↑

Properties

Label 6.0.812017791.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{7}\cdot 13^{5}$
Root discriminant $30.54$
Ramified primes $3, 13$
Class number $12$
Class group $[12]$
Galois group $S_3\times C_3$ (as 6T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -144, 96, 4, -6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 6*x^4 + 4*x^3 + 96*x^2 - 144*x + 64)
 
gp: K = bnfinit(x^6 - 3*x^5 - 6*x^4 + 4*x^3 + 96*x^2 - 144*x + 64, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 6 x^{4} + 4 x^{3} + 96 x^{2} - 144 x + 64 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-812017791=-\,3^{7}\cdot 13^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.54$
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:  $3, 13$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{12}$, which has order $12$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $2$
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:  \( \frac{7}{16} a^{5} - \frac{17}{16} a^{4} - \frac{27}{8} a^{3} - \frac{1}{4} a^{2} + \frac{85}{2} a - 37 \),  \( \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{9}{8} a^{3} + \frac{1}{4} a^{2} + 10 a - 5 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 47.992420098 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 6T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

\(\Q(\sqrt{-39}) \)

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

Sibling algebras

Galois closure: 18.0.535422519938359574117644671.1
Twin sextic algebra: 3.3.169.1 $\times$ 3.1.351.1
Degree 9 sibling: 9.3.1235079060111.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$3.6.7.6$x^{6} + 3 x^{3} + 3 x^{2} + 3$$6$$1$$7$$S_3\times C_3$$[3/2]_{2}^{3}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_13.2t1.1c1$1$ $ 3 \cdot 13 $ $x^{2} - x + 10$ $C_2$ (as 2T1) $1$ $-1$
1.3_13.6t1.2c1$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 8 x^{4} - 9 x^{3} + 6 x^{2} + 10 x + 25$ $C_6$ (as 6T1) $0$ $-1$
1.3_13.6t1.2c2$1$ $ 3 \cdot 13 $ $x^{6} - x^{5} + 8 x^{4} - 9 x^{3} + 6 x^{2} + 10 x + 25$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
2.3e3_13.3t2.1c1$2$ $ 3^{3} \cdot 13 $ $x^{3} + 3 x - 3$ $S_3$ (as 3T2) $1$ $0$
* 2.3e3_13e2.6t5.1c1$2$ $ 3^{3} \cdot 13^{2}$ $x^{6} - 3 x^{5} - 6 x^{4} + 4 x^{3} + 96 x^{2} - 144 x + 64$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3e3_13e2.6t5.1c2$2$ $ 3^{3} \cdot 13^{2}$ $x^{6} - 3 x^{5} - 6 x^{4} + 4 x^{3} + 96 x^{2} - 144 x + 64$ $S_3\times C_3$ (as 6T5) $0$ $0$

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 *.