magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -16, -28, -24, -12, 0, 3, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 3*x^6 - 12*x^4 - 24*x^3 - 28*x^2 - 16*x - 8)
gp: K = bnfinit(x^9 - x^8 + 3*x^6 - 12*x^4 - 24*x^3 - 28*x^2 - 16*x - 8, 1)
Normalized defining polynomial
\( x^{9} - x^{8} + 3 x^{6} - 12 x^{4} - 24 x^{3} - 28 x^{2} - 16 x - 8 \)
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: | \(8291469824=2^{9}\cdot 11^{3}\cdot 23^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $12.65$ | 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: | $2, 11, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4}$, $\frac{1}{332} a^{8} - \frac{7}{332} a^{7} + \frac{21}{166} a^{6} - \frac{1}{4} a^{5} - \frac{3}{83} a^{3} - \frac{59}{166} a^{2} + \frac{4}{83} a - \frac{28}{83}$
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: | \( 23.2169845064 \) | 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
| 3.1.23.1 |
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 | R | ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.6.9.3 | $x^{6} - 4 x^{4} + 4 x^{2} + 24$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $23$ | 23.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 23.6.3.2 | $x^{6} - 529 x^{2} + 48668$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e3_11.2t1.2c1 | $1$ | $ 2^{3} \cdot 11 $ | $x^{2} + 22$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.23.2t1.1c1 | $1$ | $ 23 $ | $x^{2} - x + 6$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.2e3_11_23.2t1.1c1 | $1$ | $ 2^{3} \cdot 11 \cdot 23 $ | $x^{2} - 506$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.2e6_11e2_23.6t3.1c1 | $2$ | $ 2^{6} \cdot 11^{2} \cdot 23 $ | $x^{6} + 44 x^{4} + 484 x^{2} - 244904$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| 2.2e3_11_23.6t3.6c1 | $2$ | $ 2^{3} \cdot 11 \cdot 23 $ | $x^{6} - 6 x^{4} - 28 x^{3} + 31 x^{2} + 128 x + 218$ | $D_{6}$ (as 6T3) | $1$ | $-2$ | |
| 2.2e3_11_23.3t2.1c1 | $2$ | $ 2^{3} \cdot 11 \cdot 23 $ | $x^{3} - x^{2} - 10 x - 6$ | $S_3$ (as 3T2) | $1$ | $2$ | |
| * | 2.23.3t2.1c1 | $2$ | $ 23 $ | $x^{3} - x^{2} + 1$ | $S_3$ (as 3T2) | $1$ | $0$ |
| 4.2e6_11e2_23e2.6t9.1c1 | $4$ | $ 2^{6} \cdot 11^{2} \cdot 23^{2}$ | $x^{6} - 2 x^{5} + 7 x^{4} - 18 x^{3} + 32 x^{2} - 36 x + 47$ | $S_3^2$ (as 6T9) | $1$ | $0$ | |
| 6.2e9_11e3_23e4.18t51.1c1 | $6$ | $ 2^{9} \cdot 11^{3} \cdot 23^{4}$ | $x^{9} - x^{8} + 3 x^{6} - 12 x^{4} - 24 x^{3} - 28 x^{2} - 16 x - 8$ | $C_3^2 : D_{6} $ (as 9T18) | $1$ | $0$ | |
| * | 6.2e9_11e3_23e2.9t18.1c1 | $6$ | $ 2^{9} \cdot 11^{3} \cdot 23^{2}$ | $x^{9} - x^{8} + 3 x^{6} - 12 x^{4} - 24 x^{3} - 28 x^{2} - 16 x - 8$ | $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 *.