magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, -23, 47, -50, 31, -10, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 8*x^6 - 10*x^5 + 31*x^4 - 50*x^3 + 47*x^2 - 23*x + 6)
gp: K = bnfinit(x^8 - 4*x^7 + 8*x^6 - 10*x^5 + 31*x^4 - 50*x^3 + 47*x^2 - 23*x + 6, 1)
Normalized defining polynomial
\( x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 31 x^{4} - 50 x^{3} + 47 x^{2} - 23 x + 6 \)
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: | \(1625943765625=5^{6}\cdot 101^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $33.60$ | 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: | $5, 101$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{21} a^{7} + \frac{1}{3} a^{6} + \frac{1}{21} a^{5} + \frac{1}{21} a^{4} - \frac{8}{21} a^{2} + \frac{1}{21} a + \frac{3}{7}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{8}$, which has order $16$
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: | \( \frac{5}{7} a^{7} - 2 a^{6} + \frac{19}{7} a^{5} - \frac{16}{7} a^{4} + 18 a^{3} - \frac{103}{7} a^{2} + \frac{19}{7} a + \frac{31}{7} \), \( \frac{188}{21} a^{7} - \frac{88}{3} a^{6} + \frac{1007}{21} a^{5} - \frac{1009}{21} a^{4} + 233 a^{3} - \frac{5683}{21} a^{2} + \frac{3464}{21} a - \frac{297}{7} \), \( \frac{2}{21} a^{7} - \frac{1}{3} a^{6} + \frac{23}{21} a^{5} - \frac{19}{21} a^{4} + \frac{5}{21} a^{2} + \frac{2}{21} a - \frac{1}{7} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 526.671257635 \) | 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
| \(\Q(\sqrt{505}) \), 4.0.1275125.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $101$ | 101.4.2.1 | $x^{4} + 505 x^{2} + 91809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 101.4.2.1 | $x^{4} + 505 x^{2} + 91809$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
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 *.