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Properties

Label 6.2.17034106517.2
Degree $6$
Signature $[2, 2]$
Discriminant $31^{3}\cdot 83^{3}$
Root discriminant $50.72$
Ramified primes $31, 83$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 6T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -51, 52, -53, 3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 3*x^4 - 53*x^3 + 52*x^2 - 51*x + 7)
 
gp: K = bnfinit(x^6 - 2*x^5 + 3*x^4 - 53*x^3 + 52*x^2 - 51*x + 7, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 3 x^{4} - 53 x^{3} + 52 x^{2} - 51 x + 7 \)

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:  \(17034106517=31^{3}\cdot 83^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.72$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $31, 83$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{331} a^{5} - \frac{87}{331} a^{4} + \frac{116}{331} a^{3} + \frac{17}{331} a^{2} - \frac{69}{331} a - \frac{144}{331}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{11}{331} a^{5} + \frac{36}{331} a^{4} - \frac{48}{331} a^{3} - \frac{475}{331} a^{2} - \frac{2414}{331} a + \frac{402}{331} \),  \( \frac{35}{331} a^{5} - \frac{66}{331} a^{4} + \frac{88}{331} a^{3} - \frac{1722}{331} a^{2} + \frac{1226}{331} a - \frac{1399}{331} \),  \( \frac{57519}{331} a^{5} - \frac{130178}{331} a^{4} - \frac{477396}{331} a^{3} + \frac{527001}{331} a^{2} - \frac{597576}{331} a + \frac{83620}{331} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 529.664876766 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 6T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{2573}) \)

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

Sibling algebras

Galois closure: data not computed
Twin sextic algebra: 3.1.31.1 $\times$ 3.1.83.1
Degree 9 sibling: 9.1.17034106517.1
Degree 12 sibling: Deg 12
Degree 18 siblings: Deg 18, Deg 18, Deg 18

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ R ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.31_83.2t1.1c1$1$ $ 31 \cdot 83 $ $x^{2} - x - 643$ $C_2$ (as 2T1) $1$ $1$
1.31.2t1.1c1$1$ $ 31 $ $x^{2} - x + 8$ $C_2$ (as 2T1) $1$ $-1$
1.83.2t1.1c1$1$ $ 83 $ $x^{2} - x + 21$ $C_2$ (as 2T1) $1$ $-1$
2.31e2_83.6t3.1c1$2$ $ 31^{2} \cdot 83 $ $x^{6} - x^{5} + 25 x^{4} - 13 x^{3} + 183 x^{2} - 139 x + 410$ $D_{6}$ (as 6T3) $1$ $0$
2.83.3t2.1c1$2$ $ 83 $ $x^{3} - x^{2} + x - 2$ $S_3$ (as 3T2) $1$ $0$
2.31.3t2.1c1$2$ $ 31 $ $x^{3} + x - 1$ $S_3$ (as 3T2) $1$ $0$
2.31_83e2.6t3.1c1$2$ $ 31 \cdot 83^{2}$ $x^{6} - 2 x^{5} + 43 x^{4} + 145 x^{3} + 254 x^{2} + 3927 x - 274931$ $D_{6}$ (as 6T3) $1$ $0$
* 4.31e2_83e2.6t9.1c1$4$ $ 31^{2} \cdot 83^{2}$ $x^{6} - 2 x^{5} + 3 x^{4} - 53 x^{3} + 52 x^{2} - 51 x + 7$ $S_3^2$ (as 6T9) $1$ $0$

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 *.