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Properties

Label 6.2.6718464.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{10}\cdot 3^{8}$
Root discriminant $13.74$
Ramified primes $2, 3$
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![1, 0, -9, -12, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^4 - 12*x^3 - 9*x^2 + 1)
 
gp: K = bnfinit(x^6 - 3*x^4 - 12*x^3 - 9*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1 \)

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:  \(6718464=2^{10}\cdot 3^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.74$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(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:  \( a^{5} - 3 a^{3} - 12 a^{2} - 9 a \),  \( a^{5} - 3 a^{3} - 12 a^{2} - 8 a - 1 \),  \( 6 a^{5} - 3 a^{4} - 16 a^{3} - 63 a^{2} - 21 a + 10 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 24.4670136709 \)
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: 6.2.1679616.2
Degree 6 sibling: 6.2.1679616.2
Degree 10 sibling: 10.2.11284439629824.1
Degree 15 siblings: Deg 15, 15.3.18953525353286467584.1
Degree 20 sibling: 20.4.127338577759142414150270976.1
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 R R ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$2$2.6.10.1$x^{6} + 2 x^{5} + 2 x^{4} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.6.8.2$x^{6} + 6 x^{5} + 9 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.2e8_3e8.6t15.2c1$5$ $ 2^{8} \cdot 3^{8}$ $x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1$ $A_6$ (as 6T15) $1$ $1$
* 5.2e10_3e8.6t15.2c1$5$ $ 2^{10} \cdot 3^{8}$ $x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1$ $A_6$ (as 6T15) $1$ $1$
8.2e18_3e16.36t555.2c1$8$ $ 2^{18} \cdot 3^{16}$ $x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1$ $A_6$ (as 6T15) $1$ $0$
8.2e18_3e16.36t555.2c2$8$ $ 2^{18} \cdot 3^{16}$ $x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1$ $A_6$ (as 6T15) $1$ $0$
9.2e18_3e16.10t26.2c1$9$ $ 2^{18} \cdot 3^{16}$ $x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1$ $A_6$ (as 6T15) $1$ $1$
10.2e24_3e16.30t88.2c1$10$ $ 2^{24} \cdot 3^{16}$ $x^{6} - 3 x^{4} - 12 x^{3} - 9 x^{2} + 1$ $A_6$ (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 *.