Normalized defining polynomial
\( x^{9} - 9x^{7} + 27x^{5} - 30x^{3} + 9x - 25 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[1, 4]$ |
| |
| Discriminant: |
\(148718980881\)
\(\medspace = 3^{12}\cdot 23^{4}\)
|
| |
| Root discriminant: | \(17.43\) |
| |
| Galois root discriminant: | $3^{16/9}23^{1/2}\approx 33.81267124538259$ | ||
| Ramified primes: |
\(3\), \(23\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{5}-\frac{2}{9}a^{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{3}a^{3}-a-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-a^{2}+\frac{2}{3}a+\frac{4}{3}$, $\frac{1}{9}a^{6}-\frac{2}{3}a^{4}+\frac{1}{9}a^{3}+a^{2}-\frac{4}{3}a+\frac{7}{9}$, $\frac{1}{9}a^{8}+\frac{2}{9}a^{7}-\frac{5}{9}a^{6}-\frac{14}{9}a^{5}+\frac{8}{9}a^{4}+\frac{22}{9}a^{3}+\frac{4}{9}a^{2}-\frac{1}{9}a+\frac{19}{9}$
|
| |
| Regulator: | \( 82.3148996917 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 82.3148996917 \cdot 1}{2\cdot\sqrt{148718980881}}\cr\approx \mathstrut & 0.332670833548 \end{aligned}\]
Galois group
| A solvable group of order 54 |
| The 10 conjugacy class representatives for $(C_9:C_3):C_2$ |
| Character table for $(C_9:C_3):C_2$ |
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 18 sibling: | data not computed |
| Degree 27 sibling: | 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 | ${\href{/padicField/2.9.0.1}{9} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.3.12a7.3 | $x^{9} + 3 x^{8} + 12 x^{7} + 21 x^{6} + 42 x^{5} + 60 x^{4} + 59 x^{3} + 63 x^{2} + 54 x + 10$ | $3$ | $3$ | $12$ | $C_9:C_3$ | $$[2, 2]^{3}$$ |
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.3.2.3a1.2 | $x^{6} + 4 x^{4} + 36 x^{3} + 4 x^{2} + 72 x + 347$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *54 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.207.6t1.b.a | $1$ | $ 3^{2} \cdot 23 $ | 6.0.79827687.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.207.6t1.b.b | $1$ | $ 3^{2} \cdot 23 $ | 6.0.79827687.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| *54 | 2.23.3t2.b.a | $2$ | $ 23 $ | 3.1.23.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| 2.1863.6t5.d.a | $2$ | $ 3^{4} \cdot 23 $ | 6.0.79827687.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| 2.1863.6t5.d.b | $2$ | $ 3^{4} \cdot 23 $ | 6.0.79827687.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ | |
| *54 | 6.6466042647.9t10.b.a | $6$ | $ 3^{12} \cdot 23^{3}$ | 9.1.148718980881.3 | $(C_9:C_3):C_2$ (as 9T10) | $1$ | $0$ |
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 *.