magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -3, 0, 4, 0, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + x^6 + 4*x^4 - 3*x^2 + 1)
gp: K = bnfinit(x^8 + x^6 + 4*x^4 - 3*x^2 + 1, 1)
Normalized defining polynomial
\( x^{8} + x^{6} + 4 x^{4} - 3 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(131974144=2^{10}\cdot 359^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $10.35$ | 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: | $2, 359$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a - \frac{1}{4}$
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: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( a^{7} + a^{5} + 4 a^{3} - 3 a \), \( \frac{3}{4} a^{7} + \frac{3}{4} a^{6} + a^{5} + a^{4} + 3 a^{3} + 3 a^{2} - \frac{5}{4} a - \frac{5}{4} \), \( \frac{1}{2} a^{6} + a^{4} + 2 a^{2} - a - \frac{1}{2} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 5.34214964994 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3:S_4$ (as 8T39):
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 $C_2^3:S_4$ |
| Character table for $C_2^3:S_4$ |
Intermediate fields
| 4.0.1436.1 |
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 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.6.9 | $x^{4} + 2 x^{3} + 6$ | $4$ | $1$ | $6$ | $D_{4}$ | $[2, 2]^{2}$ |
| 2.4.4.2 | $x^{4} - x^{2} + 5$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 359 | Data not computed | ||||||
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_359.2t1.1c1 | $1$ | $ 2^{2} \cdot 359 $ | $x^{2} - 359$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.2e2_359.3t2.1c1 | $2$ | $ 2^{2} \cdot 359 $ | $x^{3} - 11 x - 12$ | $S_3$ (as 3T2) | $1$ | $2$ | |
| 3.2e4_359.4t5.1c1 | $3$ | $ 2^{4} \cdot 359 $ | $x^{4} - 5 x^{2} - 2 x + 1$ | $S_4$ (as 4T5) | $1$ | $3$ | |
| 3.2e6_359e2.6t8.2c1 | $3$ | $ 2^{6} \cdot 359^{2}$ | $x^{4} + x^{2} - 4 x + 3$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 3.2e2_359.4t5.1c1 | $3$ | $ 2^{2} \cdot 359 $ | $x^{4} - x^{3} + 3 x^{2} + 2$ | $S_4$ (as 4T5) | $1$ | $-1$ |
| 3.2e4_359e2.6t8.1c1 | $3$ | $ 2^{4} \cdot 359^{2}$ | $x^{4} - x^{3} + 3 x^{2} + 2$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e4_359.4t5.2c1 | $3$ | $ 2^{4} \cdot 359 $ | $x^{4} + x^{2} - 4 x + 3$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e6_359e2.6t8.1c1 | $3$ | $ 2^{6} \cdot 359^{2}$ | $x^{4} - 5 x^{2} - 2 x + 1$ | $S_4$ (as 4T5) | $1$ | $3$ | |
| * | 4.2e8_359.8t39.2c1 | $4$ | $ 2^{8} \cdot 359 $ | $x^{8} + x^{6} + 4 x^{4} - 3 x^{2} + 1$ | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
| 4.2e4_359e3.8t39.2c1 | $4$ | $ 2^{4} \cdot 359^{3}$ | $x^{8} + x^{6} + 4 x^{4} - 3 x^{2} + 1$ | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ | |
| 6.2e10_359e3.8t34.1c1 | $6$ | $ 2^{10} \cdot 359^{3}$ | $x^{8} + 8 x^{6} - 24 x^{5} + 118 x^{4} - 96 x^{3} + 552 x^{2} - 1224 x + 1165$ | $V_4^2:S_3$ (as 8T34) | $1$ | $-2$ | |
| 8.2e12_359e4.24t333.3c1 | $8$ | $ 2^{12} \cdot 359^{4}$ | $x^{8} + x^{6} + 4 x^{4} - 3 x^{2} + 1$ | $C_2^3:S_4$ (as 8T39) | $1$ | $0$ |
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 *.