Properties

Label 6.2.1278983549.1
Degree $6$
Signature $[2, 2]$
Discriminant $29^{3}\cdot 229^{2}$
Root discriminant $32.95$
Ramified primes $29, 229$
Class number $8$
Class group $[8]$
Galois Group $S_4\times C_2$ (as 6T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-538, -415, -211, -18, -1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - x^4 - 18*x^3 - 211*x^2 - 415*x - 538)
gp: K = bnfinit(x^6 - 2*x^5 - x^4 - 18*x^3 - 211*x^2 - 415*x - 538, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut -\mathstrut 18 x^{3} \) \(\mathstrut -\mathstrut 211 x^{2} \) \(\mathstrut -\mathstrut 415 x \) \(\mathstrut -\mathstrut 538 \)

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:  \(1278983549=29^{3}\cdot 229^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $32.95$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $29, 229$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{158033} a^{5} - \frac{34718}{158033} a^{4} - \frac{47604}{158033} a^{3} + \frac{69365}{158033} a^{2} + \frac{1361}{6871} a + \frac{77578}{158033}$

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

Class group and class number

$C_{8}$, which has order $8$

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 46.9529526652 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_2\times S_4$ (as 6T11):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

3.3.229.1

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

Sibling algebras

Twin sextic algebra: 4.0.229.1 $\times$ \(\Q(\sqrt{6641}) \)
Degree 6 sibling: 6.2.292887232721.1
Degree 8 siblings: 8.0.1945064112500161.2, 8.0.37090522921.1
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 16 sibling: Deg 16
Degree 24 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 ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
229Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.29_229.2t1.1c1$1$ $ 29 \cdot 229 $ $x^{2} - x - 1660$ $C_2$ (as 2T1) $1$ $1$
1.229.2t1.1c1$1$ $ 229 $ $x^{2} - x - 57$ $C_2$ (as 2T1) $1$ $1$
1.29.2t1.1c1$1$ $ 29 $ $x^{2} - x - 7$ $C_2$ (as 2T1) $1$ $1$
2.29e2_229.6t3.2c1$2$ $ 29^{2} \cdot 229 $ $x^{6} - 176 x^{4} - 65 x^{3} + 7744 x^{2} + 5720 x - 80296$ $D_{6}$ (as 6T3) $1$ $2$
* 2.229.3t2.1c1$2$ $ 229 $ $x^{3} - 4 x - 1$ $S_3$ (as 3T2) $1$ $2$
3.229.4t5.1c1$3$ $ 229 $ $x^{4} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
* 3.29e3_229.6t11.2c1$3$ $ 29^{3} \cdot 229 $ $x^{6} - 2 x^{5} - x^{4} - 18 x^{3} - 211 x^{2} - 415 x - 538$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.29e3_229e2.6t11.2c1$3$ $ 29^{3} \cdot 229^{2}$ $x^{6} - 2 x^{5} - x^{4} - 18 x^{3} - 211 x^{2} - 415 x - 538$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.229e2.6t8.1c1$3$ $ 229^{2}$ $x^{4} - x + 1$ $S_4$ (as 4T5) $1$ $-1$

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 *.