Normalized defining polynomial
\( x^{9} - x^{8} - 2x^{7} - x^{6} + 3x^{5} + x^{4} - x^{3} + 8x^{2} - 8x - 7 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[3, 3]$ |
| |
| Discriminant: |
\(-1180287062528\)
\(\medspace = -\,2^{9}\cdot 7^{2}\cdot 19^{6}\)
|
| |
| Root discriminant: | \(21.94\) |
| |
| Galois root discriminant: | $2^{9/4}7^{1/2}19^{2/3}\approx 89.61256551149627$ | ||
| Ramified primes: |
\(2\), \(7\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{259}a^{8}+\frac{85}{259}a^{7}+\frac{8}{37}a^{6}-\frac{106}{259}a^{5}-\frac{48}{259}a^{4}+\frac{17}{259}a^{3}-\frac{93}{259}a^{2}+\frac{39}{259}a-\frac{3}{37}$
| 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: |
$\frac{54}{259}a^{8}-\frac{72}{259}a^{7}-\frac{12}{37}a^{6}-\frac{26}{259}a^{5}+\frac{257}{259}a^{4}-\frac{118}{259}a^{3}-\frac{101}{259}a^{2}+\frac{552}{259}a-\frac{88}{37}$, $\frac{9}{259}a^{8}-\frac{12}{259}a^{7}-\frac{2}{37}a^{6}+\frac{82}{259}a^{5}+\frac{86}{259}a^{4}-\frac{106}{259}a^{3}-\frac{319}{259}a^{2}+\frac{92}{259}a+\frac{10}{37}$, $\frac{4}{259}a^{8}+\frac{81}{259}a^{7}-\frac{5}{37}a^{6}-\frac{165}{259}a^{5}-\frac{192}{259}a^{4}+\frac{68}{259}a^{3}+\frac{146}{259}a^{2}+\frac{156}{259}a+\frac{136}{37}$, $\frac{27}{259}a^{8}-\frac{36}{259}a^{7}-\frac{6}{37}a^{6}-\frac{13}{259}a^{5}-\frac{1}{259}a^{4}-\frac{59}{259}a^{3}+\frac{79}{259}a^{2}+\frac{17}{259}a-\frac{7}{37}$, $\frac{30}{259}a^{8}-\frac{40}{259}a^{7}-\frac{19}{37}a^{6}-\frac{72}{259}a^{5}+\frac{114}{259}a^{4}+\frac{251}{259}a^{3}+\frac{318}{259}a^{2}+\frac{652}{259}a+\frac{58}{37}$
|
| |
| Regulator: | \( 572.550634428 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 572.550634428 \cdot 1}{2\cdot\sqrt{1180287062528}}\cr\approx \mathstrut & 0.522901241886 \end{aligned}\]
Galois group
$S_3\wr C_3$ (as 9T28):
| A solvable group of order 648 |
| The 17 conjugacy class representatives for $S_3 \wr C_3 $ |
| Character table for $S_3 \wr C_3 $ |
Intermediate fields
| 3.3.361.1 |
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 | ${\href{/padicField/3.9.0.1}{9} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.9.0.1}{9} }$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.1.0a1.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 2.3.2.9a1.3 | $x^{6} + 6 x^{4} + 2 x^{3} + 5 x^{2} + 6 x + 3$ | $2$ | $3$ | $9$ | $A_4\times C_2$ | $$[2, 2, 3]^{3}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.1.2.1a1.2 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(19\)
| 19.3.3.6a1.3 | $x^{9} + 12 x^{7} + 51 x^{6} + 48 x^{5} + 408 x^{4} + 931 x^{3} + 816 x^{2} + 3468 x + 4932$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |