↑

Properties

Label 6.2.6400000.3
Degree $6$
Signature $[2, 2]$
Discriminant $2^{11}\cdot 5^{5}$
Root discriminant $13.63$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $S_6$ (as 6T16)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 10, -5, 0, 5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + 5*x^4 - 5*x^2 + 10*x - 5)
 
gp: K = bnfinit(x^6 - 2*x^5 + 5*x^4 - 5*x^2 + 10*x - 5, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6400000=2^{11}\cdot 5^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.63$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$

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:  \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{3}{2} \),  \( a^{5} - \frac{3}{2} a^{4} + 4 a^{3} + 2 a^{2} - 4 a + \frac{11}{2} \),  \( a^{5} - \frac{3}{2} a^{4} + \frac{9}{2} a^{3} + \frac{5}{2} a^{2} - \frac{7}{2} a + 9 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 32.3726917122 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_6$ (as 6T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 720
The 11 conjugacy class representatives for $S_6$
Character table for $S_6$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling algebras

Twin sextic algebra: 6.2.6400000.4
Degree 6 sibling: 6.2.6400000.4
Degree 10 sibling: 10.2.25600000000000.5
Degree 12 siblings: 12.4.65536000000000000.2, 12.4.16384000000000000.3
Degree 15 siblings: Deg 15, Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.6.11.8$x^{6} + 4 x^{4} + 2 x^{2} + 6$$6$$1$$11$$S_4\times C_2$$[8/3, 8/3, 3]_{3}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.5.3$x^{5} + 15 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_5.2t1.1c1$1$ $ 2^{3} \cdot 5 $ $x^{2} - 10$ $C_2$ (as 2T1) $1$ $1$
* 5.2e11_5e5.6t16.5c1$5$ $ 2^{11} \cdot 5^{5}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $1$
5.2e12_5e6.12t183.2c1$5$ $ 2^{12} \cdot 5^{6}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $1$
5.2e14_5e6.12t183.4c1$5$ $ 2^{14} \cdot 5^{6}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $1$
5.2e11_5e5.6t16.6c1$5$ $ 2^{11} \cdot 5^{5}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $1$
9.2e19_5e11.10t32.2c1$9$ $ 2^{19} \cdot 5^{11}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $1$
9.2e26_5e10.20t145.4c1$9$ $ 2^{26} \cdot 5^{10}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $1$
10.2e28_5e12.30t176.2c1$10$ $ 2^{28} \cdot 5^{12}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $-2$
10.2e26_5e12.30t176.2c1$10$ $ 2^{26} \cdot 5^{12}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $-2$
16.2e42_5e18.36t1252.2c1$16$ $ 2^{42} \cdot 5^{18}$ $x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{2} + 10 x - 5$ $S_6$ (as 6T16) $1$ $0$

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 *.