Normalized defining polynomial
\( x^{9} - 86 x^{7} - 116 x^{6} + 2617 x^{5} + 6772 x^{4} - 23565 x^{3} - 118784 x^{2} - 185040 x - 71616 \)
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: | \(-373943606076754623=-\,3^{3}\cdot 7^{6}\cdot 13^{6}\cdot 29^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $89.65$ | 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: | $3, 7, 13, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{36} a^{7} - \frac{1}{18} a^{6} - \frac{1}{18} a^{5} - \frac{2}{9} a^{4} - \frac{5}{36} a^{3} - \frac{7}{18} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{11587224567888} a^{8} - \frac{769920193}{482801023662} a^{7} - \frac{69010477345}{1931204094648} a^{6} + \frac{43600911515}{965602047324} a^{5} - \frac{218421189173}{3862408189296} a^{4} + \frac{62494002797}{321867349108} a^{3} - \frac{1848903222349}{11587224567888} a^{2} + \frac{9163217372}{80466837277} a - \frac{91966940780}{241400511831}$
Class group and class number
$C_{129}$, which has order $129$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 13410.4294657 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times S_3$ (as 9T4):
| 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
| 3.3.8281.2, 3.1.87.1 |
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 | ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ | R | ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $7$ | 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| $13$ | 13.9.6.1 | $x^{9} + 234 x^{6} + 16900 x^{3} + 474552$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| $29$ | 29.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 29.6.3.2 | $x^{6} - 841 x^{2} + 73167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |