magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -4, -2, -8, 1, 0, -2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^6 + x^4 - 8*x^3 - 2*x^2 - 4*x - 2)
gp: K = bnfinit(x^8 - 2*x^6 + x^4 - 8*x^3 - 2*x^2 - 4*x - 2, 1)
Normalized defining polynomial
\( x^{8} - 2 x^{6} + x^{4} - 8 x^{3} - 2 x^{2} - 4 x - 2 \)
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: | \(-488095744=-\,2^{14}\cdot 31^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{76} a^{7} - \frac{15}{76} a^{6} - \frac{5}{76} a^{5} - \frac{1}{76} a^{4} + \frac{4}{19} a^{3} - \frac{5}{19} a^{2} - \frac{3}{38} a + \frac{5}{38}$
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: | 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: | \( 61.8892710819 \) | 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 64 |
| The 16 conjugacy class representatives for $(C_4^2 : C_2):C_2$ |
| Character table for $(C_4^2 : C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.2.1984.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 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 | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.1 | $x^{2} + 14$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.1 | $x^{2} + 14$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| $31$ | $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.3.1 | $x^{4} + 217$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2_31.2t1.1c1 | $1$ | $ 2^{2} \cdot 31 $ | $x^{2} - 31$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.2e3.2t1.1c1 | $1$ | $ 2^{3}$ | $x^{2} - 2$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.2e3_31.2t1.1c1 | $1$ | $ 2^{3} \cdot 31 $ | $x^{2} - 62$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e3.2t1.2c1 | $1$ | $ 2^{3}$ | $x^{2} + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e3_31.2t1.2c1 | $1$ | $ 2^{3} \cdot 31 $ | $x^{2} + 62$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.31.2t1.1c1 | $1$ | $ 31 $ | $x^{2} - x + 8$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.2e2_31e2.4t3.1c1 | $2$ | $ 2^{2} \cdot 31^{2}$ | $x^{4} + 31$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.2e6_31e2.4t3.2c1 | $2$ | $ 2^{6} \cdot 31^{2}$ | $x^{4} + 124$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.2e3_31e2.4t3.1c1 | $2$ | $ 2^{3} \cdot 31^{2}$ | $x^{4} - 2 x^{3} + 17 x^{2} - 16 x + 2$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.2e5_31.4t3.1c1 | $2$ | $ 2^{5} \cdot 31 $ | $x^{4} - 8 x - 7$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| * | 2.2e3_31.4t3.1c1 | $2$ | $ 2^{3} \cdot 31 $ | $x^{4} - 2 x^{3} + x^{2} - 2$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.2e5_31e2.4t3.2c1 | $2$ | $ 2^{5} \cdot 31^{2}$ | $x^{4} - 248 x + 713$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 4.2e8_31e4.8t26.3c1 | $4$ | $ 2^{8} \cdot 31^{4}$ | $x^{8} - 2 x^{6} + x^{4} - 8 x^{3} - 2 x^{2} - 4 x - 2$ | $(C_4^2 : C_2):C_2$ (as 8T26) | $1$ | $0$ | |
| * | 4.2e8_31e2.8t26.3c1 | $4$ | $ 2^{8} \cdot 31^{2}$ | $x^{8} - 2 x^{6} + x^{4} - 8 x^{3} - 2 x^{2} - 4 x - 2$ | $(C_4^2 : C_2):C_2$ (as 8T26) | $1$ | $0$ |
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 *.