magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -8, -8, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 8*x^3 - 8*x^2 + 1)
gp: K = bnfinit(x^8 - 8*x^3 - 8*x^2 + 1, 1)
Normalized defining polynomial
\( x^{8} - 8 x^{3} - 8 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1644167168=2^{25}\cdot 7^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.19$ | 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: | $2, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{17} a^{7} - \frac{2}{17} a^{6} + \frac{4}{17} a^{5} - \frac{8}{17} a^{4} - \frac{1}{17} a^{3} - \frac{6}{17} a^{2} + \frac{4}{17} a - \frac{8}{17}$
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: | $5$ | 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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 71.5321173773 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$A_4^2:C_4$ (as 8T46):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 576 |
| The 13 conjugacy class representatives for $A_4^2:C_4$ |
| Character table for $A_4^2:C_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
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 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 36 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 | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | ${\href{/LocalNumberField/5.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.25.2 | $x^{8} + 10 x^{4} + 20 x^{2} + 2$ | $8$ | $1$ | $25$ | $C_2^3: C_4$ | $[2, 3, 7/2, 4, 17/4]$ |
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.3.2.1 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.2e3.2t1.1c1 | $1$ | $ 2^{3}$ | $x^{2} - 2$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.2e4.4t1.2c1 | $1$ | $ 2^{4}$ | $x^{4} + 4 x^{2} + 2$ | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 1.2e4.4t1.2c2 | $1$ | $ 2^{4}$ | $x^{4} + 4 x^{2} + 2$ | $C_4$ (as 4T1) | $0$ | $-1$ | |
| 4.2e11_7e2.6t10.2c1 | $4$ | $ 2^{11} \cdot 7^{2}$ | $x^{6} - 2 x^{5} + x^{4} - 4 x^{3} - 11 x^{2} + 10 x + 41$ | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ | |
| 4.2e11_7e4.6t10.2c1 | $4$ | $ 2^{11} \cdot 7^{4}$ | $x^{6} - 2 x^{5} + x^{4} - 4 x^{3} - 11 x^{2} + 10 x + 41$ | $C_3^2:C_4$ (as 6T10) | $1$ | $0$ | |
| 6.2e25_7e2.12t160.2c1 | $6$ | $ 2^{25} \cdot 7^{2}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $1$ | $-2$ | |
| * | 6.2e22_7e2.8t46.1c1 | $6$ | $ 2^{22} \cdot 7^{2}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $1$ | $2$ |
| 9.2e32_7e6.16t1030.1c1 | $9$ | $ 2^{32} \cdot 7^{6}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $1$ | $1$ | |
| 9.2e35_7e6.18t184.2c1 | $9$ | $ 2^{35} \cdot 7^{6}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $1$ | $1$ | |
| 9.2e36_7e6.36t766.1c1 | $9$ | $ 2^{36} \cdot 7^{6}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $0$ | $-1$ | |
| 9.2e36_7e6.36t766.1c2 | $9$ | $ 2^{36} \cdot 7^{6}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $0$ | $-1$ | |
| 12.2e47_7e10.24t1506.1c1 | $12$ | $ 2^{47} \cdot 7^{10}$ | $x^{8} - 8 x^{3} - 8 x^{2} + 1$ | $A_4^2:C_4$ (as 8T46) | $1$ | $0$ |
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 *.