magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13924, 0, 3984, 0, 245, 0, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 6*x^6 + 245*x^4 + 3984*x^2 + 13924)
gp: K = bnfinit(x^8 - 6*x^6 + 245*x^4 + 3984*x^2 + 13924, 1)
Normalized defining polynomial
\( x^{8} - 6 x^{6} + 245 x^{4} + 3984 x^{2} + 13924 \)
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: | \(484654236938496=2^{8}\cdot 3^{4}\cdot 17^{4}\cdot 23^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $68.50$ | 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: | $2, 3, 17, 23$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4692=2^{2}\cdot 3\cdot 17\cdot 23\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4692}(2209,·)$, $\chi_{4692}(1,·)$, $\chi_{4692}(3265,·)$, $\chi_{4692}(3911,·)$, $\chi_{4692}(1427,·)$, $\chi_{4692}(781,·)$, $\chi_{4692}(4691,·)$, $\chi_{4692}(2483,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{20} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{40} a^{5} - \frac{1}{8} a^{3} - \frac{7}{20} a - \frac{1}{2}$, $\frac{1}{1400} a^{6} + \frac{1}{200} a^{4} + \frac{49}{100} a^{2} - \frac{1}{2} a - \frac{6}{175}$, $\frac{1}{82600} a^{7} + \frac{2}{1475} a^{5} - \frac{3977}{11800} a^{3} - \frac{1}{2} a^{2} - \frac{9159}{41300} a - \frac{1}{2}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{6}\times C_{84}$, which has order $1008$
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: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{33}{41300} a^{7} - \frac{31}{2950} a^{5} + \frac{1509}{5900} a^{3} + \frac{36413}{20650} a + 2 \), \( \frac{3}{1475} a^{7} - \frac{131}{5900} a^{5} + \frac{3707}{5900} a^{3} + \frac{12517}{2950} a + 4 \), \( \frac{69}{11800} a^{7} - \frac{1}{1400} a^{6} - \frac{827}{11800} a^{5} - \frac{1}{200} a^{4} + \frac{5421}{2950} a^{3} + \frac{1}{100} a^{2} + \frac{18491}{1475} a - \frac{1219}{175} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 203.260298886 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
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 $C_2^3$ |
| Character table for $C_2^3$ |
Intermediate fields
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 | R | ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $17$ | 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $23$ | 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |