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Properties

Label 6.0.42342458427.2
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{3}\cdot 199^{4}$
Root discriminant $59.04$
Ramified primes $3, 199$
Class number $9$
Class group $[3, 3]$
Galois group $C_6$ (as 6T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3481, 3894, 4297, 184, 67, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 67*x^4 + 184*x^3 + 4297*x^2 + 3894*x + 3481)
 
gp: K = bnfinit(x^6 - x^5 + 67*x^4 + 184*x^3 + 4297*x^2 + 3894*x + 3481, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} + 67 x^{4} + 184 x^{3} + 4297 x^{2} + 3894 x + 3481 \)

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:  \(-42342458427=-\,3^{3}\cdot 199^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.04$
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:  $3, 199$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(597=3\cdot 199\)
Dirichlet character group:    $\lbrace$$\chi_{597}(1,·)$, $\chi_{597}(305,·)$, $\chi_{597}(200,·)$, $\chi_{597}(106,·)$, $\chi_{597}(92,·)$, $\chi_{597}(490,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{86078935} a^{5} - \frac{5835927}{86078935} a^{4} - \frac{22171779}{86078935} a^{3} + \frac{39387758}{86078935} a^{2} - \frac{40854914}{86078935} a + \frac{4422}{1458965}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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:  \( \frac{4422}{17215787} a^{5} - \frac{4481}{17215787} a^{4} + \frac{300227}{17215787} a^{3} + \frac{548797}{17215787} a^{2} + \frac{19254857}{17215787} a + \frac{295746}{291793} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  \( \frac{569382}{17215787} a^{5} - \frac{1090883}{17215787} a^{4} + \frac{38657587}{17215787} a^{3} + \frac{70663757}{17215787} a^{2} + \frac{2346177219}{17215787} a + \frac{508993}{291793} \),  \( \frac{234}{1458965} a^{5} - \frac{15678}{1458965} a^{4} - \frac{116746}{1458965} a^{3} - \frac{1005498}{1458965} a^{2} - \frac{911196}{1458965} a - \frac{4603293}{1458965} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 114.571471826 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 6
The 6 conjugacy class representatives for $C_6$
Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.39601.1

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

Sibling algebras

Twin sextic algebra: \(\Q\) $\times$ \(\Q(\sqrt{-3}) \) $\times$ 3.3.39601.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$199$199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.199.3t1.1c1$1$ $ 199 $ $x^{3} - x^{2} - 66 x - 59$ $C_3$ (as 3T1) $0$ $1$
* 1.3_199.6t1.1c1$1$ $ 3 \cdot 199 $ $x^{6} - x^{5} + 67 x^{4} + 184 x^{3} + 4297 x^{2} + 3894 x + 3481$ $C_6$ (as 6T1) $0$ $-1$
* 1.199.3t1.1c2$1$ $ 199 $ $x^{3} - x^{2} - 66 x - 59$ $C_3$ (as 3T1) $0$ $1$
* 1.3_199.6t1.1c2$1$ $ 3 \cdot 199 $ $x^{6} - x^{5} + 67 x^{4} + 184 x^{3} + 4297 x^{2} + 3894 x + 3481$ $C_6$ (as 6T1) $0$ $-1$

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 *.