magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 6, 0, 12, 0, 8, 0, 1, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 2*x^10 + x^8 + 8*x^6 + 12*x^4 + 6*x^2 + 1)
gp: K = bnfinit(x^12 + 2*x^10 + x^8 + 8*x^6 + 12*x^4 + 6*x^2 + 1, 1)
Normalized defining polynomial
\( x^{12} + 2 x^{10} + x^{8} + 8 x^{6} + 12 x^{4} + 6 x^{2} + 1 \)
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: | \(15494111297536=2^{24}\cdot 31^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.57$ | 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: | $2, 31$ | 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: | \( 8 a^{10} + 12 a^{8} + 2 a^{6} + 63 a^{4} + 65 a^{2} + 16 \), \( a \), \( 9 a^{10} + 13 a^{8} + 2 a^{6} + 71 a^{4} + 68 a^{2} + 17 \), \( 15 a^{11} - 4 a^{10} + 22 a^{9} - 8 a^{8} + 3 a^{7} - 3 a^{6} + 118 a^{5} - 31 a^{4} + 117 a^{3} - 48 a^{2} + 26 a - 16 \), \( 4 a^{11} + a^{10} + 7 a^{9} + 2 a^{8} + 2 a^{7} + a^{6} + 31 a^{5} + 8 a^{4} + 40 a^{3} + 12 a^{2} + 13 a + 4 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 37.8290071512 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_4^2:C_3:C_2$ (as 12T95):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 192 |
| The 20 conjugacy class representatives for $C_2\times C_4^2:C_3:C_2$ |
| Character table for $C_2\times C_4^2:C_3:C_2$ |
Intermediate fields
| 3.1.31.1, 6.2.61504.1 |
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 32 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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.24.169 | $x^{12} - 8 x^{11} - 4 x^{10} - 4 x^{8} + 16 x^{7} + 12 x^{6} + 8 x^{5} + 4 x^{4} - 8$ | $4$ | $3$ | $24$ | 12T55 | $[2, 2, 3, 3, 3]^{3}$ |
| $31$ | $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.8.4.1 | $x^{8} + 32674 x^{4} - 119164 x^{2} + 266897569$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |