Properties

Label 6.6.979146657.1
Degree $6$
Signature $[6, 0]$
Discriminant $3^{3}\cdot 331^{3}$
Root discriminant $31.51$
Ramified primes $3, 331$
Class number $1$
Class group Trivial
Galois Group $S_3$ (as 6T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 127, 122, -29, -34, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 34*x^4 - 29*x^3 + 122*x^2 + 127*x + 31)
gp: K = bnfinit(x^6 - 2*x^5 - 34*x^4 - 29*x^3 + 122*x^2 + 127*x + 31, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut 34 x^{4} \) \(\mathstrut -\mathstrut 29 x^{3} \) \(\mathstrut +\mathstrut 122 x^{2} \) \(\mathstrut +\mathstrut 127 x \) \(\mathstrut +\mathstrut 31 \)

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:  \(979146657=3^{3}\cdot 331^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $31.51$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 331$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{36} a^{5} - \frac{1}{36} a^{4} + \frac{1}{36} a^{3} + \frac{2}{9} a^{2} - \frac{7}{18} a + \frac{5}{36}$

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

Class group and class number

Trivial Abelian group, 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:  $5$
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:  \( \frac{5}{12} a^{5} - \frac{5}{4} a^{4} - \frac{151}{12} a^{3} - \frac{3}{2} a^{2} + \frac{143}{3} a + \frac{81}{4} \),  \( \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{35}{6} a^{3} - \frac{23}{6} a^{2} + \frac{133}{6} a + \frac{38}{3} \),  \( \frac{7}{12} a^{5} - \frac{19}{12} a^{4} - \frac{221}{12} a^{3} - \frac{16}{3} a^{2} + \frac{425}{6} a + \frac{395}{12} \),  \( \frac{4}{9} a^{5} - \frac{23}{18} a^{4} - \frac{122}{9} a^{3} - \frac{59}{18} a^{2} + \frac{887}{18} a + \frac{475}{18} \),  \( \frac{41}{36} a^{5} - \frac{113}{36} a^{4} - \frac{1327}{36} a^{3} - \frac{44}{9} a^{2} + \frac{2881}{18} a + \frac{2293}{36} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 793.658860006 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_3$ (as 6T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 6
The 3 conjugacy class representatives for $S_3$
Character table for $S_3$

Intermediate fields

\(\Q(\sqrt{993}) \), 3.3.993.1 x3

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

Sibling algebras

Twin sextic algebra: 3.3.993.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\)
Degree 3 sibling: 3.3.993.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
331Data not computed