magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 0, 2, 1, -1, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - x^5 + x^4 + 2*x^3 - 2*x + 1)
gp: K = bnfinit(x^8 - x^7 - x^5 + x^4 + 2*x^3 - 2*x + 1, 1)
Normalized defining polynomial
\( x^{8} - x^{7} - x^{5} + x^{4} + 2 x^{3} - 2 x + 1 \)
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: | \(6543125=5^{4}\cdot 19^{2}\cdot 29\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $7.11$ | 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: | $5, 19, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$
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: | \( \frac{2}{5} a^{7} - \frac{3}{5} a^{6} - \frac{1}{5} a^{5} - \frac{4}{5} a^{4} + \frac{4}{5} a^{3} + \frac{7}{5} a^{2} - \frac{1}{5} a - \frac{6}{5} \), \( a \), \( \frac{4}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{3}{5} a^{4} - \frac{2}{5} a^{3} + \frac{9}{5} a^{2} + \frac{8}{5} a - \frac{7}{5} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1.00719516812 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$D_4^2.C_2$ (as 8T35):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.475.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }$ | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.1.1 | $x^{2} - 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.5_29.2t1.1c1 | $1$ | $ 5 \cdot 29 $ | $x^{2} - x - 36$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.29.2t1.1c1 | $1$ | $ 29 $ | $x^{2} - x - 7$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.5_19_29.2t1.1c1 | $1$ | $ 5 \cdot 19 \cdot 29 $ | $x^{2} - x + 689$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.19.2t1.1c1 | $1$ | $ 19 $ | $x^{2} - x + 5$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.5_19.2t1.1c1 | $1$ | $ 5 \cdot 19 $ | $x^{2} - x + 24$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.19_29.2t1.1c1 | $1$ | $ 19 \cdot 29 $ | $x^{2} - x + 138$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.5_19_29.4t3.1c1 | $2$ | $ 5 \cdot 19 \cdot 29 $ | $x^{4} - x^{3} + 5 x^{2} + 12 x - 36$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.5_19_29.4t3.2c1 | $2$ | $ 5 \cdot 19 \cdot 29 $ | $x^{4} - x^{3} - 13 x - 31$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| * | 2.5_19.4t3.1c1 | $2$ | $ 5 \cdot 19 $ | $x^{4} - 2 x^{3} + 2 x^{2} - x - 1$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| 2.5_19_29e2.4t3.1c1 | $2$ | $ 5 \cdot 19 \cdot 29^{2}$ | $x^{4} - 2 x^{3} + 16 x^{2} - 15 x - 995$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.5_19e2_29.4t3.1c1 | $2$ | $ 5 \cdot 19^{2} \cdot 29 $ | $x^{4} - x^{3} + 30 x^{2} + 54 x + 161$ | $D_{4}$ (as 4T3) | $1$ | $-2$ | |
| 2.5_29.4t3.1c1 | $2$ | $ 5 \cdot 29 $ | $x^{4} - x^{3} - 5 x^{2} - x + 1$ | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| 4.5_19e2_29e2.8t29.2c1 | $4$ | $ 5 \cdot 19^{2} \cdot 29^{2}$ | $x^{8} - 3 x^{7} - x^{6} + 22 x^{5} - 17 x^{4} - 106 x^{3} + 6 x^{2} + 236 x + 161$ | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
| 4.5e2_19e3_29.8t35.3c1 | $4$ | $ 5^{2} \cdot 19^{3} \cdot 29 $ | $x^{8} - x^{7} - x^{5} + x^{4} + 2 x^{3} - 2 x + 1$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $2$ | |
| 4.5e2_19_29e3.8t35.3c1 | $4$ | $ 5^{2} \cdot 19 \cdot 29^{3}$ | $x^{8} - x^{7} - x^{5} + x^{4} + 2 x^{3} - 2 x + 1$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $-2$ | |
| 4.5e3_19e2_29e2.8t29.2c1 | $4$ | $ 5^{3} \cdot 19^{2} \cdot 29^{2}$ | $x^{8} - 3 x^{7} - x^{6} + 22 x^{5} - 17 x^{4} - 106 x^{3} + 6 x^{2} + 236 x + 161$ | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $0$ | |
| 4.5e2_19e3_29e3.8t35.3c1 | $4$ | $ 5^{2} \cdot 19^{3} \cdot 29^{3}$ | $x^{8} - x^{7} - x^{5} + x^{4} + 2 x^{3} - 2 x + 1$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $2$ | |
| * | 4.5e2_19_29.8t35.3c1 | $4$ | $ 5^{2} \cdot 19 \cdot 29 $ | $x^{8} - x^{7} - x^{5} + x^{4} + 2 x^{3} - 2 x + 1$ | $C_2 \wr C_2\wr C_2$ (as 8T35) | $1$ | $-2$ |
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 *.