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Properties

Label 6.2.16189331472.5
Degree $6$
Signature $[2, 2]$
Discriminant $2^{4}\cdot 3^{2}\cdot 17^{2}\cdot 73^{3}$
Root discriminant $50.30$
Ramified primes $2, 3, 17, 73$
Class number $3$
Class group $[3]$
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![-1962, 1312, -370, 91, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 7*x^4 + 91*x^3 - 370*x^2 + 1312*x - 1962)
 
gp: K = bnfinit(x^6 - x^5 + 7*x^4 + 91*x^3 - 370*x^2 + 1312*x - 1962, 1)
 

Normalized defining polynomial

\( x^{6} - x^{5} + 7 x^{4} + 91 x^{3} - 370 x^{2} + 1312 x - 1962 \)

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:  \(16189331472=2^{4}\cdot 3^{2}\cdot 17^{2}\cdot 73^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.30$
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:  $2, 3, 17, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{4} - \frac{2}{9} a^{2} + \frac{4}{27} a + \frac{1}{3}$, $\frac{1}{1107} a^{5} - \frac{8}{1107} a^{4} - \frac{20}{369} a^{3} + \frac{511}{1107} a^{2} + \frac{112}{1107} a + \frac{4}{123}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{584}{1107} a^{5} + \frac{412}{1107} a^{4} + \frac{1850}{369} a^{3} + \frac{65708}{1107} a^{2} - \frac{103922}{1107} a + \frac{70027}{123} \),  \( \frac{76}{369} a^{5} - \frac{512}{1107} a^{4} - \frac{167}{123} a^{3} + \frac{5216}{369} a^{2} - \frac{107591}{1107} a + \frac{18091}{123} \),  \( \frac{250}{27} a^{4} - \frac{500}{9} a^{2} + \frac{14500}{27} a - \frac{9329}{3} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2239.61705252 \)
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

\(\Q(\sqrt{73}) \)

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: 3.1.3723.1 $\times$ 3.1.204.1
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ R ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.73.2t1.1c1$1$ $ 73 $ $x^{2} - x - 18$ $C_2$ (as 2T1) $1$ $1$
1.3_17.2t1.1c1$1$ $ 3 \cdot 17 $ $x^{2} - x + 13$ $C_2$ (as 2T1) $1$ $-1$
1.3_17_73.2t1.1c1$1$ $ 3 \cdot 17 \cdot 73 $ $x^{2} - x + 931$ $C_2$ (as 2T1) $1$ $-1$
2.3_17_73.6t3.5c1$2$ $ 3 \cdot 17 \cdot 73 $ $x^{6} - x^{5} + 10 x^{4} + 15 x^{3} + 21 x^{2} + 18 x + 36$ $D_{6}$ (as 6T3) $1$ $0$
2.3_17_73.3t2.1c1$2$ $ 3 \cdot 17 \cdot 73 $ $x^{3} - x^{2} + x - 12$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3_17.3t2.1c1$2$ $ 2^{2} \cdot 3 \cdot 17 $ $x^{3} - x^{2} + x - 3$ $S_3$ (as 3T2) $1$ $0$
2.2e2_3_17_73e2.6t3.2c1$2$ $ 2^{2} \cdot 3 \cdot 17 \cdot 73^{2}$ $x^{6} - x^{5} - 53 x^{4} + 41 x^{3} + 988 x^{2} - 28 x - 6546$ $D_{6}$ (as 6T3) $1$ $0$
* 4.2e4_3e2_17e2_73e2.6t9.1c1$4$ $ 2^{4} \cdot 3^{2} \cdot 17^{2} \cdot 73^{2}$ $x^{6} - x^{5} + 7 x^{4} + 91 x^{3} - 370 x^{2} + 1312 x - 1962$ $S_3^2$ (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 *.