magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 48, 24, -18, -18, 24, 0, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 24*x^6 - 18*x^5 - 18*x^4 + 24*x^3 + 48*x^2 - 3*x + 1)
gp: K = bnfinit(x^9 - 3*x^8 + 24*x^6 - 18*x^5 - 18*x^4 + 24*x^3 + 48*x^2 - 3*x + 1, 1)
Normalized defining polynomial
\( x^{9} - 3 x^{8} + 24 x^{6} - 18 x^{5} - 18 x^{4} + 24 x^{3} + 48 x^{2} - 3 x + 1 \)
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: | \(2641807540224=2^{27}\cdot 3^{9}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.00$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{3}{8} a - \frac{3}{8}$, $\frac{1}{1256} a^{8} - \frac{59}{1256} a^{7} + \frac{7}{1256} a^{6} + \frac{103}{1256} a^{5} - \frac{291}{1256} a^{4} + \frac{421}{1256} a^{3} + \frac{155}{1256} a^{2} + \frac{3}{1256} a + \frac{75}{314}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}$, which has order $3$
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: | \( 689.945765083 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$PSU(3,2):C_2$ (as 9T19):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 144 |
| The 9 conjugacy class representatives for $(C_3^2:C_8):C_2$ |
| Character table for $(C_3^2:C_8):C_2$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.8.27.113 | $x^{8} + 8 x^{5} + 10 x^{4} + 22$ | $8$ | $1$ | $27$ | $QD_{16}$ | $[2, 3, 7/2, 9/2]$ | |
| $3$ | 3.9.9.12 | $x^{9} + 6 x + 6$ | $9$ | $1$ | $9$ | $(C_3^2:C_8):C_2$ | $[9/8, 9/8]_{8}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2_3.2t1.1c1 | $1$ | $ 2^{2} \cdot 3 $ | $x^{2} - 3$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e3.2t1.1c1 | $1$ | $ 2^{3}$ | $x^{2} - 2$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.2e3_3.2t1.1c1 | $1$ | $ 2^{3} \cdot 3 $ | $x^{2} - 6$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.2e7_3e2.4t3.4c1 | $2$ | $ 2^{7} \cdot 3^{2}$ | $x^{4} - 6 x^{2} + 6$ | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| 2.2e9_3e2.8t8.5c1 | $2$ | $ 2^{9} \cdot 3^{2}$ | $x^{8} + 12 x^{6} + 30 x^{4} + 24 x^{2} + 6$ | $QD_{16}$ (as 8T8) | $0$ | $-2$ | |
| 2.2e9_3e2.8t8.5c2 | $2$ | $ 2^{9} \cdot 3^{2}$ | $x^{8} + 12 x^{6} + 30 x^{4} + 24 x^{2} + 6$ | $QD_{16}$ (as 8T8) | $0$ | $-2$ | |
| 8.2e31_3e9.18t68.1c1 | $8$ | $ 2^{31} \cdot 3^{9}$ | $x^{9} - 3 x^{8} + 24 x^{6} - 18 x^{5} - 18 x^{4} + 24 x^{3} + 48 x^{2} - 3 x + 1$ | $(C_3^2:C_8):C_2$ (as 9T19) | $1$ | $0$ | |
| * | 8.2e27_3e9.9t19.1c1 | $8$ | $ 2^{27} \cdot 3^{9}$ | $x^{9} - 3 x^{8} + 24 x^{6} - 18 x^{5} - 18 x^{4} + 24 x^{3} + 48 x^{2} - 3 x + 1$ | $(C_3^2:C_8):C_2$ (as 9T19) | $1$ | $0$ |
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 *.