magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, -4, 0, 12, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 12*x^4 - 4*x^2 - 1)
gp: K = bnfinit(x^8 + 12*x^4 - 4*x^2 - 1, 1)
Normalized defining polynomial
\( x^{8} + 12 x^{4} - 4 x^{2} - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-959512576=-\,2^{16}\cdot 11^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.27$ | 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: | $2, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a + \frac{1}{4}$
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: | $4$ | 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: | \( a \), \( \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{13}{4} a^{3} - \frac{13}{4} a^{2} + \frac{9}{4} a - \frac{5}{4} \), \( a^{7} + \frac{1}{4} a^{6} + \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + 12 a^{3} + \frac{13}{4} a^{2} + \frac{5}{2} a + \frac{1}{4} \), \( a^{7} - \frac{1}{4} a^{6} + \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + 12 a^{3} - \frac{13}{4} a^{2} + \frac{5}{2} a - \frac{1}{4} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 96.1320913675 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3:A_4:C_2$ (as 8T40):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 192 |
| The 13 conjugacy class representatives for $Q_8:S_4$ |
| Character table for $Q_8:S_4$ |
Intermediate fields
| 4.2.7744.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.8.0.1}{8} }$ | ${\href{/LocalNumberField/23.8.0.1}{8} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.4.10.1 | $x^{4} + 2 x^{2} - 9$ | $4$ | $1$ | $10$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2.2t1.1c1 | $1$ | $ 2^{2}$ | $x^{2} + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.2e2_11e2.3t2.1c1 | $2$ | $ 2^{2} \cdot 11^{2}$ | $x^{3} - x^{2} + 4 x + 2$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| 3.2e8_11e2.6t8.2c1 | $3$ | $ 2^{8} \cdot 11^{2}$ | $x^{4} - 2 x^{3} - 3 x^{2} + 2$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e10_11e2.4t5.1c1 | $3$ | $ 2^{10} \cdot 11^{2}$ | $x^{4} - 6 x^{2} - 8 x - 25$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e10_11e2.6t8.2c1 | $3$ | $ 2^{10} \cdot 11^{2}$ | $x^{4} - 8 x^{2} - 16 x + 24$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e10_11e2.4t5.2c1 | $3$ | $ 2^{10} \cdot 11^{2}$ | $x^{4} - 8 x^{2} - 16 x + 24$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e10_11e2.6t8.1c1 | $3$ | $ 2^{10} \cdot 11^{2}$ | $x^{4} - 6 x^{2} - 8 x - 25$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 3.2e6_11e2.4t5.1c1 | $3$ | $ 2^{6} \cdot 11^{2}$ | $x^{4} - 2 x^{3} - 3 x^{2} + 2$ | $S_4$ (as 4T5) | $1$ | $1$ |
| 4.2e12_11e2.8t40.3c1 | $4$ | $ 2^{12} \cdot 11^{2}$ | $x^{8} + 12 x^{4} - 4 x^{2} - 1$ | $Q_8:S_4$ (as 8T40) | $1$ | $0$ | |
| * | 4.2e10_11e2.8t40.3c1 | $4$ | $ 2^{10} \cdot 11^{2}$ | $x^{8} + 12 x^{4} - 4 x^{2} - 1$ | $Q_8:S_4$ (as 8T40) | $1$ | $0$ |
| 6.2e16_11e4.8t34.1c1 | $6$ | $ 2^{16} \cdot 11^{4}$ | $x^{8} - 4 x^{7} + 4 x^{6} + 5 x^{4} - 16 x^{3} + 28 x^{2} - 16 x + 8$ | $V_4^2:S_3$ (as 8T34) | $1$ | $0$ | |
| 8.2e22_11e6.24t332.3c1 | $8$ | $ 2^{22} \cdot 11^{6}$ | $x^{8} + 12 x^{4} - 4 x^{2} - 1$ | $Q_8:S_4$ (as 8T40) | $1$ | $0$ |
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 *.