Properties

Label 6.2.22325625.3
Degree $6$
Signature $[2, 2]$
Discriminant $3^{6}\cdot 5^{4}\cdot 7^{2}$
Root discriminant $16.78$
Ramified primes $3, 5, 7$
Class number $1$
Class group Trivial
Galois Group $C_3^2:C_4$ (as 6T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -21, 9, -7, 6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 + 6*x^4 - 7*x^3 + 9*x^2 - 21*x + 11)
gp: K = bnfinit(x^6 + 6*x^4 - 7*x^3 + 9*x^2 - 21*x + 11, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut +\mathstrut 6 x^{4} \) \(\mathstrut -\mathstrut 7 x^{3} \) \(\mathstrut +\mathstrut 9 x^{2} \) \(\mathstrut -\mathstrut 21 x \) \(\mathstrut +\mathstrut 11 \)

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:  \(22325625=3^{6}\cdot 5^{4}\cdot 7^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $16.78$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$

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:  \( a^{3} + 3 a - 4 \),  \( a^{4} - a^{3} + 3 a^{2} - 6 a + 3 \),  \( 23 a^{5} - 5 a^{4} + 142 a^{3} - 109 a^{2} + 204 a - 310 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 32.4844814116 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3:S_3.C_2$ (as 6T10):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 36
The 6 conjugacy class representatives for $C_3^2:C_4$
Character table for $C_3^2:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Galois closure: data not computed
Twin sextic algebra: 6.2.22325625.4
Degree 6 sibling: 6.2.22325625.4
Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
Degree 18 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.3_5_7.4t1.1c1$1$ $ 3 \cdot 5 \cdot 7 $ $x^{4} - x^{3} + 26 x^{2} - 26 x + 151$ $C_4$ (as 4T1) $0$ $-1$
1.3_5_7.4t1.1c2$1$ $ 3 \cdot 5 \cdot 7 $ $x^{4} - x^{3} + 26 x^{2} - 26 x + 151$ $C_4$ (as 4T1) $0$ $-1$
4.3e6_5e3_7e2.6t10.1c1$4$ $ 3^{6} \cdot 5^{3} \cdot 7^{2}$ $x^{6} + 6 x^{4} - 7 x^{3} + 9 x^{2} - 21 x + 11$ $C_3^2:C_4$ (as 6T10) $1$ $0$
4.3e6_5e3_7e2.6t10.2c1$4$ $ 3^{6} \cdot 5^{3} \cdot 7^{2}$ $x^{6} + 6 x^{4} - 7 x^{3} + 9 x^{2} - 21 x + 11$ $C_3^2:C_4$ (as 6T10) $1$ $0$

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 *.