Properties

Label 6.0.19208000.1
Degree $6$
Signature $[0, 3]$
Discriminant $-\,2^{6}\cdot 5^{3}\cdot 7^{4}$
Root discriminant $16.36$
Ramified primes $2, 5, 7$
Class number $6$
Class group $[6]$
Galois Group $C_6$ (as 6T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![281, -64, 87, -14, 12, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 12*x^4 - 14*x^3 + 87*x^2 - 64*x + 281)
gp: K = bnfinit(x^6 - 2*x^5 + 12*x^4 - 14*x^3 + 87*x^2 - 64*x + 281, 1)

Normalized defining polynomial

\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut +\mathstrut 12 x^{4} \) \(\mathstrut -\mathstrut 14 x^{3} \) \(\mathstrut +\mathstrut 87 x^{2} \) \(\mathstrut -\mathstrut 64 x \) \(\mathstrut +\mathstrut 281 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 3]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-19208000=-\,2^{6}\cdot 5^{3}\cdot 7^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $16.36$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(99,·)$, $\chi_{140}(81,·)$, $\chi_{140}(39,·)$, $\chi_{140}(121,·)$, $\chi_{140}(79,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14629} a^{5} - \frac{4041}{14629} a^{4} - \frac{4353}{14629} a^{3} - \frac{2305}{14629} a^{2} + \frac{5938}{14629} a - \frac{6715}{14629}$

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

Class group and class number

Multiplicative Abelian group isomorphic to C6, order $6$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $2$
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:  \( \frac{94}{14629} a^{5} + \frac{500}{14629} a^{4} + \frac{430}{14629} a^{3} + \frac{2765}{14629} a^{2} + \frac{2270}{14629} a + \frac{27095}{14629} \),  \( \frac{40}{14629} a^{5} - \frac{721}{14629} a^{4} + \frac{1428}{14629} a^{3} - \frac{4426}{14629} a^{2} + \frac{3456}{14629} a - \frac{34536}{14629} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 2.10181872849 \)
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

\(\Q(\sqrt{-5}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra: \(\Q(\zeta_{7})^+\) $\times$ \(\Q(\sqrt{-5}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_5.2t1.1c1$1$ $ 2^{2} \cdot 5 $ $x^{2} + 5$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_5_7.6t1.2c1$1$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.2e2_5_7.6t1.2c2$1$ $ 2^{2} \cdot 5 \cdot 7 $ $x^{6} - 2 x^{5} + 12 x^{4} - 14 x^{3} + 87 x^{2} - 64 x + 281$ $C_6$ (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 *.