Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 8*x^6 - 8*x^5 + 7*x^4 - 8*x^3 + 8*x^2 - 4*x + 1)
gp: K = bnfinit(x^8 - 4*x^7 + 8*x^6 - 8*x^5 + 7*x^4 - 8*x^3 + 8*x^2 - 4*x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 8, -8, 7, -8, 8, -4, 1]);
\( x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 7 x^{4} - 8 x^{3} + 8 x^{2} - 4 x + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(3211264\)\(\medspace = 2^{16}\cdot 7^{2}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $6.51$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $4$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( \frac{2}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + 2 a^{4} - a^{3} - \frac{2}{7} a^{2} - \frac{10}{7} a + \frac{9}{7} \) (order $8$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | \( \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{8}{7} a^{5} + 2 a^{4} + \frac{6}{7} a^{2} - \frac{12}{7} a + \frac{8}{7} \), \( \frac{8}{7} a^{7} - \frac{33}{7} a^{6} + \frac{62}{7} a^{5} - 8 a^{4} + 6 a^{3} - \frac{64}{7} a^{2} + \frac{51}{7} a - \frac{20}{7} \), \( \frac{12}{7} a^{7} - \frac{39}{7} a^{6} + \frac{65}{7} a^{5} - 6 a^{4} + 6 a^{3} - \frac{54}{7} a^{2} + \frac{38}{7} a - \frac{9}{7} \) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 2.5180422861 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2\times D_4$ (as 8T9):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $D_4\times C_2$ |
| Character table for $D_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 16.0.24759631762948096.2 |
| Degree 8 siblings: | 8.4.157351936.1, 8.0.9834496.1, 8.0.39337984.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |