magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 4, 0, 18, 0, -3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^6 + 18*x^4 + 4*x^2 + 9)
gp: K = bnfinit(x^8 - 3*x^6 + 18*x^4 + 4*x^2 + 9, 1)
Normalized defining polynomial
\( x^{8} - 3 x^{6} + 18 x^{4} + 4 x^{2} + 9 \)
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: | \(1235663104=2^{8}\cdot 13^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.69$ | 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: | $2, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(52=2^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{52}(1,·)$, $\chi_{52}(5,·)$, $\chi_{52}(47,·)$, $\chi_{52}(51,·)$, $\chi_{52}(21,·)$, $\chi_{52}(25,·)$, $\chi_{52}(27,·)$, $\chi_{52}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{27} a^{6} + \frac{2}{27} a^{4} + \frac{1}{27} a^{2} + \frac{1}{3}$, $\frac{1}{27} a^{7} + \frac{2}{27} a^{5} + \frac{1}{27} a^{3} + \frac{1}{3} a$
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: | $3$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{2}{27} a^{7} - \frac{5}{27} a^{5} + \frac{29}{27} a^{3} + a \) (order $4$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{2}{27} a^{6} - \frac{5}{27} a^{4} + \frac{20}{27} a^{2} + a + \frac{2}{3} \), \( \frac{2}{27} a^{7} - \frac{1}{27} a^{6} - \frac{5}{27} a^{5} + \frac{7}{27} a^{4} + \frac{38}{27} a^{3} - \frac{19}{27} a^{2} + \frac{1}{3} a + \frac{2}{3} \), \( \frac{1}{27} a^{6} - \frac{7}{27} a^{4} + \frac{1}{3} a^{3} + \frac{19}{27} a^{2} - \frac{2}{3} a + \frac{1}{3} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 35.7224800402 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_4$ (as 8T2):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| An abelian group of order 8 |
| The 8 conjugacy class representatives for $C_4\times C_2$ |
| Character table for $C_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(i, \sqrt{13})\), 4.4.35152.1, 4.0.2197.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| $13$ | 13.4.3.2 | $x^{4} - 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.4.3.2 | $x^{4} - 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |