magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![50, -80, 8, 16, 22, -12, -4, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^6 - 12*x^5 + 22*x^4 + 16*x^3 + 8*x^2 - 80*x + 50)
gp: K = bnfinit(x^8 - 4*x^6 - 12*x^5 + 22*x^4 + 16*x^3 + 8*x^2 - 80*x + 50, 1)
Normalized defining polynomial
\( x^{8} - 4 x^{6} - 12 x^{5} + 22 x^{4} + 16 x^{3} + 8 x^{2} - 80 x + 50 \)
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: | \(462422016=2^{20}\cdot 3^{2}\cdot 7^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.11$ | 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: | $2, 3, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{15} a^{6} - \frac{2}{15} a^{5} + \frac{1}{3} a^{4} - \frac{2}{15} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{3}$, $\frac{1}{345} a^{7} + \frac{4}{345} a^{6} - \frac{172}{345} a^{5} + \frac{13}{345} a^{4} - \frac{52}{115} a^{3} + \frac{7}{23} a^{2} + \frac{14}{345} a + \frac{3}{23}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}$, which has order $2$
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: | \( -\frac{7}{345} a^{7} + \frac{6}{115} a^{6} + \frac{77}{345} a^{5} + \frac{139}{345} a^{4} - \frac{76}{69} a^{3} - \frac{153}{115} a^{2} - \frac{374}{345} a + \frac{259}{69} \) (order $8$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{73}{345} a^{7} + \frac{36}{115} a^{6} - \frac{113}{345} a^{5} - \frac{1006}{345} a^{4} + \frac{4}{69} a^{3} + \frac{347}{115} a^{2} + \frac{2126}{345} a - \frac{493}{69} \), \( a - 1 \), \( \frac{4}{115} a^{7} + \frac{16}{115} a^{6} + \frac{2}{115} a^{5} - \frac{63}{115} a^{4} - \frac{49}{115} a^{3} + \frac{38}{23} a^{2} + \frac{56}{115} a - \frac{33}{23} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 34.4656575278 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$Q_8:C_2^2$ (as 8T22):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 32 |
| The 17 conjugacy class representatives for $C_2^3 : D_4 $ |
| Character table for $C_2^3 : D_4 $ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{8})\) |
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 | R | R | ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.20.55 | $x^{8} + 4 x^{6} + 4 x^{5} + 6 x^{4} + 2$ | $8$ | $1$ | $20$ | $Q_8:C_2$ | $[2, 3, 3]^{2}$ |
| $3$ | 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 3.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.2e3.2t1.2c1 | $1$ | $ 2^{3}$ | $x^{2} + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.2e3_7.2t1.1c1 | $1$ | $ 2^{3} \cdot 7 $ | $x^{2} - 14$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.7.2t1.1c1 | $1$ | $ 7 $ | $x^{2} - x + 2$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e2_3_7.2t1.1c1 | $1$ | $ 2^{2} \cdot 3 \cdot 7 $ | $x^{2} + 21$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e3_3_7.2t1.1c1 | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | $x^{2} - 42$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e3_3.2t1.2c1 | $1$ | $ 2^{3} \cdot 3 $ | $x^{2} + 6$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e2_3.2t1.1c1 | $1$ | $ 2^{2} \cdot 3 $ | $x^{2} - 3$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e3_3_7.2t1.2c1 | $1$ | $ 2^{3} \cdot 3 \cdot 7 $ | $x^{2} + 42$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3_7.2t1.1c1 | $1$ | $ 3 \cdot 7 $ | $x^{2} - x - 5$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e3_3.2t1.1c1 | $1$ | $ 2^{3} \cdot 3 $ | $x^{2} - 6$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.2e3.2t1.1c1 | $1$ | $ 2^{3}$ | $x^{2} - 2$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.2e2_7.2t1.1c1 | $1$ | $ 2^{2} \cdot 7 $ | $x^{2} - 7$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e3_7.2t1.2c1 | $1$ | $ 2^{3} \cdot 7 $ | $x^{2} + 14$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 4.2e12_3e2_7e2.8t22.5c1 | $4$ | $ 2^{12} \cdot 3^{2} \cdot 7^{2}$ | $x^{8} - 4 x^{6} - 12 x^{5} + 22 x^{4} + 16 x^{3} + 8 x^{2} - 80 x + 50$ | $C_2^3 : D_4 $ (as 8T22) | $1$ | $0$ |
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 *.