Normalized defining polynomial
\( x^{9} - 4x^{8} - 4x^{7} + 28x^{6} - 14x^{5} - 29x^{4} + 15x^{3} + 10x^{2} - 3x - 1 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[9, 0]$ |
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| Discriminant: |
\(144872805473\)
\(\medspace = 11^{3}\cdot 37^{2}\cdot 43^{3}\)
|
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| Root discriminant: | \(17.38\) |
| |
| Galois root discriminant: | $11^{1/2}37^{2/3}43^{1/2}\approx 241.48957456928824$ | ||
| Ramified primes: |
\(11\), \(37\), \(43\)
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| Discriminant root field: | \(\Q(\sqrt{473}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$
| 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 $1$ |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{8}-4a^{7}-4a^{6}+28a^{5}-14a^{4}-29a^{3}+15a^{2}+10a-3$, $\frac{4}{3}a^{8}-6a^{7}-\frac{7}{3}a^{6}+\frac{116}{3}a^{5}-\frac{115}{3}a^{4}-21a^{3}+\frac{97}{3}a^{2}+\frac{1}{3}a-4$, $\frac{4}{3}a^{8}-6a^{7}-\frac{7}{3}a^{6}+\frac{116}{3}a^{5}-\frac{115}{3}a^{4}-21a^{3}+\frac{97}{3}a^{2}+\frac{1}{3}a-5$, $\frac{1}{3}a^{8}-2a^{7}+\frac{4}{3}a^{6}+\frac{34}{3}a^{5}-\frac{64}{3}a^{4}+\frac{10}{3}a^{3}+12a^{2}-4a-\frac{2}{3}$, $\frac{2}{3}a^{8}-3a^{7}-\frac{4}{3}a^{6}+\frac{59}{3}a^{5}-\frac{53}{3}a^{4}-\frac{37}{3}a^{3}+13a^{2}-\frac{1}{3}$, $a^{8}-\frac{13}{3}a^{7}-\frac{7}{3}a^{6}+28a^{5}-\frac{74}{3}a^{4}-\frac{46}{3}a^{3}+\frac{62}{3}a^{2}-\frac{5}{3}a-4$, $\frac{4}{3}a^{8}-\frac{16}{3}a^{7}-\frac{14}{3}a^{6}+\frac{104}{3}a^{5}-21a^{4}-\frac{64}{3}a^{3}+10a^{2}+\frac{8}{3}a-1$, $\frac{1}{3}a^{8}-2a^{7}+\frac{2}{3}a^{6}+\frac{41}{3}a^{5}-\frac{55}{3}a^{4}-11a^{3}+\frac{55}{3}a^{2}+\frac{4}{3}a-4$
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| Regulator: | \( 335.764904866 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 335.764904866 \cdot 1}{2\cdot\sqrt{144872805473}}\cr\approx \mathstrut & 0.225830101944 \end{aligned}\]
Galois group
$C_3\wr S_3$ (as 9T20):
| A solvable group of order 162 |
| The 22 conjugacy class representatives for $C_3 \wr S_3 $ |
| Character table for $C_3 \wr S_3 $ |
Intermediate fields
| 3.3.473.1 |
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 siblings: | 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 | ${\href{/padicField/2.9.0.1}{9} }$ | ${\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | ${\href{/padicField/7.9.0.1}{9} }$ | R | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.9.0.1}{9} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | R | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.9.0.1}{9} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 11.3.2.3a1.2 | $x^{6} + 4 x^{4} + 18 x^{3} + 4 x^{2} + 36 x + 92$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(37\)
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.2.1.0a1.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 37.1.3.2a1.1 | $x^{3} + 37$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
|
\(43\)
| 43.3.1.0a1.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 43.3.2.3a1.1 | $x^{6} + 2 x^{4} + 80 x^{3} + x^{2} + 123 x + 1600$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |