magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12996, 0, 2492, 0, 669, 0, 42, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 42*x^6 + 669*x^4 + 2492*x^2 + 12996)
gp: K = bnfinit(x^8 + 42*x^6 + 669*x^4 + 2492*x^2 + 12996, 1)
Normalized defining polynomial
\( x^{8} + 42 x^{6} + 669 x^{4} + 2492 x^{2} + 12996 \)
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: | \(444638964023296=2^{16}\cdot 7^{4}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $67.76$ | 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: | $2, 7, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2296=2^{3}\cdot 7\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2296}(1,·)$, $\chi_{2296}(1723,·)$, $\chi_{2296}(2213,·)$, $\chi_{2296}(1639,·)$, $\chi_{2296}(1805,·)$, $\chi_{2296}(1231,·)$, $\chi_{2296}(1721,·)$, $\chi_{2296}(1147,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{4} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{3} - \frac{5}{12} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{7920} a^{6} + \frac{28}{495} a^{4} - \frac{1}{6} a^{3} - \frac{3563}{7920} a^{2} + \frac{1}{6} a + \frac{73}{440}$, $\frac{1}{150480} a^{7} - \frac{769}{18810} a^{5} - \frac{15443}{150480} a^{3} + \frac{8579}{25080} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{4}\times C_{168}$, which has order $1344$
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: | \( \frac{1}{25080} a^{7} - \frac{1}{1584} a^{6} + \frac{59}{12540} a^{5} - \frac{13}{396} a^{4} + \frac{3367}{25080} a^{3} - \frac{793}{1584} a^{2} + \frac{4399}{4180} a - \frac{117}{88} \), \( \frac{1}{360} a^{6} + \frac{7}{90} a^{4} + \frac{97}{360} a^{2} - \frac{327}{20} \), \( \frac{1}{3135} a^{7} - \frac{4}{495} a^{6} + \frac{118}{3135} a^{5} - \frac{142}{495} a^{4} + \frac{3367}{3135} a^{3} - \frac{2248}{495} a^{2} + \frac{8798}{1045} a - \frac{639}{55} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 451.659027468 \) | 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 | ${\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ |
| 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| $7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |