magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 6, -6, 0, 3, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 3*x^4 - 6*x^2 + 6*x - 2)
gp: K = bnfinit(x^6 - 3*x^5 + 3*x^4 - 6*x^2 + 6*x - 2, 1)
Normalized defining polynomial
\( x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2 \)
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: | \(1679616=2^{8}\cdot 3^{8}\) | magma: Discriminant(K);
sage: K.disc()
gp: K.disc
| |
Root discriminant: | $10.90$ | magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
Ramified primes: | $2, 3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
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: | \( a^{5} - 2 a^{4} + a^{3} + a^{2} - 5 a + 1 \), \( a^{5} - 3 a^{4} + 3 a^{3} - 5 a + 5 \), \( a^{4} - 2 a^{3} + a^{2} + a - 5 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
Regulator: | \( 12.2562739075 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
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: | 6.2.6718464.1 |
Degree 6 sibling: | 6.2.6718464.1 |
Degree 10 sibling: | 10.2.11284439629824.1 |
Degree 15 siblings: | Deg 15, 15.3.18953525353286467584.1 |
Degree 20 sibling: | 20.4.127338577759142414150270976.1 |
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 | R | R | ${\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
$2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.8.7 | $x^{4} + 4 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
$3$ | 3.6.8.1 | $x^{6} + 6 x^{5} + 18 x^{2} + 9$ | $3$ | $2$ | $8$ | $C_3^2:C_4$ | $[2, 2]^{4}$ |
Artin representations
Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
5.2e10_3e8.6t15.1c1 | $5$ | $ 2^{10} \cdot 3^{8}$ | $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ | $A_6$ (as 6T15) | $1$ | $1$ | |
* | 5.2e8_3e8.6t15.1c1 | $5$ | $ 2^{8} \cdot 3^{8}$ | $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ | $A_6$ (as 6T15) | $1$ | $1$ |
8.2e18_3e16.36t555.1c1 | $8$ | $ 2^{18} \cdot 3^{16}$ | $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ | $A_6$ (as 6T15) | $1$ | $0$ | |
8.2e18_3e16.36t555.1c2 | $8$ | $ 2^{18} \cdot 3^{16}$ | $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ | $A_6$ (as 6T15) | $1$ | $0$ | |
9.2e18_3e16.10t26.1c1 | $9$ | $ 2^{18} \cdot 3^{16}$ | $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ | $A_6$ (as 6T15) | $1$ | $1$ | |
10.2e24_3e16.30t88.1c1 | $10$ | $ 2^{24} \cdot 3^{16}$ | $x^{6} - 3 x^{5} + 3 x^{4} - 6 x^{2} + 6 x - 2$ | $A_6$ (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 *.