Properties

Label 6.6.3570125.1
Degree $6$
Signature $[6, 0]$
Discriminant $5^{3}\cdot 13^{4}$
Root discriminant $12.36$
Ramified primes $5, 13$
Class number $1$
Class group Trivial
Galois Group $C_6$ (as 6T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -10, 19, 13, -12, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 12*x^4 + 13*x^3 + 19*x^2 - 10*x - 5)
gp: K = bnfinit(x^6 - x^5 - 12*x^4 + 13*x^3 + 19*x^2 - 10*x - 5, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut 12 x^{4} \) \(\mathstrut +\mathstrut 13 x^{3} \) \(\mathstrut +\mathstrut 19 x^{2} \) \(\mathstrut -\mathstrut 10 x \) \(\mathstrut -\mathstrut 5 \)

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:  \(3570125=5^{3}\cdot 13^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.36$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $5, 13$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(16,·)$, $\chi_{65}(1,·)$, $\chi_{65}(9,·)$, $\chi_{65}(29,·)$, $\chi_{65}(14,·)$, $\chi_{65}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{151} a^{5} - \frac{44}{151} a^{4} + \frac{68}{151} a^{3} - \frac{42}{151} a^{2} + \frac{13}{151} a + \frac{35}{151}$

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:  \( \frac{4}{151} a^{5} - \frac{25}{151} a^{4} - \frac{30}{151} a^{3} + \frac{285}{151} a^{2} - \frac{99}{151} a - \frac{313}{151} \),  \( \frac{4}{151} a^{5} - \frac{25}{151} a^{4} - \frac{30}{151} a^{3} + \frac{285}{151} a^{2} - \frac{250}{151} a - \frac{162}{151} \),  \( \frac{38}{151} a^{5} - \frac{11}{151} a^{4} - \frac{436}{151} a^{3} + \frac{216}{151} a^{2} + \frac{645}{151} a + \frac{122}{151} \),  \( \frac{21}{151} a^{5} - \frac{18}{151} a^{4} - \frac{233}{151} a^{3} + \frac{326}{151} a^{2} + \frac{273}{151} a - \frac{322}{151} \),  \( \frac{10}{151} a^{5} + \frac{13}{151} a^{4} - \frac{75}{151} a^{3} - \frac{118}{151} a^{2} - \frac{21}{151} a + \frac{48}{151} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 19.2357657825 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_6$ (as 6T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A cyclic group of order 6
The 6 conjugacy class representatives for $C_6$
Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.169.1

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

Sibling algebras

Twin sextic algebra: 3.3.169.1 $\times$ \(\Q(\sqrt{5}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.1c1$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
* 1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.5_13.6t1.1c1$1$ $ 5 \cdot 13 $ $x^{6} - x^{5} - 12 x^{4} + 13 x^{3} + 19 x^{2} - 10 x - 5$ $C_6$ (as 6T1) $0$ $1$
* 1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
* 1.5_13.6t1.1c2$1$ $ 5 \cdot 13 $ $x^{6} - x^{5} - 12 x^{4} + 13 x^{3} + 19 x^{2} - 10 x - 5$ $C_6$ (as 6T1) $0$ $1$

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 *.