Properties

Label 6.0.25969216.3
Degree $6$
Signature $[0, 3]$
Discriminant $-25969216$
Root discriminant \(17.21\)
Ramified primes see page
Class number $3$
Class group $[3]$
Galois group $S_3\times C_3$ (as 6T5)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 12*x^4 + 46*x^3 + 23*x^2 - 240*x + 225)
 
gp: K = bnfinit(x^6 - 2*x^5 - 12*x^4 + 46*x^3 + 23*x^2 - 240*x + 225, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, -240, 23, 46, -12, -2, 1]);
 

\( x^{6} - 2x^{5} - 12x^{4} + 46x^{3} + 23x^{2} - 240x + 225 \) Copy content Toggle raw display

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:   \(-25969216\) \(\medspace = -\,2^{6}\cdot 7^{4}\cdot 13^{2}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  \(17.21\)
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:   \(2\), \(7\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$\card{ \Aut(K/\Q) }$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11}a^{4}+\frac{5}{11}a^{3}-\frac{1}{11}a^{2}-\frac{4}{11}a-\frac{3}{11}$, $\frac{1}{165}a^{5}-\frac{2}{165}a^{4}+\frac{21}{55}a^{3}-\frac{74}{165}a^{2}-\frac{52}{165}a-\frac{3}{11}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{3}$, which has order $3$

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:   \( -\frac{2}{33} a^{5} + \frac{1}{33} a^{4} + \frac{8}{11} a^{3} - \frac{47}{33} a^{2} - \frac{115}{33} a + 7 \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:   $\frac{2}{165}a^{5}-\frac{4}{165}a^{4}-\frac{13}{55}a^{3}+\frac{17}{165}a^{2}+\frac{226}{165}a-\frac{17}{11}$, $\frac{29}{165}a^{5}+\frac{2}{165}a^{4}-\frac{116}{55}a^{3}+\frac{599}{165}a^{2}+\frac{1882}{165}a-19$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 19.0957855121 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 19.0957855121 \cdot 3}{4\sqrt{25969216}}\approx 0.697122302609$

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-1}) \)

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

Sibling algebras

Galois closure: 18.0.84535014172552012147112280064.2
Twin sextic algebra: 3.3.8281.2 $\times$ 3.1.676.1
Degree 9 sibling: 9.3.36343632130624.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{3}$ ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{3}$ R ${\href{/padicField/11.2.0.1}{2} }^{3}$ R ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }$ ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }$ ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.6.0.1}{6} }$ ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }$

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
\(2\) Copy content Toggle raw display 2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
\(7\) Copy content Toggle raw display 7.6.4.1$x^{6} + 35 x^{3} + 441$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(13\) Copy content Toggle raw display 13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.364.6t1.g.a$1$ $ 2^{2} \cdot 7 \cdot 13 $ 6.0.4388797504.1 $C_6$ (as 6T1) $0$ $-1$
1.364.6t1.g.b$1$ $ 2^{2} \cdot 7 \cdot 13 $ 6.0.4388797504.1 $C_6$ (as 6T1) $0$ $-1$
1.91.3t1.a.a$1$ $ 7 \cdot 13 $ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
1.91.3t1.a.b$1$ $ 7 \cdot 13 $ 3.3.8281.2 $C_3$ (as 3T1) $0$ $1$
2.676.3t2.b.a$2$ $ 2^{2} \cdot 13^{2}$ 3.1.676.1 $S_3$ (as 3T2) $1$ $0$
* 2.2548.6t5.d.a$2$ $ 2^{2} \cdot 7^{2} \cdot 13 $ 6.0.25969216.3 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2548.6t5.d.b$2$ $ 2^{2} \cdot 7^{2} \cdot 13 $ 6.0.25969216.3 $S_3\times C_3$ (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 *.