magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, 8, 0, 0, 0, -4, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 2*x^8 - 4*x^6 + 8*x^2 + 4)
gp: K = bnfinit(x^10 + 2*x^8 - 4*x^6 + 8*x^2 + 4, 1)
Normalized defining polynomial
\( x^{10} + 2 x^{8} - 4 x^{6} + 8 x^{2} + 4 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-316329754624=-\,2^{16}\cdot 13^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.13$ | 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 not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{4}$
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: | $4$ | 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: | \( \frac{1}{2} a^{8} + \frac{1}{2} a^{6} - 3 a^{4} + 2 a^{2} + 3 \), \( a^{9} + \frac{3}{2} a^{7} - \frac{9}{2} a^{5} + 3 a^{3} + 6 a \), \( \frac{3}{4} a^{9} - a^{8} + a^{7} - \frac{3}{2} a^{6} - \frac{7}{2} a^{5} + \frac{9}{2} a^{4} + 3 a^{3} - 3 a^{2} + 4 a - 6 \), \( \frac{9}{2} a^{9} - 3 a^{8} + \frac{11}{2} a^{7} - 4 a^{6} - 22 a^{5} + 15 a^{4} + 18 a^{3} - 9 a^{2} + 22 a - 19 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 76.8693294589 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$((C_2^4 : C_5):C_4)\times C_2$ (as 10T29):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 640 |
| The 22 conjugacy class representatives for $((C_2^4 : C_5):C_4)\times C_2$ |
| Character table for $((C_2^4 : C_5):C_4)\times C_2$ is not computed |
Intermediate fields
| 5.1.35152.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 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 | ${\href{/LocalNumberField/3.10.0.1}{10} }$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.16.10 | $x^{10} - 2 x^{9} + 2 x^{7} + 2 x^{6} - 6 x^{4} - 4 x^{3} - 6 x^{2} - 2$ | $10$ | $1$ | $16$ | $((C_2^4 : C_5):C_4)\times C_2$ | $[2, 12/5, 12/5, 12/5, 12/5]_{5}^{4}$ |
| $13$ | $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |