magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -1, 7, 10, -21, -21, 18, 15, -4, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 6*x^10 - 4*x^9 + 15*x^8 + 18*x^7 - 21*x^6 - 21*x^5 + 10*x^4 + 7*x^3 - x^2 + 2*x + 1)
gp: K = bnfinit(x^12 - 6*x^10 - 4*x^9 + 15*x^8 + 18*x^7 - 21*x^6 - 21*x^5 + 10*x^4 + 7*x^3 - x^2 + 2*x + 1, 1)
Normalized defining polynomial
\( x^{12} - 6 x^{10} - 4 x^{9} + 15 x^{8} + 18 x^{7} - 21 x^{6} - 21 x^{5} + 10 x^{4} + 7 x^{3} - x^{2} + 2 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $12$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-15691065834027=-\,3\cdot 349^{2}\cdot 6553^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.58$ | 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: | $3, 349, 6553$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{31} a^{11} - \frac{2}{31} a^{10} - \frac{2}{31} a^{9} + \frac{15}{31} a^{7} - \frac{12}{31} a^{6} + \frac{3}{31} a^{5} + \frac{4}{31} a^{4} + \frac{2}{31} a^{3} + \frac{3}{31} a^{2} - \frac{7}{31} a - \frac{15}{31}$
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: | $8$ | 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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 114.337032061 \) | 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 non-solvable group of order 46080 |
| The 65 conjugacy class representatives for [2^6]S(6)=2wrS(6) are not computed |
| Character table for [2^6]S(6)=2wrS(6) is not computed |
Intermediate fields
| 6.6.2286997.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 40 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.12.0.1}{12} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.12.0.1}{12} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.5.0.1 | $x^{5} - x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 3.5.0.1 | $x^{5} - x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 349 | Data not computed | ||||||
| 6553 | Data not computed | ||||||