Properties

Label 6.2.95004009.3
Degree $6$
Signature $[2, 2]$
Discriminant $3^{6}\cdot 19^{4}$
Root discriminant $21.36$
Ramified primes $3, 19$
Class number $1$
Class group Trivial
Galois Group $A_6$ (as 6T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, -9, 9, 8, -3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 3*x^4 + 8*x^3 + 9*x^2 - 9*x + 6)
gp: K = bnfinit(x^6 - 3*x^5 - 3*x^4 + 8*x^3 + 9*x^2 - 9*x + 6, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 3 x^{5} \) \(\mathstrut -\mathstrut 3 x^{4} \) \(\mathstrut +\mathstrut 8 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 9 x \) \(\mathstrut +\mathstrut 6 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $6$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(95004009=3^{6}\cdot 19^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $21.36$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 19$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$

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:  $3$
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:  \( a^{4} - a^{3} - 5 a^{2} - a + 7 \),  \( a^{2} - 3 a + 1 \),  \( \frac{104}{3} a^{5} - \frac{220}{3} a^{4} - \frac{562}{3} a^{3} + 153 a^{2} + 541 a + 73 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 201.334350946 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$A_6$ (as 6T15):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: 6.2.7695324729.3
Degree 6 sibling: 6.2.7695324729.3
Degree 10 sibling: 10.2.731086699811838561.1
Degree 15 siblings: Deg 15, Deg 15
Degree 20 sibling: Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.3e10_19e4.6t15.2c1$5$ $ 3^{10} \cdot 19^{4}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $1$
* 5.3e6_19e4.6t15.2c1$5$ $ 3^{6} \cdot 19^{4}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $1$
8.3e16_19e6.36t555.2c1$8$ $ 3^{16} \cdot 19^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $0$
8.3e16_19e6.36t555.2c2$8$ $ 3^{16} \cdot 19^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $0$
9.3e16_19e8.10t26.2c1$9$ $ 3^{16} \cdot 19^{8}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $1$
10.3e18_19e8.30t88.2c1$10$ $ 3^{18} \cdot 19^{8}$ $x^{6} - 3 x^{5} - 3 x^{4} + 8 x^{3} + 9 x^{2} - 9 x + 6$ $A_6$ (as 6T15) $1$ $-2$

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 *.