magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 15, 0, 3, 0, -5, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 5*x^6 + 3*x^4 + 15*x^2 - 19)
gp: K = bnfinit(x^8 - 5*x^6 + 3*x^4 + 15*x^2 - 19, 1)
Normalized defining polynomial
\( x^{8} - 5 x^{6} + 3 x^{4} + 15 x^{2} - 19 \)
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: | \(-1097440000=-\,2^{8}\cdot 5^{4}\cdot 19^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.49$ | 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: | $2, 5, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a$
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: | \( \frac{2}{5} a^{6} - \frac{7}{5} a^{4} - \frac{2}{5} a^{2} + \frac{17}{5} \), \( \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{6}{5} a^{2} + \frac{6}{5} \), \( a^{7} + \frac{3}{5} a^{6} - 3 a^{5} - \frac{13}{5} a^{4} - 3 a^{3} - \frac{8}{5} a^{2} + 10 a + \frac{43}{5} \), \( \frac{4}{5} a^{7} - \frac{7}{5} a^{6} - \frac{9}{5} a^{5} + \frac{17}{5} a^{4} - \frac{14}{5} a^{3} + \frac{22}{5} a^{2} + \frac{24}{5} a - \frac{47}{5} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 32.1649665828 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_8:C_2^2$ (as 8T15):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $Z_8 : Z_8^\times$ |
| Character table for $Z_8 : Z_8^\times$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.475.1 |
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 16 siblings: | data not computed |
| Arithmetically equvalently sibling: | 8.2.1097440000.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.5 | $x^{8} + 2 x^{7} + 8 x^{2} + 16$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.2e2_5.2t1.1c1 | $1$ | $ 2^{2} \cdot 5 $ | $x^{2} + 5$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e2_5_19.2t1.1c1 | $1$ | $ 2^{2} \cdot 5 \cdot 19 $ | $x^{2} - 95$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e2_19.2t1.1c1 | $1$ | $ 2^{2} \cdot 19 $ | $x^{2} - 19$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.19.2t1.1c1 | $1$ | $ 19 $ | $x^{2} - x + 5$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.5_19.2t1.1c1 | $1$ | $ 5 \cdot 19 $ | $x^{2} - x + 24$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.5_19.4t3.2c1 | $2$ | $ 5 \cdot 19 $ | $x^{4} - x^{3} + 3 x^{2} + x + 1$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.2e4_5_19.4t3.3c1 | $2$ | $ 2^{4} \cdot 5 \cdot 19 $ | $x^{4} - x^{2} + 5$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| * | 4.2e8_5e2_19e2.8t15.2c1 | $4$ | $ 2^{8} \cdot 5^{2} \cdot 19^{2}$ | $x^{8} - 5 x^{6} + 3 x^{4} + 15 x^{2} - 19$ | $Z_8 : Z_8^\times$ (as 8T15) | $1$ | $0$ |
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 *.