magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -33, 21, 13, -11, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 11*x^4 + 13*x^3 + 21*x^2 - 33*x + 11)
gp: K = bnfinit(x^6 - x^5 - 11*x^4 + 13*x^3 + 21*x^2 - 33*x + 11, 1)
Normalized defining polynomial
\( x^{6} - x^{5} - 11 x^{4} + 13 x^{3} + 21 x^{2} - 33 x + 11 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(226773481=11^{2}\cdot 37^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.69$ | 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: | $11, 37$ | 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: | $5$ | 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: | \( a - 1 \), \( a^{5} - 11 a^{3} + 2 a^{2} + 22 a - 12 \), \( 3 a^{5} - a^{4} - 34 a^{3} + 17 a^{2} + 77 a - 53 \), \( 5 a^{5} - a^{4} - 56 a^{3} + 21 a^{2} + 123 a - 73 \), \( a^{4} + a^{3} - 9 a^{2} - 8 a + 10 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 155.023450972 \) | 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 solvable group of order 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| 3.3.1369.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | data not computed |
| Twin sextic algebra: | \(\Q\) $\times$ \(\Q\) $\times$ 4.4.165649.1 |
| Degree 4 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 | ${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | R | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $37$ | 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |