Normalized defining polynomial
\( x^{9} - 2x^{8} - 7x^{7} - x^{6} + 4x^{5} + 24x^{4} + 208x^{3} + 16x^{2} + 381x - 103 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[3, 3]$ |
| |
| Discriminant: |
\(-6240683160531\)
\(\medspace = -\,3^{3}\cdot 17^{3}\cdot 19^{6}\)
|
| |
| Root discriminant: | \(26.41\) |
| |
| Galois root discriminant: | $3^{1/2}17^{1/2}19^{2/3}\approx 50.849593878531266$ | ||
| Ramified primes: |
\(3\), \(17\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-51}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{1}{7}a^{5}+\frac{1}{7}a^{4}-\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}$, $\frac{1}{49}a^{7}+\frac{1}{49}a^{6}+\frac{6}{49}a^{5}-\frac{22}{49}a^{4}-\frac{2}{49}a^{3}-\frac{6}{49}a^{2}+\frac{23}{49}a-\frac{24}{49}$, $\frac{1}{88837}a^{8}-\frac{424}{88837}a^{7}+\frac{1247}{88837}a^{6}+\frac{6787}{88837}a^{5}-\frac{21326}{88837}a^{4}+\frac{27059}{88837}a^{3}+\frac{41283}{88837}a^{2}-\frac{9358}{88837}a+\frac{40629}{88837}$
| 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: |
$\frac{3}{12691}a^{8}+\frac{282}{12691}a^{7}-\frac{144}{12691}a^{6}-\frac{2949}{12691}a^{5}-\frac{2077}{12691}a^{4}+\frac{5549}{12691}a^{3}+\frac{3932}{12691}a^{2}+\frac{33050}{12691}a+\frac{35640}{12691}$, $\frac{53}{12691}a^{8}+\frac{61}{12691}a^{7}-\frac{213}{12691}a^{6}-\frac{40}{12691}a^{5}-\frac{1556}{12691}a^{4}-\frac{6949}{12691}a^{3}-\frac{3141}{12691}a^{2}-\frac{15788}{12691}a-\frac{11903}{12691}$, $\frac{1093}{88837}a^{8}-\frac{2930}{88837}a^{7}-\frac{4031}{88837}a^{6}+\frac{15712}{88837}a^{5}+\frac{423}{88837}a^{4}-\frac{65250}{88837}a^{3}+\frac{47513}{88837}a^{2}-\frac{347444}{88837}a+\frac{206557}{88837}$, $\frac{155}{12691}a^{8}-\frac{452}{12691}a^{7}-\frac{706}{12691}a^{6}+\frac{2258}{12691}a^{5}-\frac{431}{12691}a^{4}-\frac{6576}{12691}a^{3}+\frac{20731}{12691}a^{2}-\frac{38163}{12691}a+\frac{11824}{12691}$, $\frac{7645}{88837}a^{8}-\frac{19779}{88837}a^{7}-\frac{62894}{88837}a^{6}+\frac{83766}{88837}a^{5}+\frac{145421}{88837}a^{4}-\frac{95147}{88837}a^{3}+\frac{1237961}{88837}a^{2}-\frac{463245}{88837}a+\frac{39992}{88837}$
|
| |
| Regulator: | \( 856.582672097 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 856.582672097 \cdot 1}{2\cdot\sqrt{6240683160531}}\cr\approx \mathstrut & 0.340214495911 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 9T4):
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3\times C_3$ |
| Character table for $S_3\times C_3$ |
Intermediate fields
| 3.3.361.1, 3.1.18411.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 6 sibling: | 6.0.47887011.1 |
| Minimal sibling: | 6.0.47887011.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{3}$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{3}$ | R | R | ${\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.1.0a1.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 3.3.2.3a1.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(17\)
| 17.3.1.0a1.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 17.3.2.3a1.1 | $x^{6} + 2 x^{4} + 28 x^{3} + x^{2} + 45 x + 196$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(19\)
| 19.3.3.6a1.3 | $x^{9} + 12 x^{7} + 51 x^{6} + 48 x^{5} + 408 x^{4} + 931 x^{3} + 816 x^{2} + 3468 x + 4932$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *18 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.51.2t1.a.a | $1$ | $ 3 \cdot 17 $ | \(\Q(\sqrt{-51}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.969.6t1.a.a | $1$ | $ 3 \cdot 17 \cdot 19 $ | 6.0.17287210971.4 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.969.6t1.a.b | $1$ | $ 3 \cdot 17 \cdot 19 $ | 6.0.17287210971.4 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| *18 | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *18 | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *18 | 2.18411.3t2.a.a | $2$ | $ 3 \cdot 17 \cdot 19^{2}$ | 3.1.18411.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *18 | 2.969.6t5.c.a | $2$ | $ 3 \cdot 17 \cdot 19 $ | 9.3.6240683160531.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
| *18 | 2.969.6t5.c.b | $2$ | $ 3 \cdot 17 \cdot 19 $ | 9.3.6240683160531.1 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
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 *.