magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -5, 5, 12, 1, 7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 7*x^6 + x^5 + 12*x^4 + 5*x^3 - 5*x^2 - 2*x + 1)
gp: K = bnfinit(x^8 - x^7 + 7*x^6 + x^5 + 12*x^4 + 5*x^3 - 5*x^2 - 2*x + 1, 1)
Normalized defining polynomial
\( x^{8} - x^{7} + 7 x^{6} + x^{5} + 12 x^{4} + 5 x^{3} - 5 x^{2} - 2 x + 1 \)
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: | \(169130025=3^{4}\cdot 5^{2}\cdot 17^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $10.68$ | 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: | $3, 5, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{86} a^{7} - \frac{19}{86} a^{6} - \frac{19}{43} a^{5} - \frac{3}{86} a^{4} - \frac{10}{43} a^{3} - \frac{11}{43} a^{2} + \frac{2}{43} a - \frac{31}{86}$
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: | $3$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{30}{43} a^{7} + \frac{11}{43} a^{6} - \frac{193}{43} a^{5} - \frac{168}{43} a^{4} - \frac{389}{43} a^{3} - \frac{415}{43} a^{2} + \frac{9}{43} a + \frac{113}{43} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( a \), \( \frac{35}{43} a^{7} - \frac{83}{86} a^{6} + \frac{261}{43} a^{5} - \frac{19}{43} a^{4} + \frac{965}{86} a^{3} + \frac{223}{86} a^{2} - \frac{193}{86} a + \frac{23}{86} \), \( \frac{22}{43} a^{7} - \frac{19}{86} a^{6} + \frac{153}{43} a^{5} + \frac{106}{43} a^{4} + \frac{711}{86} a^{3} + \frac{623}{86} a^{2} + \frac{219}{86} a + \frac{55}{86} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 17.1474065936 \) | 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 10 conjugacy class representatives for $Q_8:C_2$ |
| Character table for $Q_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{17})\) |
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 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| $5$ | 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $17$ | 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.5_17.2t1.1c1 | $1$ | $ 5 \cdot 17 $ | $x^{2} - x - 21$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.3_5_17.2t1.1c1 | $1$ | $ 3 \cdot 5 \cdot 17 $ | $x^{2} - x + 64$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3_5.2t1.1c1 | $1$ | $ 3 \cdot 5 $ | $x^{2} - x + 4$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.17.2t1.1c1 | $1$ | $ 17 $ | $x^{2} - x - 4$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.3_17.2t1.1c1 | $1$ | $ 3 \cdot 17 $ | $x^{2} - x + 13$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 2.3_5_17.8t11.2c1 | $2$ | $ 3 \cdot 5 \cdot 17 $ | $x^{8} - x^{7} + 7 x^{6} + x^{5} + 12 x^{4} + 5 x^{3} - 5 x^{2} - 2 x + 1$ | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
| * | 2.3_5_17.8t11.2c2 | $2$ | $ 3 \cdot 5 \cdot 17 $ | $x^{8} - x^{7} + 7 x^{6} + x^{5} + 12 x^{4} + 5 x^{3} - 5 x^{2} - 2 x + 1$ | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
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 *.