# Properties

 Label 6.0.27971.1 Degree $6$ Signature $[0, 3]$ Discriminant $-27971$ Root discriminant $5.51$ Ramified primes $83, 337$ 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^5 + x^4 - 2*x^3 + 3*x^2 - 2*x + 1)

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

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

$$x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$$

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: $$-27971$$$$\medspace = -\,83\cdot 337$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $5.51$ 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: $83, 337$ 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: $$a$$,  $$a^{4} - a + 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$0.436494860807$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.436494860807 \cdot 1}{2\sqrt{27971}}\approx 0.3236940731829$

## 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: 6.4.21883862619611.1 Degree 6 sibling: 6.4.21883862619611.1 Degree 10 sibling: 10.4.21883862619611.1 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 ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
83Data not computed
337Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.27971.2t1.a.a$1$ $83 \cdot 337$ $x^{2} - x + 6993$ $C_2$ (as 2T1) $1$ $-1$
* 5.27971.6t16.a.a$5$ $83 \cdot 337$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $-1$
5.782376841.12t183.a.a$5$ $83^{2} \cdot 337^{2}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $-3$
5.612...281.12t183.a.a$5$ $83^{4} \cdot 337^{4}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.218...611.6t16.a.a$5$ $83^{3} \cdot 337^{3}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $3$
9.218...611.10t32.a.a$9$ $83^{3} \cdot 337^{3}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $3$
9.478...321.20t145.a.a$9$ $83^{6} \cdot 337^{6}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $-3$
10.478...321.30t164.a.a$10$ $83^{6} \cdot 337^{6}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $2$
10.612...281.30t164.a.a$10$ $83^{4} \cdot 337^{4}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.374...961.36t1252.a.a$16$ $83^{8} \cdot 337^{8}$ $x^{6} - x^{5} + x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$ $S_6$ (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 *.