magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 1, 1, -3, 2, -2, -1, 0, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + x^10 - x^8 - 2*x^7 + 2*x^6 - 3*x^5 + x^4 + x^3 + x^2 + 1)
gp: K = bnfinit(x^12 - x^11 + x^10 - x^8 - 2*x^7 + 2*x^6 - 3*x^5 + x^4 + x^3 + x^2 + 1, 1)
Normalized defining polynomial
\( x^{12} - x^{11} + x^{10} - x^{8} - 2 x^{7} + 2 x^{6} - 3 x^{5} + x^{4} + x^{3} + x^{2} + 1 \)
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: | \(546564453125=5^{9}\cdot 23^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $9.51$ | 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: | $5, 23$ | 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: | \( a^{10} - a^{9} - a^{6} - 2 a^{5} + 2 a^{4} - a^{3} + a^{2} + a + 1 \) (order $10$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( a^{5} - 1 \), \( a^{11} - 3 a^{6} + 2 a + 1 \), \( a^{11} - a^{10} + a^{9} - a^{7} - a^{6} + 2 a^{5} - 3 a^{4} + a^{3} - a \), \( a^{11} - a^{7} - 2 a^{6} - a^{4} + a^{3} + 2 a^{2} + 1 \), \( a^{10} - a^{9} - a^{6} - 3 a^{5} + 2 a^{4} + 2 a + 2 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 32.3656237465 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4\times S_3$ (as 12T11):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 24 |
| The 12 conjugacy class representatives for $S_3 \times C_4$ |
| Character table for $S_3 \times C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.1.23.1, \(\Q(\zeta_{5})\), 6.2.66125.1 |
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 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }$ | ${\href{/LocalNumberField/3.12.0.1}{12} }$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $23$ | 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 23.8.4.1 | $x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |