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Properties

Label 6.2.23066015625.3
Degree $6$
Signature $[2, 2]$
Discriminant $3^{10}\cdot 5^{8}$
Root discriminant $53.35$
Ramified primes $3, 5$
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![900, -405, 0, -75, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 75*x^3 - 405*x + 900)
 
gp: K = bnfinit(x^6 - 75*x^3 - 405*x + 900, 1)
 

Normalized defining polynomial

\( x^{6} - 75 x^{3} - 405 x + 900 \)

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:  \(23066015625=3^{10}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.35$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{9} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{4} + \frac{1}{3} a$, $\frac{1}{855} a^{5} + \frac{2}{57} a^{4} + \frac{1}{19} a^{3} - \frac{10}{57} a^{2} - \frac{5}{19} a - \frac{7}{19}$

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:  \( \frac{2}{57} a^{5} + \frac{47}{171} a^{4} - \frac{53}{171} a^{3} - \frac{81}{19} a^{2} - \frac{925}{57} a + \frac{1099}{57} \),  \( \frac{130}{171} a^{5} - \frac{128}{171} a^{4} - \frac{1465}{171} a^{3} - \frac{59}{57} a^{2} - \frac{5171}{57} a + \frac{9929}{57} \),  \( \frac{784}{855} a^{5} + \frac{1208}{171} a^{4} - \frac{764}{57} a^{3} - \frac{6529}{57} a^{2} - \frac{21868}{57} a + \frac{16647}{19} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2000.25434019 \)
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: 6.2.2562890625.1
Degree 6 sibling: 6.2.2562890625.1
Degree 10 sibling: 10.2.59115675201416015625.1
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 R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.3.5.2$x^{3} + 21$$3$$1$$5$$S_3$$[5/2]_{2}$
3.3.5.3$x^{3} + 12$$3$$1$$5$$S_3$$[5/2]_{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.8.3$x^{5} - 5 x^{4} + 30$$5$$1$$8$$C_5$$[2]$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
5.3e8_5e8.6t15.2c1$5$ $ 3^{8} \cdot 5^{8}$ $x^{6} - 75 x^{3} - 405 x + 900$ $A_6$ (as 6T15) $1$ $1$
* 5.3e10_5e8.6t15.4c1$5$ $ 3^{10} \cdot 5^{8}$ $x^{6} - 75 x^{3} - 405 x + 900$ $A_6$ (as 6T15) $1$ $1$
8.3e18_5e12.36t555.2c1$8$ $ 3^{18} \cdot 5^{12}$ $x^{6} - 75 x^{3} - 405 x + 900$ $A_6$ (as 6T15) $1$ $0$
8.3e18_5e12.36t555.2c2$8$ $ 3^{18} \cdot 5^{12}$ $x^{6} - 75 x^{3} - 405 x + 900$ $A_6$ (as 6T15) $1$ $0$
9.3e18_5e16.10t26.2c1$9$ $ 3^{18} \cdot 5^{16}$ $x^{6} - 75 x^{3} - 405 x + 900$ $A_6$ (as 6T15) $1$ $1$
10.3e20_5e16.30t88.2c1$10$ $ 3^{20} \cdot 5^{16}$ $x^{6} - 75 x^{3} - 405 x + 900$ $A_6$ (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 *.