Normalized defining polynomial
\( x^{9} - x^{8} - x^{7} - 4x^{6} + 4x^{5} + 23x^{4} - 35x^{3} + 5x^{2} + 6x + 1 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[3, 3]$ |
| |
| Discriminant: |
\(-4616841152\)
\(\medspace = -\,2^{6}\cdot 7^{2}\cdot 11^{2}\cdot 23^{3}\)
|
| |
| Root discriminant: | \(11.85\) |
| |
| Galois root discriminant: | $2^{3/2}7^{1/2}11^{2/3}23^{1/2}\approx 177.5087319632081$ | ||
| Ramified primes: |
\(2\), \(7\), \(11\), \(23\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{14}a^{7}-\frac{3}{14}a^{6}-\frac{3}{14}a^{5}+\frac{5}{14}a^{4}+\frac{2}{7}a^{3}-\frac{2}{7}a^{2}-\frac{3}{14}a+\frac{1}{14}$, $\frac{1}{154}a^{8}+\frac{1}{154}a^{7}+\frac{17}{77}a^{6}-\frac{5}{22}a^{5}-\frac{1}{14}a^{4}+\frac{6}{77}a^{3}-\frac{3}{14}a^{2}-\frac{39}{154}a-\frac{17}{154}$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{61}{154}a^{8}-\frac{7}{22}a^{7}-\frac{30}{77}a^{6}-\frac{265}{154}a^{5}+\frac{15}{14}a^{4}+\frac{685}{77}a^{3}-\frac{171}{14}a^{2}+\frac{261}{154}a+\frac{239}{154}$, $\frac{6}{77}a^{8}+\frac{1}{154}a^{7}-\frac{3}{22}a^{6}-\frac{79}{154}a^{5}-\frac{3}{14}a^{4}+\frac{127}{77}a^{3}-\frac{2}{7}a^{2}-\frac{127}{154}a-\frac{61}{154}$, $\frac{23}{77}a^{8}-\frac{43}{77}a^{7}-\frac{3}{11}a^{6}-\frac{68}{77}a^{5}+\frac{17}{7}a^{4}+\frac{551}{77}a^{3}-\frac{115}{7}a^{2}+\frac{456}{77}a+\frac{236}{77}$, $\frac{29}{154}a^{8}+\frac{31}{77}a^{7}-\frac{37}{154}a^{6}-\frac{95}{77}a^{5}-2a^{4}+\frac{317}{77}a^{3}+\frac{83}{14}a^{2}-\frac{692}{77}a-\frac{153}{77}$, $\frac{69}{154}a^{8}-\frac{37}{77}a^{7}-\frac{37}{77}a^{6}-\frac{146}{77}a^{5}+\frac{27}{14}a^{4}+\frac{821}{77}a^{3}-\frac{211}{14}a^{2}+\frac{255}{77}a+\frac{43}{22}$
|
| |
| Regulator: | \( 20.4253836621 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 20.4253836621 \cdot 1}{2\cdot\sqrt{4616841152}}\cr\approx \mathstrut & 0.298261606756 \end{aligned}\]
Galois group
$C_3^3:S_4$ (as 9T29):
| A solvable group of order 648 |
| The 17 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
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 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 27 siblings: | data not computed |
| Degree 36 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 | R | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3}$ | R | R | ${\href{/padicField/13.9.0.1}{9} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.1.0a1.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 2.3.2.6a3.1 | $x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 6 x + 5$ | $2$ | $3$ | $6$ | $A_4$ | $$[2, 2]^{3}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.2.2.2a1.1 | $x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(11\)
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 11.1.3.2a1.1 | $x^{3} + 11$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 11.4.1.0a1.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(23\)
| 23.3.1.0a1.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 23.3.2.3a1.2 | $x^{6} + 4 x^{4} + 36 x^{3} + 4 x^{2} + 72 x + 347$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |