magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -8, 0, 20, 0, -4, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^6 + 20*x^4 - 8*x^2 + 1)
gp: K = bnfinit(x^8 - 4*x^6 + 20*x^4 - 8*x^2 + 1, 1)
Normalized defining polynomial
\( x^{8} - 4 x^{6} + 20 x^{4} - 8 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5435817984=2^{26}\cdot 3^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.48$ | 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: | $2, 3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{4} + \frac{2}{7} a^{2} + \frac{3}{7}$, $\frac{1}{7} a^{7} - \frac{2}{7} a^{5} + \frac{2}{7} a^{3} + \frac{3}{7} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}$, which has order $2$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $3$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{4}{7} a^{6} + \frac{15}{7} a^{4} - \frac{78}{7} a^{2} + \frac{16}{7} \) (order $4$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{4}{7} a^{7} - \frac{15}{7} a^{5} + \frac{78}{7} a^{3} - \frac{16}{7} a \), \( \frac{4}{7} a^{7} - \frac{3}{7} a^{6} - \frac{15}{7} a^{5} + \frac{13}{7} a^{4} + \frac{78}{7} a^{3} - \frac{62}{7} a^{2} - \frac{23}{7} a + \frac{19}{7} \), \( 10 a^{7} + \frac{25}{7} a^{6} - 39 a^{5} - \frac{99}{7} a^{4} + 196 a^{3} + \frac{498}{7} a^{2} - 59 a - \frac{177}{7} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 76.4628919919 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.D_4$ (as 8T19):
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 $C_2^3 : C_4 $ |
| Character table for $C_2^3 : C_4 $ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.4608.1 |
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 8 siblings: | data not computed |
| Degree 16 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 | R | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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]])
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.2e3.2t1.2c1 | $1$ | $ 2^{3}$ | $x^{2} + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e3.2t1.1c1 | $1$ | $ 2^{3}$ | $x^{2} - 2$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e4_3.4t1.2c1 | $1$ | $ 2^{4} \cdot 3 $ | $x^{4} + 12 x^{2} + 18$ | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.2e4_3.4t1.1c1 | $1$ | $ 2^{4} \cdot 3 $ | $x^{4} - 12 x^{2} + 18$ | $C_4$ (as 4T1) | $0$ | $1$ | |
| 1.2e4_3.4t1.1c2 | $1$ | $ 2^{4} \cdot 3 $ | $x^{4} - 12 x^{2} + 18$ | $C_4$ (as 4T1) | $0$ | $1$ | |
| 1.2e4_3.4t1.2c2 | $1$ | $ 2^{4} \cdot 3 $ | $x^{4} + 12 x^{2} + 18$ | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 2.2e8.4t3.2c1 | $2$ | $ 2^{8}$ | $x^{4} + 2$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| * | 2.2e7_3e2.4t3.6c1 | $2$ | $ 2^{7} \cdot 3^{2}$ | $x^{4} - 6 x^{2} + 18$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 4.2e17_3e2.8t21.3c1 | $4$ | $ 2^{17} \cdot 3^{2}$ | $x^{8} - 4 x^{6} + 20 x^{4} - 8 x^{2} + 1$ | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |
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 *.