Properties

Label 6.6.1169905924153.1
Degree $6$
Signature $[6, 0]$
Discriminant $41^{3}\cdot 257^{3}$
Root discriminant $102.65$
Ramified primes $41, 257$
Class number $2$
Class group $[2]$
Galois Group $D_{6}$ (as 6T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-249501, -31624, 17720, 504, -267, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 267*x^4 + 504*x^3 + 17720*x^2 - 31624*x - 249501)
gp: K = bnfinit(x^6 - 2*x^5 - 267*x^4 + 504*x^3 + 17720*x^2 - 31624*x - 249501, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut 267 x^{4} \) \(\mathstrut +\mathstrut 504 x^{3} \) \(\mathstrut +\mathstrut 17720 x^{2} \) \(\mathstrut -\mathstrut 31624 x \) \(\mathstrut -\mathstrut 249501 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[6, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1169905924153=41^{3}\cdot 257^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $102.65$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{10} a^{3} - \frac{1}{10} a^{2} - \frac{2}{5} a + \frac{3}{10}$, $\frac{1}{10} a^{4} - \frac{1}{2} a^{2} - \frac{1}{10} a + \frac{3}{10}$, $\frac{1}{13514130} a^{5} + \frac{9237}{214510} a^{4} - \frac{54277}{2252355} a^{3} - \frac{1155913}{4504710} a^{2} + \frac{436907}{1351413} a + \frac{19231}{321765}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C2, order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $5$
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:  \( \frac{353}{13514130} a^{5} + \frac{109}{214510} a^{4} - \frac{29527}{4504710} a^{3} - \frac{180517}{2252355} a^{2} + \frac{2186858}{6757065} a + \frac{1285663}{643530} \),  \( \frac{1549}{13514130} a^{5} + \frac{148}{107255} a^{4} - \frac{124463}{4504710} a^{3} - \frac{247685}{900942} a^{2} + \frac{3476299}{2702826} a + \frac{2052676}{321765} \),  \( \frac{22333}{13514130} a^{5} - \frac{2173}{107255} a^{4} - \frac{1252973}{4504710} a^{3} + \frac{17704937}{4504710} a^{2} - \frac{36769267}{13514130} a - \frac{18861428}{321765} \),  \( \frac{1193}{150157} a^{5} + \frac{5578}{107255} a^{4} - \frac{1193977}{750785} a^{3} - \frac{6041723}{750785} a^{2} + \frac{53709522}{750785} a + \frac{28251671}{107255} \),  \( \frac{46074353996}{355635} a^{5} - \frac{12565732908}{5645} a^{4} - \frac{2402777101841}{237090} a^{3} + \frac{84777422348387}{237090} a^{2} - \frac{715085595178457}{355635} a + \frac{109670932958657}{33870} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 17492.7422128 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_6$ (as 6T3):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 12
The 6 conjugacy class representatives for $D_{6}$
Character table for $D_{6}$

Intermediate fields

\(\Q(\sqrt{10537}) \), 3.3.257.1

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

Sibling algebras

Galois closure: Deg 12
Twin sextic algebra: 3.3.257.1 $\times$ \(\Q(\sqrt{41}) \) $\times$ \(\Q\)
Degree 6 sibling: 6.6.4552163129.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
257Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.257.2t1.1c1$1$ $ 257 $ $x^{2} - x - 64$ $C_2$ (as 2T1) $1$ $1$
* 1.41_257.2t1.1c1$1$ $ 41 \cdot 257 $ $x^{2} - x - 2634$ $C_2$ (as 2T1) $1$ $1$
1.41.2t1.1c1$1$ $ 41 $ $x^{2} - x - 10$ $C_2$ (as 2T1) $1$ $1$
* 2.257.3t2.1c1$2$ $ 257 $ $x^{3} - x^{2} - 4 x + 3$ $S_3$ (as 3T2) $1$ $2$
* 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) $1$ $2$

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 *.