Galois group

$S_6$ (as 6T16):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 720 The 11 conjugacy class representatives for $S_6$ Character table for $S_6$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Sibling algebras

 Twin sextic algebra: 6.4.29240885274112.1 Degree 6 sibling: 6.4.29240885274112.1 Degree 10 sibling: 10.4.29240885274112.1 Degree 12 siblings: Deg 12, Deg 12 Degree 15 siblings: Deg 15, Deg 15 Degree 20 siblings: Deg 20, Deg 20, Deg 20 Degree 30 siblings: data not computed Degree 36 sibling: data not computed Degree 40 siblings: data not computed Degree 45 sibling: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3] 2.4.0.1x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
3851Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.30808.2t1.a.a$1$ $2^{3} \cdot 3851$ $x^{2} + 7702$ $C_2$ (as 2T1) $1$ $-1$
* 5.30808.6t16.a.a$5$ $2^{3} \cdot 3851$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-1$
5.949132864.12t183.a.a$5$ $2^{6} \cdot 3851^{2}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-3$
5.900...496.12t183.a.a$5$ $2^{12} \cdot 3851^{4}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
5.292...112.6t16.a.a$5$ $2^{9} \cdot 3851^{3}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $3$
9.292...112.10t32.a.a$9$ $2^{9} \cdot 3851^{3}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $3$
9.855...544.20t145.a.a$9$ $2^{18} \cdot 3851^{6}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-3$
10.855...544.30t164.a.a$10$ $2^{18} \cdot 3851^{6}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $2$
10.900...496.30t164.a.a$10$ $2^{12} \cdot 3851^{4}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.811...016.36t1252.a.a$16$ $2^{24} \cdot 3851^{8}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $0$

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 *.