magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, -28, 16, -4, 4, -7, 4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 4*x^6 - 7*x^5 + 4*x^4 - 4*x^3 + 16*x^2 - 28*x + 16)
gp: K = bnfinit(x^8 - x^7 + 4*x^6 - 7*x^5 + 4*x^4 - 4*x^3 + 16*x^2 - 28*x + 16, 1)
Normalized defining polynomial
\( x^{8} - x^{7} + 4 x^{6} - 7 x^{5} + 4 x^{4} - 4 x^{3} + 16 x^{2} - 28 x + 16 \)
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: | \(6561000000=2^{6}\cdot 3^{8}\cdot 5^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.87$ | 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: | $2, 3, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{36} a^{7} + \frac{1}{36} a^{6} + \frac{1}{6} a^{5} + \frac{5}{36} a^{4} + \frac{7}{18} a^{3} - \frac{1}{3} a^{2} - \frac{2}{9} a - \frac{2}{9}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{4}$, which has order $4$
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: | \( a^{7} + 4 a^{5} - 3 a^{4} + a^{3} - 3 a^{2} + 13 a - 15 \), \( \frac{1}{3} a^{7} - \frac{1}{12} a^{6} + \frac{5}{4} a^{5} - \frac{4}{3} a^{4} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} + \frac{23}{6} a - \frac{13}{3} \), \( \frac{2}{3} a^{7} + \frac{1}{12} a^{6} + \frac{11}{4} a^{5} - \frac{5}{3} a^{4} + \frac{11}{12} a^{3} - \frac{5}{2} a^{2} + \frac{55}{6} a - \frac{29}{3} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 54.7187652998 \) | 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
| \(\Q(\sqrt{-15}) \) |
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 | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.6.7.1 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
| $5$ | 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
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_5.2t1.1c1 | $1$ | $ 3 \cdot 5 $ | $x^{2} - x + 4$ | $C_2$ (as 2T1) | $1$ | $-1$ |
| 2.3e3_5.3t2.1c1 | $2$ | $ 3^{3} \cdot 5 $ | $x^{3} + 3 x - 1$ | $S_3$ (as 3T2) | $1$ | $0$ | |
| 3.2e6_3e3_5e3.4t5.1c1 | $3$ | $ 2^{6} \cdot 3^{3} \cdot 5^{3}$ | $x^{4} - 2 x^{3} - 6 x^{2} - 8 x - 14$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e6_3e3_5.4t5.1c1 | $3$ | $ 2^{6} \cdot 3^{3} \cdot 5 $ | $x^{4} - 2 x^{3} + 6 x^{2} - 6$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e6_3e3_5e3.4t5.2c1 | $3$ | $ 2^{6} \cdot 3^{3} \cdot 5^{3}$ | $x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 11$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| 3.2e6_3e4_5e2.6t8.3c1 | $3$ | $ 2^{6} \cdot 3^{4} \cdot 5^{2}$ | $x^{4} - 2 x^{3} - 6 x^{2} - 8 x - 14$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e6_3e4_5e2.6t8.9c1 | $3$ | $ 2^{6} \cdot 3^{4} \cdot 5^{2}$ | $x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 11$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| 3.2e6_3e4_5e2.6t8.4c1 | $3$ | $ 2^{6} \cdot 3^{4} \cdot 5^{2}$ | $x^{4} - 2 x^{3} + 6 x^{2} - 6$ | $S_4$ (as 4T5) | $1$ | $-1$ | |
| * | 6.2e6_3e7_5e5.8t34.1c1 | $6$ | $ 2^{6} \cdot 3^{7} \cdot 5^{5}$ | $x^{8} - x^{7} + 4 x^{6} - 7 x^{5} + 4 x^{4} - 4 x^{3} + 16 x^{2} - 28 x + 16$ | $V_4^2:S_3$ (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 *.