magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, -1, 1, 3, -7, 6, 3, -9, 5, 2, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 + 2*x^10 + 5*x^9 - 9*x^8 + 3*x^7 + 6*x^6 - 7*x^5 + 3*x^4 + x^3 - x^2 + x - 1)
gp: K = bnfinit(x^12 - 3*x^11 + 2*x^10 + 5*x^9 - 9*x^8 + 3*x^7 + 6*x^6 - 7*x^5 + 3*x^4 + x^3 - x^2 + x - 1, 1)
Normalized defining polynomial
\( x^{12} - 3 x^{11} + 2 x^{10} + 5 x^{9} - 9 x^{8} + 3 x^{7} + 6 x^{6} - 7 x^{5} + 3 x^{4} + x^{3} - x^{2} + x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-233544944483=-\,23^{4}\cdot 101\cdot 8263\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $8.86$ | 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: | $23, 101, 8263$ | 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: | $6$ | 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: | \( a^{8} - a^{7} - a^{6} + 3 a^{5} - a^{4} + 2 a^{2} - a \), \( a \), \( a^{11} - 2 a^{10} + 5 a^{8} - 4 a^{7} - 2 a^{6} + 6 a^{5} - 2 a^{4} - a^{3} + 2 a^{2} + 1 \), \( a^{8} - a^{7} - a^{6} + 3 a^{5} - a^{3} + 2 a^{2} \), \( a^{9} - 2 a^{8} + a^{7} + 3 a^{6} - 4 a^{5} + 2 a^{4} + 2 a^{3} - 2 a^{2} + 2 a - 1 \), \( a^{11} - 3 a^{10} + 2 a^{9} + 4 a^{8} - 7 a^{7} + 2 a^{6} + 4 a^{5} - 4 a^{4} + a^{3} + a \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 4.29199444222 \) | 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 82944 |
| The 65 conjugacy class representatives for [S(4)^3]S(3)=S(4)wrS(3) are not computed |
| Character table for [S(4)^3]S(3)=S(4)wrS(3) is not computed |
Intermediate fields
| 3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }$ | ${\href{/LocalNumberField/3.9.0.1}{9} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.12.0.1}{12} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ | ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $101$ | $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{101}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 101.4.0.1 | $x^{4} - x + 12$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 8263 | Data not computed | ||||||