magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 16, -8, -52, 16, 24, -8, -10, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 10*x^7 - 8*x^6 + 24*x^5 + 16*x^4 - 52*x^3 - 8*x^2 + 16*x + 8)
gp: K = bnfinit(x^9 - 10*x^7 - 8*x^6 + 24*x^5 + 16*x^4 - 52*x^3 - 8*x^2 + 16*x + 8, 1)
Normalized defining polynomial
\( x^{9} - 10 x^{7} - 8 x^{6} + 24 x^{5} + 16 x^{4} - 52 x^{3} - 8 x^{2} + 16 x + 8 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $9$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-51645087424=-\,2^{6}\cdot 7^{6}\cdot 19^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.50$ | 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: | $2, 7, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{5036} a^{8} - \frac{14}{1259} a^{7} + \frac{152}{1259} a^{6} + \frac{299}{1259} a^{5} + \frac{517}{2518} a^{4} + \frac{13}{2518} a^{3} - \frac{377}{1259} a^{2} - \frac{293}{1259} a + \frac{45}{1259}$
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: | $5$ | 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: | \( 60.1226217748 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times S_3$ (as 9T4):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3\times C_3$ |
| Character table for $S_3\times C_3$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 3.1.76.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 6 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 | ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.9.6.1 | $x^{9} - 4 x^{3} + 8$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ |
| $7$ | 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| $19$ | 19.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 19.6.3.2 | $x^{6} - 361 x^{2} + 27436$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.19.2t1.1c1 | $1$ | $ 19 $ | $x^{2} - x + 5$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.7.3t1.1c1 | $1$ | $ 7 $ | $x^{3} - x^{2} - 2 x + 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.7_19.6t1.2c1 | $1$ | $ 7 \cdot 19 $ | $x^{6} - x^{5} + 10 x^{4} - 7 x^{3} + 75 x^{2} - 16 x + 239$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.7_19.6t1.2c2 | $1$ | $ 7 \cdot 19 $ | $x^{6} - x^{5} + 10 x^{4} - 7 x^{3} + 75 x^{2} - 16 x + 239$ | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.7.3t1.1c2 | $1$ | $ 7 $ | $x^{3} - x^{2} - 2 x + 1$ | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 2.2e2_19.3t2.1c1 | $2$ | $ 2^{2} \cdot 19 $ | $x^{3} - 2 x - 2$ | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.2e2_7e2_19.6t5.1c1 | $2$ | $ 2^{2} \cdot 7^{2} \cdot 19 $ | $x^{9} - 10 x^{7} - 8 x^{6} + 24 x^{5} + 16 x^{4} - 52 x^{3} - 8 x^{2} + 16 x + 8$ | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
| * | 2.2e2_7e2_19.6t5.1c2 | $2$ | $ 2^{2} \cdot 7^{2} \cdot 19 $ | $x^{9} - 10 x^{7} - 8 x^{6} + 24 x^{5} + 16 x^{4} - 52 x^{3} - 8 x^{2} + 16 x + 8$ | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
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 *.