magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 3, 0, 4, 0, 3, 0, 4, 0, 3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 3*x^10 + 4*x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1)
gp: K = bnfinit(x^12 + 3*x^10 + 4*x^8 + 3*x^6 + 4*x^4 + 3*x^2 + 1, 1)
Normalized defining polynomial
\( x^{12} + 3 x^{10} + 4 x^{8} + 3 x^{6} + 4 x^{4} + 3 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(49331181715456=2^{20}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.84$ | 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: | $2, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$
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: | \( a^{11} + 3 a^{9} + 4 a^{7} + 3 a^{5} + 4 a^{3} + 3 a \), \( a^{11} + 2 a^{9} + 2 a^{7} + a^{5} + 3 a^{3} \), \( 2 a^{10} - a^{9} + 5 a^{8} - 2 a^{7} + 5 a^{6} - 2 a^{5} + 3 a^{4} - a^{3} + 6 a^{2} - 2 a + 3 \), \( a^{11} - a^{10} + 2 a^{9} - 3 a^{8} + a^{7} - 4 a^{6} - a^{5} - 3 a^{4} + 2 a^{3} - 3 a^{2} + a - 1 \), \( a^{11} + 2 a^{10} + 2 a^{9} + 6 a^{8} + 2 a^{7} + 7 a^{6} + a^{5} + 4 a^{4} + 3 a^{3} + 6 a^{2} - a + 4 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 84.7546117944 \) | 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 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 3.1.76.1 x3, 6.0.7023616.2, 6.2.369664.1, 6.0.109744.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 4 sibling: | data not computed |
| Degree 6 siblings: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 12 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.20.37 | $x^{12} - 6 x^{10} - x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} - 7$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ |
| $19$ | 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |