magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 27, 0, 27, 0, 8, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 8*x^6 + 27*x^4 + 27*x^2 + 9)
gp: K = bnfinit(x^8 + 8*x^6 + 27*x^4 + 27*x^2 + 9, 1)
Normalized defining polynomial
\( x^{8} + 8 x^{6} + 27 x^{4} + 27 x^{2} + 9 \)
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: | \(35070801984=2^{6}\cdot 3^{8}\cdot 17^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.80$ | 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: | $2, 3, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a - \frac{1}{2}$
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: | \( \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + 2 a^{5} + \frac{8}{3} a^{4} + \frac{17}{3} a^{3} + 6 a^{2} + 3 a + 3 \), \( \frac{8}{3} a^{7} + \frac{16}{3} a^{6} + 21 a^{5} + \frac{116}{3} a^{4} + \frac{205}{3} a^{3} + 115 a^{2} + 57 a + 57 \), \( \frac{1}{3} a^{6} + \frac{8}{3} a^{4} + 3 a^{2} + 1 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 485.351335699 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3:A_4$ (as 8T32):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 96 |
| The 11 conjugacy class representatives for $((C_2 \times D_4): C_2):C_3$ |
| Character table for $((C_2 \times D_4): C_2):C_3$ |
Intermediate fields
| 4.0.23409.1 |
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 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 | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 2.6.6.1 | $x^{6} + x^{2} - 1$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.3.4.2 | $x^{3} - 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| 3.3.4.2 | $x^{3} - 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| $17$ | 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.3e2.3t1.1c1 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.3e2.3t1.1c2 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 3.2e6_3e4.4t4.1c1 | $3$ | $ 2^{6} \cdot 3^{4}$ | $x^{4} - 2 x^{3} + 6 x^{2} - 4 x + 2$ | $A_4$ (as 4T4) | $1$ | $-1$ | |
| 3.2e6_3e4_17e2.4t4.3c1 | $3$ | $ 2^{6} \cdot 3^{4} \cdot 17^{2}$ | $x^{4} - 2 x^{3} + 18 x^{2} + 306$ | $A_4$ (as 4T4) | $1$ | $-1$ | |
| 3.2e6_3e4_17e2.4t4.1c1 | $3$ | $ 2^{6} \cdot 3^{4} \cdot 17^{2}$ | $x^{4} - 2 x^{3} - 18 x^{2} + 2 x + 35$ | $A_4$ (as 4T4) | $1$ | $3$ | |
| 3.2e6_3e4_17e2.4t4.2c1 | $3$ | $ 2^{6} \cdot 3^{4} \cdot 17^{2}$ | $x^{4} - 2 x^{3} - 6 x^{2} - 10 x + 59$ | $A_4$ (as 4T4) | $1$ | $-1$ | |
| * | 3.3e4_17e2.4t4.1c1 | $3$ | $ 3^{4} \cdot 17^{2}$ | $x^{4} - x^{3} + 6 x^{2} - 5 x + 8$ | $A_4$ (as 4T4) | $1$ | $-1$ |
| * | 4.2e6_3e4_17e2.8t32.2c1 | $4$ | $ 2^{6} \cdot 3^{4} \cdot 17^{2}$ | $x^{8} + 8 x^{6} + 27 x^{4} + 27 x^{2} + 9$ | $((C_2 \times D_4): C_2):C_3$ (as 8T32) | $1$ | $0$ |
| 4.2e6_3e6_17e2.24t97.2c1 | $4$ | $ 2^{6} \cdot 3^{6} \cdot 17^{2}$ | $x^{8} + 8 x^{6} + 27 x^{4} + 27 x^{2} + 9$ | $((C_2 \times D_4): C_2):C_3$ (as 8T32) | $0$ | $0$ | |
| 4.2e6_3e6_17e2.24t97.2c2 | $4$ | $ 2^{6} \cdot 3^{6} \cdot 17^{2}$ | $x^{8} + 8 x^{6} + 27 x^{4} + 27 x^{2} + 9$ | $((C_2 \times D_4): C_2):C_3$ (as 8T32) | $0$ | $0$ |
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 *.