magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, 84, 112, 96, 30, -16, -12, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 12*x^6 - 16*x^5 + 30*x^4 + 96*x^3 + 112*x^2 + 84*x + 36)
gp: K = bnfinit(x^8 - 12*x^6 - 16*x^5 + 30*x^4 + 96*x^3 + 112*x^2 + 84*x + 36, 1)
Normalized defining polynomial
\( x^{8} - 12 x^{6} - 16 x^{5} + 30 x^{4} + 96 x^{3} + 112 x^{2} + 84 x + 36 \)
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: | \(49113651456=2^{8}\cdot 3^{12}\cdot 19^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $21.70$ | 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: | $2, 3, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{18} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{9} a^{4} + \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{1}{3}$
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: | \( \frac{4}{9} a^{7} - \frac{2}{3} a^{6} - \frac{13}{3} a^{5} - \frac{11}{18} a^{4} + 14 a^{3} + \frac{65}{3} a^{2} + \frac{169}{9} a + \frac{35}{3} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( \frac{11}{18} a^{7} - \frac{1}{6} a^{6} - \frac{47}{6} a^{5} - \frac{131}{18} a^{4} + 27 a^{3} + \frac{167}{3} a^{2} + \frac{265}{9} a - \frac{1}{3} \), \( \frac{1}{2} a^{7} - a^{6} - 5 a^{5} + 4 a^{4} + 17 a^{3} + 7 a^{2} + 5 a + 11 \), \( \frac{73}{18} a^{7} - \frac{29}{6} a^{6} - \frac{271}{6} a^{5} - \frac{44}{9} a^{4} + 142 a^{3} + \frac{601}{3} a^{2} + \frac{1568}{9} a + \frac{289}{3} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 709.721743042 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2:S_4$ (as 8T34):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 96 |
| The 10 conjugacy class representatives for $V_4^2:S_3$ |
| Character table for $V_4^2:S_3$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | 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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.11 | $x^{8} + 20 x^{2} + 4$ | $4$ | $2$ | $8$ | $S_4$ | $[4/3, 4/3]_{3}^{2}$ |
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.11.8 | $x^{6} + 12$ | $6$ | $1$ | $11$ | $S_3$ | $[5/2]_{2}$ | |
| $19$ | $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 2.2e2_3e5.3t2.1c1 | $2$ | $ 2^{2} \cdot 3^{5}$ | $x^{3} - 12$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| 3.2e2_3e6_19e2.6t8.1c1 | $3$ | $ 2^{2} \cdot 3^{6} \cdot 19^{2}$ | $x^{4} - x^{3} + 6 x^{2} - 10 x - 14$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e4_3e5_19e2.4t5.2c1 | $3$ | $ 2^{4} \cdot 3^{5} \cdot 19^{2}$ | $x^{4} - 24 x^{2} - 76 x - 84$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e4_3e6_19e2.6t8.1c1 | $3$ | $ 2^{4} \cdot 3^{6} \cdot 19^{2}$ | $x^{4} - 6 x^{2} - 76 x - 219$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e2_3e5_19e2.4t5.1c1 | $3$ | $ 2^{2} \cdot 3^{5} \cdot 19^{2}$ | $x^{4} - x^{3} + 6 x^{2} - 10 x - 14$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e4_3e6_19e2.6t8.2c1 | $3$ | $ 2^{4} \cdot 3^{6} \cdot 19^{2}$ | $x^{4} - 24 x^{2} - 76 x - 84$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e4_3e5_19e2.4t5.1c1 | $3$ | $ 2^{4} \cdot 3^{5} \cdot 19^{2}$ | $x^{4} - 6 x^{2} - 76 x - 219$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| * | 6.2e8_3e11_19e2.8t34.1c1 | $6$ | $ 2^{8} \cdot 3^{11} \cdot 19^{2}$ | $x^{8} - 12 x^{6} - 16 x^{5} + 30 x^{4} + 96 x^{3} + 112 x^{2} + 84 x + 36$ | $V_4^2:S_3$ (as 8T34) | $1$ | $0$ |
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 *.