↑

Properties

Label 6.2.328050000.2
Degree $6$
Signature $[2, 2]$
Discriminant $2^{4}\cdot 3^{8}\cdot 5^{5}$
Root discriminant $26.26$
Ramified primes $2, 3, 5$
Class number $6$
Class group $[6]$
Galois group $\PGL(2,5)$ (as 6T14)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![154, -18, 30, -20, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 20*x^3 + 30*x^2 - 18*x + 154)
 
gp: K = bnfinit(x^6 - 3*x^5 - 20*x^3 + 30*x^2 - 18*x + 154, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 20 x^{3} + 30 x^{2} - 18 x + 154 \)

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:  \(328050000=2^{4}\cdot 3^{8}\cdot 5^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.26$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9}$, $\frac{1}{9} a^{4} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{5} + \frac{1}{27} a^{4} + \frac{1}{27} a^{3} + \frac{2}{27} a^{2} + \frac{11}{27} a + \frac{2}{27}$

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

Class group and class number

$C_{6}$, which has 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:  $3$
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:  \( \frac{1}{27} a^{5} - \frac{2}{27} a^{4} + \frac{1}{27} a^{3} - \frac{25}{27} a^{2} - \frac{13}{27} a - \frac{61}{27} \),  \( \frac{1}{27} a^{5} - \frac{2}{27} a^{4} - \frac{5}{27} a^{3} - \frac{16}{27} a^{2} + \frac{14}{27} a - \frac{1}{27} \),  \( \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - a - \frac{29}{9} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 62.0642135622 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_5$ (as 6T14):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 120
The 7 conjugacy class representatives for $\PGL(2,5)$
Character table for $\PGL(2,5)$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: 5.1.162000.2 $\times$ \(\Q\)
Degree 5 sibling: 5.1.162000.2
Degree 10 siblings: 10.2.10628820000000.3, 10.2.131220000000.1
Degree 12 sibling: Deg 12
Degree 15 sibling: Deg 15
Degree 20 siblings: Deg 20, Deg 20, Deg 20
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.6.8.4$x^{6} + 18 x^{2} + 63$$3$$2$$8$$C_6$$[2]^{2}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{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) $1$ $1$
4.2e4_3e4_5e3.10t12.2c1$4$ $ 2^{4} \cdot 3^{4} \cdot 5^{3}$ $x^{6} - 3 x^{5} - 20 x^{3} + 30 x^{2} - 18 x + 154$ $\PGL(2,5)$ (as 6T14) $1$ $0$
4.2e4_3e4_5e3.5t5.2c1$4$ $ 2^{4} \cdot 3^{4} \cdot 5^{3}$ $x^{6} - 3 x^{5} - 20 x^{3} + 30 x^{2} - 18 x + 154$ $\PGL(2,5)$ (as 6T14) $1$ $0$
5.2e4_3e8_5e4.10t13.2c1$5$ $ 2^{4} \cdot 3^{8} \cdot 5^{4}$ $x^{6} - 3 x^{5} - 20 x^{3} + 30 x^{2} - 18 x + 154$ $\PGL(2,5)$ (as 6T14) $1$ $1$
* 5.2e4_3e8_5e5.6t14.5c1$5$ $ 2^{4} \cdot 3^{8} \cdot 5^{5}$ $x^{6} - 3 x^{5} - 20 x^{3} + 30 x^{2} - 18 x + 154$ $\PGL(2,5)$ (as 6T14) $1$ $1$
6.2e4_3e8_5e5.20t35.2c1$6$ $ 2^{4} \cdot 3^{8} \cdot 5^{5}$ $x^{6} - 3 x^{5} - 20 x^{3} + 30 x^{2} - 18 x + 154$ $\PGL(2,5)$ (as 6T14) $1$ $-2$

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 *.