magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, -16, 0, 22, 0, -8, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 8*x^6 + 22*x^4 - 16*x^2 - 19)
gp: K = bnfinit(x^8 - 8*x^6 + 22*x^4 - 16*x^2 - 19, 1)
Normalized defining polynomial
\( x^{8} - 8 x^{6} + 22 x^{4} - 16 x^{2} - 19 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-28768731136=-\,2^{22}\cdot 19^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4}$
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: | $4$ | 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: | \( \frac{1}{4} a^{6} - \frac{5}{4} a^{4} + \frac{9}{4} a^{2} - \frac{1}{4} \), \( \frac{1}{4} a^{6} - \frac{5}{4} a^{4} + \frac{9}{4} a^{2} + \frac{7}{4} \), \( \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{7}{4} a^{5} + \frac{9}{4} a^{4} + \frac{9}{4} a^{3} - \frac{13}{4} a^{2} + \frac{9}{4} a - \frac{15}{4} \), \( \frac{1}{4} a^{7} - a^{6} - a^{5} + \frac{19}{4} a^{4} + \frac{5}{4} a^{3} - 6 a^{2} + 3 a - \frac{21}{4} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 250.192573054 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$\GL(2,3)$ (as 8T23):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 48 |
| The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
| Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
| 4.2.4864.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 16 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Arithmetically equvalently sibling: | 8.2.28768731136.8 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }$ | ${\href{/LocalNumberField/31.8.0.1}{8} }$ | ${\href{/LocalNumberField/37.8.0.1}{8} }$ | ${\href{/LocalNumberField/41.8.0.1}{8} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.22.137 | $x^{8} + 4 x^{7} + 12 x^{2} + 2$ | $8$ | $1$ | $22$ | $\textrm{GL(2,3)}$ | $[8/3, 8/3, 7/2]_{3}^{2}$ |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.19.2t1.1c1 | $1$ | $ 19 $ | $x^{2} - x + 5$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.2e2_19.3t2.1c1 | $2$ | $ 2^{2} \cdot 19 $ | $x^{3} - 2 x - 2$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| 2.2e7_19.24t22.4c1 | $2$ | $ 2^{7} \cdot 19 $ | $x^{8} - 8 x^{6} + 22 x^{4} - 16 x^{2} - 19$ | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 2.2e7_19.24t22.4c2 | $2$ | $ 2^{7} \cdot 19 $ | $x^{8} - 8 x^{6} + 22 x^{4} - 16 x^{2} - 19$ | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
| 3.2e8_19e2.6t8.5c1 | $3$ | $ 2^{8} \cdot 19^{2}$ | $x^{4} - 4 x + 2$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 3.2e8_19.4t5.1c1 | $3$ | $ 2^{8} \cdot 19 $ | $x^{4} - 4 x + 2$ | $S_4$ (as 4T5) | $1$ | $1$ |
| * | 4.2e14_19e2.8t23.4c1 | $4$ | $ 2^{14} \cdot 19^{2}$ | $x^{8} - 8 x^{6} + 22 x^{4} - 16 x^{2} - 19$ | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.