Properties

Label 6.0.7308160119.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,3^{9}\cdot 13^{5}$
Root discriminant $44.05$
Ramified primes $3, 13$
Class number $156$
Class group $[156]$
Galois group $C_6$ (as 6T1)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 26*x^3 + 351*x^2 + 234*x + 208)
 
gp: K = bnfinit(x^6 - 26*x^3 + 351*x^2 + 234*x + 208, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![208, 234, 351, -26, 0, 0, 1]);
 

Normalized defining polynomial

\( x^{6} - 26 x^{3} + 351 x^{2} + 234 x + 208 \)

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

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-7308160119=-\,3^{9}\cdot 13^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $44.05$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 13$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $6$
This field is Galois and abelian over $\Q$.
Conductor:  \(117=3^{2}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{117}(16,·)$, $\chi_{117}(1,·)$, $\chi_{117}(116,·)$, $\chi_{117}(101,·)$, $\chi_{117}(22,·)$, $\chi_{117}(95,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{88} a^{4} - \frac{1}{44} a^{3} - \frac{21}{88} a^{2} + \frac{1}{4} a + \frac{5}{11}$, $\frac{1}{704} a^{5} - \frac{3}{704} a^{4} - \frac{151}{704} a^{3} + \frac{43}{704} a^{2} + \frac{31}{352} a + \frac{3}{44}$

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

Class group and class number

$C_{156}$, which has order $156$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $2$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{7}{176} a^{5} + \frac{31}{176} a^{4} + \frac{71}{176} a^{3} - \frac{263}{176} a^{2} - \frac{91}{88} a + \frac{8}{11} \),  \( \frac{1}{44} a^{5} + \frac{15}{44} a^{4} - \frac{1}{4} a^{3} - \frac{27}{44} a^{2} - \frac{13}{22} a + \frac{1435}{11} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 106.812145281 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

$C_6$ (as 6T1):

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

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.13689.1

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

Sibling algebras

Twin sextic algebra: 3.3.13689.1 $\times$ \(\Q(\sqrt{-39}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
$13$13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.39.2t1.a.a$1$ $ 3 \cdot 13 $ $x^{2} - x + 10$ $C_2$ (as 2T1) $1$ $-1$
* 1.117.3t1.b.a$1$ $ 3^{2} \cdot 13 $ $x^{3} - 39 x - 26$ $C_3$ (as 3T1) $0$ $1$
* 1.117.6t1.d.a$1$ $ 3^{2} \cdot 13 $ $x^{6} - 26 x^{3} + 351 x^{2} + 234 x + 208$ $C_6$ (as 6T1) $0$ $-1$
* 1.117.3t1.b.b$1$ $ 3^{2} \cdot 13 $ $x^{3} - 39 x - 26$ $C_3$ (as 3T1) $0$ $1$
* 1.117.6t1.d.b$1$ $ 3^{2} \cdot 13 $ $x^{6} - 26 x^{3} + 351 x^{2} + 234 x + 208$ $C_6$ (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 *.