Properties

 Label 6.2.336069.1 Degree $6$ Signature $[2, 2]$ Discriminant $3^{6}\cdot 461$ Root discriminant $8.34$ Ramified primes $3, 461$ Class number $1$ Class group Trivial Galois group $S_6$ (as 6T16)

Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 3*x^4 + 2*x^3 - 6*x^2 + 3*x + 1)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, -6, 2, 3, -3, 1]);

Normalizeddefining polynomial

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$336069=3^{6}\cdot 461$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $8.34$ 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: $3, 461$ 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: $3$ 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: $$a^{5} - 2 a^{4} + a^{3} + 3 a^{2} - 3 a$$,  $$a$$,  $$a^{5} - 2 a^{4} + a^{3} + 2 a^{2} - 2 a - 1$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3.02833554722$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

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: 6.2.71421719949.1 Degree 6 sibling: 6.2.71421719949.1 Degree 10 sibling: 10.2.5785159315869.1 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

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.6.6.7$x^{6} + 3 x + 6$$6$$1$$6$$C_3^2:D_4$$[5/4, 5/4]_{4}^{2}$
461Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.461.2t1.a.a$1$ $461$ $x^{2} - x - 115$ $C_2$ (as 2T1) $1$ $1$
* 5.336069.6t16.a.a$5$ $3^{6} \cdot 461$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.154927809.12t183.a.a$5$ $3^{6} \cdot 461^{2}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.32925412896489.12t183.a.a$5$ $3^{6} \cdot 461^{4}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
5.71421719949.6t16.a.a$5$ $3^{6} \cdot 461^{3}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
9.5785159315869.10t32.a.a$9$ $3^{10} \cdot 461^{3}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
9.566784675608153840289.20t145.a.a$9$ $3^{10} \cdot 461^{6}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $1$
10.5101062080473384562601.30t164.a.a$10$ $3^{12} \cdot 461^{6}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $-2$
10.24002626001540481.30t164.a.a$10$ $3^{12} \cdot 461^{4}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.7112667345306508377883766440881.36t1252.a.a$16$ $3^{20} \cdot 461^{8}$ $x^{6} - 3 x^{5} + 3 x^{4} + 2 x^{3} - 6 x^{2} + 3 x + 1$ $S_6$ (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 *.