magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 8, 12, -2, -8, -3, 1, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 + x^7 - 3*x^6 - 8*x^5 - 2*x^4 + 12*x^3 + 8*x^2 + x - 1)
gp: K = bnfinit(x^9 + x^7 - 3*x^6 - 8*x^5 - 2*x^4 + 12*x^3 + 8*x^2 + x - 1, 1)
Normalized defining polynomial
\( x^{9} + x^{7} - 3 x^{6} - 8 x^{5} - 2 x^{4} + 12 x^{3} + 8 x^{2} + x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(8356098429=3^{4}\cdot 7^{3}\cdot 67^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.66$ | 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: | $3, 7, 67$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{255} a^{8} - \frac{32}{255} a^{7} + \frac{1}{51} a^{6} + \frac{92}{255} a^{5} + \frac{36}{85} a^{4} + \frac{112}{255} a^{3} - \frac{2}{255} a^{2} + \frac{24}{85} a - \frac{8}{255}$
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: | 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: | \( 36.2116696804 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3.S_3^2$ (as 9T18):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 108 |
| The 11 conjugacy class representatives for $C_3^2 : D_{6} $ |
| Character table for $C_3^2 : D_{6} $ |
Intermediate fields
| 3.1.1407.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 9 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 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 | ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $7$ | 7.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 7.6.3.1 | $x^{6} - 14 x^{4} + 49 x^{2} - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| $67$ | 67.3.0.1 | $x^{3} - x + 16$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 67.6.3.2 | $x^{6} - 4489 x^{2} + 4812208$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3_7_67.2t1.1c1 | $1$ | $ 3 \cdot 7 \cdot 67 $ | $x^{2} - x + 352$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.7_67.2t1.1c1 | $1$ | $ 7 \cdot 67 $ | $x^{2} - x - 117$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.3_7_67.6t3.2c1 | $2$ | $ 3 \cdot 7 \cdot 67 $ | $x^{6} - 19 x^{3} - 27$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| 2.3e2_7_67.6t3.3c1 | $2$ | $ 3^{2} \cdot 7 \cdot 67 $ | $x^{6} - x^{5} + 6 x^{4} - 3 x^{3} + 29 x^{2} - 20 x + 16$ | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
| 2.7_67.3t2.1c1 | $2$ | $ 7 \cdot 67 $ | $x^{3} - x^{2} - 5 x + 4$ | $S_3$ (as 3T2) | $1$ | $2$ | |
| * | 2.3_7_67.3t2.1c1 | $2$ | $ 3 \cdot 7 \cdot 67 $ | $x^{3} - x^{2} - 8 x - 9$ | $S_3$ (as 3T2) | $1$ | $0$ |
| 4.3e2_7e2_67e2.6t9.1c1 | $4$ | $ 3^{2} \cdot 7^{2} \cdot 67^{2}$ | $x^{6} - x^{5} - 6 x^{4} + 56 x^{2} - 98 x + 49$ | $S_3^2$ (as 6T9) | $1$ | $0$ | |
| 6.3e3_7e4_67e4.18t51.1c1 | $6$ | $ 3^{3} \cdot 7^{4} \cdot 67^{4}$ | $x^{9} + x^{7} - 3 x^{6} - 8 x^{5} - 2 x^{4} + 12 x^{3} + 8 x^{2} + x - 1$ | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ | |
| * | 6.3e3_7e2_67e2.9t18.1c1 | $6$ | $ 3^{3} \cdot 7^{2} \cdot 67^{2}$ | $x^{9} + x^{7} - 3 x^{6} - 8 x^{5} - 2 x^{4} + 12 x^{3} + 8 x^{2} + x - 1$ | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ |
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 *.