magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, 0, 0, 5, 0, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 5*x^3 - 5)
gp: K = bnfinit(x^6 - 3*x^5 + 5*x^3 - 5, 1)
Normalized defining polynomial
\( x^{6} - 3 x^{5} + 5 x^{3} - 5 \)
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: | \(9112500=2^{2}\cdot 3^{6}\cdot 5^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.45$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
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: | \( a^{4} - 2 a^{3} - 2 a^{2} + 4 a + 2 \), \( a^{2} - 2 \), \( 2 a^{5} - 3 a^{4} - 4 a^{3} + 4 a^{2} + 5 a + 7 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 39.6968193802 \) | 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: | 6.2.36450000.3 |
| Degree 6 sibling: | 6.2.36450000.3 |
| Degree 10 sibling: | 10.2.184528125000000.1 |
| Degree 12 siblings: | Deg 12, Deg 12 |
| Degree 15 siblings: | Deg 15, Deg 15 |
| Degree 20 siblings: | Deg 20, Deg 20, 20.4.34050628916015625000000000000.1 |
| 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 | R | R | ${\href{/LocalNumberField/7.6.0.1}{6} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }$ | ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $3$ | 3.6.6.6 | $x^{6} + 3 x + 3$ | $6$ | $1$ | $6$ | $C_3^2:D_4$ | $[5/4, 5/4]_{4}^{2}$ |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 5.2e2_3e6_5e5.6t16.1c1 | $5$ | $ 2^{2} \cdot 3^{6} \cdot 5^{5}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $1$ |
| 5.2e4_3e6_5e6.12t183.1c1 | $5$ | $ 2^{4} \cdot 3^{6} \cdot 5^{6}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $1$ | |
| 5.2e2_3e6_5e6.12t183.1c1 | $5$ | $ 2^{2} \cdot 3^{6} \cdot 5^{6}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $1$ | |
| 5.2e4_3e6_5e5.6t16.1c1 | $5$ | $ 2^{4} \cdot 3^{6} \cdot 5^{5}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $1$ | |
| 9.2e6_3e10_5e11.10t32.1c1 | $9$ | $ 2^{6} \cdot 3^{10} \cdot 5^{11}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $1$ | |
| 9.2e6_3e10_5e10.20t145.1c1 | $9$ | $ 2^{6} \cdot 3^{10} \cdot 5^{10}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $1$ | |
| 10.2e6_3e12_5e12.30t176.1c1 | $10$ | $ 2^{6} \cdot 3^{12} \cdot 5^{12}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $-2$ | |
| 10.2e6_3e12_5e12.30t176.2c1 | $10$ | $ 2^{6} \cdot 3^{12} \cdot 5^{12}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (as 6T16) | $1$ | $-2$ | |
| 16.2e12_3e20_5e18.36t1252.1c1 | $16$ | $ 2^{12} \cdot 3^{20} \cdot 5^{18}$ | $x^{6} - 3 x^{5} + 5 x^{3} - 5$ | $S_6$ (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 *.