magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 0, -27, -9, 9, -3, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^6 + 9*x^5 - 9*x^4 - 27*x^3 + 9*x + 1)
gp: K = bnfinit(x^9 - 3*x^6 + 9*x^5 - 9*x^4 - 27*x^3 + 9*x + 1, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[5, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(24794911296=2^{6}\cdot 3^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.29$ | 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: | $2, 3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5923} a^{8} - \frac{109}{5923} a^{7} + \frac{35}{5923} a^{6} + \frac{2105}{5923} a^{5} + \frac{1561}{5923} a^{4} + \frac{1609}{5923} a^{3} + \frac{2282}{5923} a^{2} + \frac{28}{5923} a + \frac{2880}{5923}$
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: | $6$ | 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: | \( 87.778393065 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3^3:C_2^2:C_3$ (as 9T25):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 324 |
| The 13 conjugacy class representatives for $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ |
| Character table for $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\) |
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 18 siblings: | data not computed |
| Degree 27 siblings: | 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 | R | ${\href{/LocalNumberField/5.9.0.1}{9} }$ | ${\href{/LocalNumberField/7.9.0.1}{9} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }$ | ${\href{/LocalNumberField/13.9.0.1}{9} }$ | ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }$ | ${\href{/LocalNumberField/31.9.0.1}{9} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }$ | ${\href{/LocalNumberField/43.9.0.1}{9} }$ | ${\href{/LocalNumberField/47.9.0.1}{9} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.6.1 | $x^{6} + x^{2} - 1$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
| $3$ | 3.9.18.13 | $x^{9} + 6 x^{6} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3 \wr C_3 $ | $[2, 2, 7/3]^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.3e2.3t1.1c1 | $1$ | $ 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) | $1$ | $-1$ | |
| 4.2e6_3e9.12t132.1c1 | $4$ | $ 2^{6} \cdot 3^{9}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $0$ | $0$ | |
| 4.2e6_3e9.12t132.1c2 | $4$ | $ 2^{6} \cdot 3^{9}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $0$ | $0$ | |
| 4.2e6_3e9.12t132.2c1 | $4$ | $ 2^{6} \cdot 3^{9}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $0$ | $0$ | |
| 4.2e6_3e7.12t133.1c1 | $4$ | $ 2^{6} \cdot 3^{7}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $0$ | $0$ | |
| 4.2e6_3e7.12t133.1c2 | $4$ | $ 2^{6} \cdot 3^{7}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $0$ | $0$ | |
| 4.2e6_3e9.12t132.2c2 | $4$ | $ 2^{6} \cdot 3^{9}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $0$ | $0$ | |
| * | 6.2e6_3e14.9t25.1c1 | $6$ | $ 2^{6} \cdot 3^{14}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $1$ | $2$ |
| 6.2e12_3e14.18t141.1c1 | $6$ | $ 2^{12} \cdot 3^{14}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $1$ | $-2$ | |
| 12.2e18_3e26.18t142.1c1 | $12$ | $ 2^{18} \cdot 3^{26}$ | $x^{9} - 3 x^{6} + 9 x^{5} - 9 x^{4} - 27 x^{3} + 9 x + 1$ | $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ (as 9T25) | $1$ | $0$ |
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 *.