Normalized defining polynomial
\( x^{9} - 3x^{8} + 16x^{7} - 30x^{6} + 69x^{5} - 79x^{4} + 59x^{3} - 15x^{2} - 25x - 1 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[3, 3]$ |
| |
| Discriminant: |
\(-6615027235648\)
\(\medspace = -\,2^{6}\cdot 13^{3}\cdot 19^{6}\)
|
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| Root discriminant: | \(26.58\) |
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| Galois root discriminant: | $2\cdot 13^{1/2}19^{2/3}\approx 51.34569922532248$ | ||
| Ramified primes: |
\(2\), \(13\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-13}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{16412}a^{8}+\frac{314}{4103}a^{7}+\frac{1442}{4103}a^{6}+\frac{3889}{8206}a^{5}-\frac{5393}{16412}a^{4}+\frac{2351}{8206}a^{3}-\frac{4855}{16412}a^{2}-\frac{1799}{4103}a-\frac{365}{16412}$
| 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: | $5$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{83}{1492}a^{8}-\frac{48}{373}a^{7}+\frac{326}{373}a^{6}-\frac{977}{746}a^{5}+\frac{5949}{1492}a^{4}-\frac{2557}{746}a^{3}+\frac{5843}{1492}a^{2}-\frac{117}{373}a-\frac{1947}{1492}$, $\frac{189}{1492}a^{8}-\frac{295}{746}a^{7}+\frac{1615}{746}a^{6}-\frac{1573}{373}a^{5}+\frac{15425}{1492}a^{4}-\frac{9229}{746}a^{3}+\frac{17889}{1492}a^{2}-\frac{3773}{746}a-\frac{5575}{1492}$, $\frac{455}{1492}a^{8}-\frac{362}{373}a^{7}+\frac{1868}{373}a^{6}-\frac{7477}{746}a^{5}+\frac{33349}{1492}a^{4}-\frac{20947}{746}a^{3}+\frac{31959}{1492}a^{2}-\frac{3167}{373}a-\frac{10907}{1492}$, $\frac{2301}{16412}a^{8}-\frac{3331}{8206}a^{7}+\frac{17945}{8206}a^{6}-\frac{16439}{4103}a^{5}+\frac{154093}{16412}a^{4}-\frac{88369}{8206}a^{3}+\frac{136513}{16412}a^{2}-\frac{19659}{8206}a-\frac{60295}{16412}$, $\frac{1337}{16412}a^{8}-\frac{1479}{8206}a^{7}+\frac{11397}{8206}a^{6}-\frac{7660}{4103}a^{5}+\frac{101105}{16412}a^{4}-\frac{40641}{8206}a^{3}+\frac{90077}{16412}a^{2}+\frac{2293}{8206}a-\frac{20263}{16412}$
|
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| Regulator: | \( 741.18348259 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 741.18348259 \cdot 1}{2\cdot\sqrt{6615027235648}}\cr\approx \mathstrut & 0.28592985927 \end{aligned}\]
Galois group
$C_3^2:C_6$ (as 9T13):
| A solvable group of order 54 |
| The 10 conjugacy class representatives for $C_3^2 : S_3 $ |
| Character table for $C_3^2 : S_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 9 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Minimal sibling: | 9.1.343981416253696.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.1.0a1.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 2.3.2.6a1.2 | $x^{6} + 2 x^{4} + 4 x^{3} + x^{2} + 4 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $$[2]^{3}$$ | |
|
\(13\)
| 13.3.1.0a1.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 13.3.2.3a1.2 | $x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(19\)
| 19.3.3.6a1.3 | $x^{9} + 12 x^{7} + 51 x^{6} + 48 x^{5} + 408 x^{4} + 931 x^{3} + 816 x^{2} + 3468 x + 4932$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |