Properties

Label 6.0.30808.1
Degree $6$
Signature $[0, 3]$
Discriminant $-30808$
Root discriminant $5.60$
Ramified primes $2, 3851$
Class number $1$
Class group trivial
Galois group $S_6$ (as 6T16)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^4 - x^3 + x^2 + x + 1)
 
gp: K = bnfinit(x^6 - x^4 - x^3 + x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 1, -1, -1, 0, 1]);
 

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

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:  \(-30808\)\(\medspace = -\,2^{3}\cdot 3851\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $5.60$
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:  $2, 3851$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( a^{5} - a^{3} - a^{2} + a + 1 \),  \( a^{5} - a^{3} + a \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 0.643679055537 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.643679055537 \cdot 1}{2\sqrt{30808}}\approx 0.4548280438136$

Galois group

$S_6$ (as 6T16):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 720
The 11 conjugacy class representatives for $S_6$
Character table for $S_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling algebras

Twin sextic algebra: 6.4.29240885274112.1
Degree 6 sibling: 6.4.29240885274112.1
Degree 10 sibling: 10.4.29240885274112.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
3851Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.30808.2t1.a.a$1$ $ 2^{3} \cdot 3851 $ $x^{2} + 7702$ $C_2$ (as 2T1) $1$ $-1$
* 5.30808.6t16.a.a$5$ $ 2^{3} \cdot 3851 $ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-1$
5.949132864.12t183.a.a$5$ $ 2^{6} \cdot 3851^{2}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-3$
5.900...496.12t183.a.a$5$ $ 2^{12} \cdot 3851^{4}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $1$
5.292...112.6t16.a.a$5$ $ 2^{9} \cdot 3851^{3}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $3$
9.292...112.10t32.a.a$9$ $ 2^{9} \cdot 3851^{3}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $3$
9.855...544.20t145.a.a$9$ $ 2^{18} \cdot 3851^{6}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-3$
10.855...544.30t164.a.a$10$ $ 2^{18} \cdot 3851^{6}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $2$
10.900...496.30t164.a.a$10$ $ 2^{12} \cdot 3851^{4}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $-2$
16.811...016.36t1252.a.a$16$ $ 2^{24} \cdot 3851^{8}$ $x^{6} - x^{4} - x^{3} + x^{2} + x + 1$ $S_6$ (as 6T16) $1$ $0$

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 *.