Properties

Label 6.2.515607849.1
Degree $6$
Signature $[2, 2]$
Discriminant $3^{6}\cdot 29^{4}$
Root discriminant $28.32$
Ramified primes $3, 29$
Class number $1$
Class group Trivial
Galois group $A_6$ (as 6T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50, 75, 6, 11, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 3*x^4 + 11*x^3 + 6*x^2 + 75*x + 50)
 
gp: K = bnfinit(x^6 - 3*x^5 - 3*x^4 + 11*x^3 + 6*x^2 + 75*x + 50, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50 \)

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:  \(515607849=3^{6}\cdot 29^{4}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.32$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 29$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1405} a^{5} - \frac{453}{1405} a^{4} + \frac{122}{1405} a^{3} - \frac{94}{1405} a^{2} + \frac{156}{1405} a + \frac{25}{281}$

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:  \( \frac{37}{1405} a^{5} + \frac{99}{1405} a^{4} - \frac{1106}{1405} a^{3} + \frac{737}{1405} a^{2} + \frac{4367}{1405} a + \frac{363}{281} \),  \( \frac{9}{1405} a^{5} + \frac{138}{1405} a^{4} - \frac{307}{1405} a^{3} - \frac{846}{1405} a^{2} + \frac{1404}{1405} a + \frac{225}{281} \),  \( \frac{24}{281} a^{5} - \frac{194}{281} a^{4} + \frac{118}{281} a^{3} + \frac{1678}{281} a^{2} - \frac{1876}{281} a - \frac{5149}{281} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 179.773819629 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_6$ (as 6T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: 6.2.613089.1
Degree 6 sibling: 6.2.613089.1
Degree 10 sibling: 10.2.316113500535561.1
Degree 15 siblings: Deg 15, Deg 15
Degree 20 sibling: Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
$29$29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.3e6_29e2.6t15.2c1$5$ $ 3^{6} \cdot 29^{2}$ $x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ $A_6$ (as 6T15) $1$ $1$
* 5.3e6_29e4.6t15.2c1$5$ $ 3^{6} \cdot 29^{4}$ $x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ $A_6$ (as 6T15) $1$ $1$
8.3e12_29e6.36t555.2c1$8$ $ 3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ $A_6$ (as 6T15) $1$ $0$
8.3e12_29e6.36t555.2c2$8$ $ 3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ $A_6$ (as 6T15) $1$ $0$
9.3e12_29e6.10t26.2c1$9$ $ 3^{12} \cdot 29^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ $A_6$ (as 6T15) $1$ $1$
10.3e14_29e6.30t88.2c1$10$ $ 3^{14} \cdot 29^{6}$ $x^{6} - 3 x^{5} - 3 x^{4} + 11 x^{3} + 6 x^{2} + 75 x + 50$ $A_6$ (as 6T15) $1$ $-2$

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 *.