Properties

Label 6.2.4552163129.1
Degree $6$
Signature $[2, 2]$
Discriminant $41^{3}\cdot 257^{2}$
Root discriminant $40.71$
Ramified primes $41, 257$
Class number $4$
Class group $[4]$
Galois Group $S_4\times C_2$ (as 6T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1671, -794, -257, 56, 18, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 18*x^4 + 56*x^3 - 257*x^2 - 794*x - 1671)
gp: K = bnfinit(x^6 - 2*x^5 + 18*x^4 + 56*x^3 - 257*x^2 - 794*x - 1671, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 18 x^{4} \) \(\mathstrut +\mathstrut 56 x^{3} \) \(\mathstrut -\mathstrut 257 x^{2} \) \(\mathstrut -\mathstrut 794 x \) \(\mathstrut -\mathstrut 1671 \)

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:  \(4552163129=41^{3}\cdot 257^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $40.71$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $41, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{479242} a^{5} - \frac{42969}{239621} a^{4} + \frac{24383}{239621} a^{3} - \frac{23251}{479242} a^{2} + \frac{137781}{479242} a + \frac{43663}{479242}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C4, order $4$

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:  \( \frac{251}{479242} a^{5} - \frac{2274}{239621} a^{4} + \frac{19595}{479242} a^{3} - \frac{85097}{479242} a^{2} - \frac{81007}{239621} a + \frac{567476}{239621} \),  \( \frac{541}{479242} a^{5} - \frac{2992}{239621} a^{4} + \frac{12048}{239621} a^{3} - \frac{118499}{479242} a^{2} - \frac{222231}{479242} a - \frac{340417}{479242} \),  \( \frac{940250126}{239621} a^{5} + \frac{1714073384}{239621} a^{4} + \frac{46954804189}{479242} a^{3} + \frac{142408078383}{239621} a^{2} + \frac{605562883593}{479242} a + \frac{821949480463}{479242} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 135.397028597 \)
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

3.3.257.1

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

Sibling algebras

Twin sextic algebra: 4.0.257.1 $\times$ \(\Q(\sqrt{10537}) \)
Degree 6 sibling: 6.2.1169905924153.1
Degree 8 siblings: Deg 8, Deg 8
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
41Data not computed
257Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.41_257.2t1.1c1$1$ $ 41 \cdot 257 $ $x^{2} - x - 2634$ $C_2$ (as 2T1) $1$ $1$
1.257.2t1.1c1$1$ $ 257 $ $x^{2} - x - 64$ $C_2$ (as 2T1) $1$ $1$
1.41.2t1.1c1$1$ $ 41 $ $x^{2} - x - 10$ $C_2$ (as 2T1) $1$ $1$
2.41e2_257.6t3.2c1$2$ $ 41^{2} \cdot 257 $ $x^{6} - 2 x^{5} - 267 x^{4} + 504 x^{3} + 17720 x^{2} - 31624 x - 249501$ $D_{6}$ (as 6T3) $1$ $2$
* 2.257.3t2.1c1$2$ $ 257 $ $x^{3} - x^{2} - 4 x + 3$ $S_3$ (as 3T2) $1$ $2$
3.257.4t5.1c1$3$ $ 257 $ $x^{4} + x^{2} - x + 1$ $S_4$ (as 4T5) $1$ $-1$
* 3.41e3_257.6t11.2c1$3$ $ 41^{3} \cdot 257 $ $x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.41e3_257e2.6t11.2c1$3$ $ 41^{3} \cdot 257^{2}$ $x^{6} - 2 x^{5} + 18 x^{4} + 56 x^{3} - 257 x^{2} - 794 x - 1671$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.257e2.6t8.1c1$3$ $ 257^{2}$ $x^{4} + x^{2} - x + 1$ $S_4$ (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 *.