Properties

Label 6.2.1924456983552.4
Degree 6
Signature $[2, 2]$
Discriminant $2^{10}\cdot 3^{11}\cdot 103^{2}$
Ramified primes $2, 3, 103$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group $S_6$

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3732, -3024, 189, -72, -18, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 18*x^4 - 72*x^3 + 189*x^2 - 3024*x + 3732)
gp: K = bnfinit(x^6 - 18*x^4 - 72*x^3 + 189*x^2 - 3024*x + 3732, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 18 x^{4} \) \(\mathstrut -\mathstrut 72 x^{3} \) \(\mathstrut +\mathstrut 189 x^{2} \) \(\mathstrut -\mathstrut 3024 x \) \(\mathstrut +\mathstrut 3732 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 2]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1924456983552=2^{10}\cdot 3^{11}\cdot 103^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $2, 3, 103$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{344684} a^{5} - \frac{113}{344684} a^{4} + \frac{12751}{344684} a^{3} - \frac{62199}{344684} a^{2} + \frac{33749}{86171} a - \frac{22869}{86171}$

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

Class group and class number

Trivial Abelian group, order 1 (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $3$
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:  \( \frac{232946}{86171} a^{5} - \frac{1505650}{86171} a^{4} + \frac{2392864}{86171} a^{3} - \frac{2629102}{86171} a^{2} - \frac{60096856}{86171} a + \frac{77726023}{86171} \),  \( \frac{1607863371697}{172342} a^{5} + \frac{2045310515857}{172342} a^{4} - \frac{13169883972463}{86171} a^{3} - \frac{149272121755057}{172342} a^{2} + \frac{114001983790615}{172342} a - \frac{2358580317647326}{86171} \),  \( \frac{1633659404629}{172342} a^{5} - \frac{5890787834679}{86171} a^{4} + \frac{14161671039417}{86171} a^{3} - \frac{32371973442625}{86171} a^{2} - \frac{269744638647457}{172342} a + \frac{207503025891490}{86171} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 34903.5557251 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_6$:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
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: 6.2.2639858688.2
Degree 6 sibling: 6.2.2639858688.2
Degree 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 15 siblings: data not computed
Degree 20 siblings: 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 R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.8.8$x^{4} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.6.11.3$x^{6} + 6$$6$$1$$11$$S_3\times C_2$$[5/2]_{2}^{2}$
$103$$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{103}$$x + 2$$1$$1$$0$Trivial$[\ ]$
103.4.2.2$x^{4} - 103 x^{2} + 53045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$