Normalized defining polynomial
\( x^{9} - 42x^{6} + 132x^{3} - 2744 \)
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1166498446237171776\) \(\medspace = 2^{6}\cdot 3^{18}\cdot 19^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(101.73\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{2/3}3^{37/18}19^{2/3}\approx 108.12803488848313$ | ||
Ramified primes: | \(2\), \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{68}a^{6}+\frac{1}{34}a^{3}+\frac{4}{17}$, $\frac{1}{952}a^{7}-\frac{3}{17}a^{4}+\frac{89}{238}a$, $\frac{1}{6664}a^{8}+\frac{39}{476}a^{5}-\frac{253}{833}a^{2}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
$C_{6}\times C_{18}$, which has order $108$ (assuming GRH)
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{3}{3332}a^{8}-\frac{3}{952}a^{7}-\frac{1}{119}a^{5}+\frac{1}{34}a^{4}+\frac{148}{833}a^{2}-\frac{29}{238}a$, $\frac{23}{6664}a^{8}-\frac{1}{119}a^{7}+\frac{1}{34}a^{6}-\frac{55}{476}a^{5}+\frac{7}{17}a^{4}-\frac{16}{17}a^{3}-\frac{821}{833}a^{2}-\frac{356}{119}a-\frac{162}{17}$, $\frac{1}{238}a^{7}-\frac{7}{34}a^{4}+\frac{416}{119}a-7$, $\frac{21339}{3332}a^{8}+\frac{309}{34}a^{7}-\frac{961}{34}a^{6}-\frac{35911}{119}a^{5}-\frac{4315}{17}a^{4}+\frac{21479}{17}a^{3}+\frac{1431739}{833}a^{2}-\frac{69064}{17}a-\frac{110351}{17}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 27113.40870458052 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 27113.40870458052 \cdot 108}{2\cdot\sqrt{1166498446237171776}}\cr\approx \mathstrut & 4.22556967575424 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 18 |
The 6 conjugacy class representatives for $C_3^2:C_2$ |
Character table for $C_3^2:C_2$ |
Intermediate fields
3.1.972.2, 3.1.87723.2, 3.1.38988.1, 3.1.350892.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.1.0.1}{1} }^{9}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(3\) | 3.9.18.2 | $x^{9} + 3 x^{3} + 9 x^{2} + 9 x + 3$ | $9$ | $1$ | $18$ | $C_3^2:C_2$ | $[3/2, 5/2]_{2}$ |
\(19\) | 19.9.6.2 | $x^{9} + 981 x^{7} + 108 x^{6} + 316911 x^{5} + 20529 x^{4} + 34115982 x^{3} + 10990188 x^{2} + 130942880 x + 566550143$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |