magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, -21, 23, -20, 20, -12, 5, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 5*x^6 - 12*x^5 + 20*x^4 - 20*x^3 + 23*x^2 - 21*x + 11)
gp: K = bnfinit(x^8 - 3*x^7 + 5*x^6 - 12*x^5 + 20*x^4 - 20*x^3 + 23*x^2 - 21*x + 11, 1)
Normalized defining polynomial
\( x^{8} - 3 x^{7} + 5 x^{6} - 12 x^{5} + 20 x^{4} - 20 x^{3} + 23 x^{2} - 21 x + 11 \)
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: | \(433680625=5^{4}\cdot 7^{4}\cdot 17^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.01$ | 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: | $5, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$
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: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{2} a^{4} - \frac{3}{2} a^{3} + \frac{3}{2} a^{2} - \frac{1}{2} a + \frac{1}{4} \), \( \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + a^{5} - \frac{5}{2} a^{4} + 3 a^{3} - 3 a^{2} + \frac{13}{4} a - \frac{5}{2} \), \( \frac{1}{2} a^{3} - a^{2} + a - \frac{1}{2} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 18.965077743 \) | 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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}, \sqrt{-7})\) |
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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| $17$ | 17.4.2.2 | $x^{4} - 17 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.7.2t1.1c1 | $1$ | $ 7 $ | $x^{2} - x + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.5_17.2t1.1c1 | $1$ | $ 5 \cdot 17 $ | $x^{2} - x - 21$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.5_7_17.2t1.1c1 | $1$ | $ 5 \cdot 7 \cdot 17 $ | $x^{2} - x + 149$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.5_7.2t1.1c1 | $1$ | $ 5 \cdot 7 $ | $x^{2} - x + 9$ | $C_2$ (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) | $1$ | $1$ |
| 1.7_17.2t1.1c1 | $1$ | $ 7 \cdot 17 $ | $x^{2} - x + 30$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.5_7_17.8t11.1c1 | $2$ | $ 5 \cdot 7 \cdot 17 $ | $x^{8} - 3 x^{7} + 5 x^{6} - 12 x^{5} + 20 x^{4} - 20 x^{3} + 23 x^{2} - 21 x + 11$ | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
| * | 2.5_7_17.8t11.1c2 | $2$ | $ 5 \cdot 7 \cdot 17 $ | $x^{8} - 3 x^{7} + 5 x^{6} - 12 x^{5} + 20 x^{4} - 20 x^{3} + 23 x^{2} - 21 x + 11$ | $Q_8:C_2$ (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 *.