magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, -117, -162, -105, -9, 14, 2, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 + 2*x^6 + 14*x^5 - 9*x^4 - 105*x^3 - 162*x^2 - 117*x - 27)
gp: K = bnfinit(x^8 - 2*x^7 + 2*x^6 + 14*x^5 - 9*x^4 - 105*x^3 - 162*x^2 - 117*x - 27, 1)
Normalized defining polynomial
\( x^{8} - 2 x^{7} + 2 x^{6} + 14 x^{5} - 9 x^{4} - 105 x^{3} - 162 x^{2} - 117 x - 27 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1391050107747=-\,3^{3}\cdot 61^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $32.95$ | 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: | $3, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{981} a^{7} + \frac{53}{981} a^{6} - \frac{15}{109} a^{5} + \frac{328}{981} a^{4} + \frac{155}{981} a^{3} - \frac{100}{327} a^{2} + \frac{5}{327} a + \frac{6}{109}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}$, which has order $3$
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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1607.44327914 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 16 |
| The 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{61}) \), 4.2.11163.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }$ | ${\href{/LocalNumberField/29.8.0.1}{8} }$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }$ | ${\href{/LocalNumberField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $61$ | 61.4.3.1 | $x^{4} - 61$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 61.4.3.1 | $x^{4} - 61$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.61.2t1.1c1 | $1$ | $ 61 $ | $x^{2} - x - 15$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3_61.2t1.1c1 | $1$ | $ 3 \cdot 61 $ | $x^{2} - x + 46$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.3_61.4t3.1c1 | $2$ | $ 3 \cdot 61 $ | $x^{4} - 7 x^{2} - 3$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 2.3_61e2.8t8.1c1 | $2$ | $ 3 \cdot 61^{2}$ | $x^{8} - 2 x^{7} + 2 x^{6} + 14 x^{5} - 9 x^{4} - 105 x^{3} - 162 x^{2} - 117 x - 27$ | $QD_{16}$ (as 8T8) | $0$ | $0$ |
| * | 2.3_61e2.8t8.1c2 | $2$ | $ 3 \cdot 61^{2}$ | $x^{8} - 2 x^{7} + 2 x^{6} + 14 x^{5} - 9 x^{4} - 105 x^{3} - 162 x^{2} - 117 x - 27$ | $QD_{16}$ (as 8T8) | $0$ | $0$ |
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 *.