Normalized defining polynomial
\( x^{9} - 2x^{8} - 25x^{7} + 40x^{6} + 314x^{5} - 654x^{4} - 1063x^{3} + 3528x^{2} - 2244x - 1832 \)
Invariants
| Degree: | $9$ |
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| Signature: | $[3, 3]$ |
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| Discriminant: |
\(-44869371487471\)
\(\medspace = -\,31^{3}\cdot 197^{4}\)
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| Root discriminant: | \(32.88\) |
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| Galois root discriminant: | $31^{1/2}197^{2/3}\approx 188.50592172874553$ | ||
| Ramified primes: |
\(31\), \(197\)
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| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{30}a^{6}-\frac{1}{5}a^{5}-\frac{1}{30}a^{4}-\frac{3}{10}a^{3}-\frac{7}{30}a^{2}+\frac{2}{5}a-\frac{2}{15}$, $\frac{1}{30}a^{7}-\frac{7}{30}a^{5}+\frac{7}{15}a^{3}-\frac{7}{30}a+\frac{1}{5}$, $\frac{1}{429540540}a^{8}+\frac{338941}{107385135}a^{7}-\frac{1820529}{143180180}a^{6}-\frac{49793969}{214770270}a^{5}-\frac{421669}{9762285}a^{4}-\frac{30866026}{107385135}a^{3}+\frac{6098863}{429540540}a^{2}+\frac{16487329}{214770270}a-\frac{21626674}{107385135}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
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| Narrow class group: | Trivial group, which has order $1$ |
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Unit group
| Rank: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{779}{7809828}a^{8}-\frac{7431}{6508190}a^{7}-\frac{211531}{39049140}a^{6}+\frac{152553}{6508190}a^{5}+\frac{977954}{9762285}a^{4}-\frac{301139}{1301638}a^{3}-\frac{20648323}{39049140}a^{2}+\frac{603540}{650819}a-\frac{564566}{3254095}$, $\frac{15638}{9762285}a^{8}+\frac{21519}{6508190}a^{7}-\frac{137563}{3904914}a^{6}-\frac{433887}{6508190}a^{5}+\frac{1212347}{3254095}a^{4}+\frac{197059}{650819}a^{3}-\frac{11231097}{6508190}a^{2}+\frac{4819641}{3254095}a+\frac{10423637}{9762285}$, $\frac{526531}{214770270}a^{8}-\frac{824371}{107385135}a^{7}-\frac{457167}{14318018}a^{6}+\frac{12418711}{107385135}a^{5}+\frac{5437607}{19524570}a^{4}-\frac{389985841}{214770270}a^{3}+\frac{596713969}{214770270}a^{2}-\frac{214084861}{214770270}a-\frac{23601430}{21477027}$, $\frac{1465787}{33041580}a^{8}-\frac{1828249}{16520790}a^{7}-\frac{52824757}{33041580}a^{6}+\frac{5403367}{16520790}a^{5}+\frac{9634401}{500630}a^{4}-\frac{6555275}{1652079}a^{3}-\frac{666693671}{11013860}a^{2}+\frac{789529688}{8260395}a+\frac{257480816}{8260395}$, $\frac{486341}{429540540}a^{8}+\frac{762101}{71590090}a^{7}-\frac{58163987}{429540540}a^{6}+\frac{8795004}{35795045}a^{5}+\frac{19066207}{19524570}a^{4}-\frac{243003411}{71590090}a^{3}+\frac{292881763}{429540540}a^{2}+\frac{558977609}{71590090}a-\frac{351120152}{35795045}$
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| Regulator: | \( 3111.74669351 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 3111.74669351 \cdot 1}{2\cdot\sqrt{44869371487471}}\cr\approx \mathstrut & 0.460923488384 \end{aligned}\]
Galois group
$C_3^2:S_3$ (as 9T12):
| A solvable group of order 54 |
| The 10 conjugacy class representatives for $(C_3^2:C_3):C_2$ |
| Character table for $(C_3^2:C_3):C_2$ |
Intermediate fields
| 3.1.31.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 9 siblings: | data not computed |
| Degree 18 siblings: | 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 | ${\href{/padicField/2.3.0.1}{3} }^{3}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | R | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(31\)
| 31.3.1.0a1.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 31.3.2.3a1.2 | $x^{6} + 2 x^{4} + 56 x^{3} + x^{2} + 56 x + 815$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(197\)
| 197.3.1.0a1.1 | $x^{3} + 3 x + 195$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 197.2.3.4a1.1 | $x^{6} + 576 x^{5} + 110598 x^{4} + 7080192 x^{3} + 221196 x^{2} + 2501 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |