Normalized defining polynomial
\( x^{9} - 4x^{8} - 3x^{7} + 56x^{6} - 70x^{5} - 196x^{4} + 462x^{3} + 152x^{2} - 972x + 608 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $[1, 4]$ |
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| Discriminant: |
\(159385218048000\)
\(\medspace = 2^{13}\cdot 3^{3}\cdot 5^{3}\cdot 7^{8}\)
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| Root discriminant: | \(37.85\) |
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| Galois root discriminant: | $2^{2}3^{1/2}5^{1/2}7^{8/9}\approx 87.35824058258169$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
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| |
| Discriminant root field: | \(\Q(\sqrt{30}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{12504820}a^{8}+\frac{974037}{6252410}a^{7}-\frac{186291}{12504820}a^{6}+\frac{2743699}{6252410}a^{5}-\frac{1804523}{6252410}a^{4}-\frac{1405451}{3126205}a^{3}-\frac{384107}{1250482}a^{2}+\frac{1314303}{3126205}a-\frac{261814}{3126205}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{9}$, which has order $9$ |
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| Narrow class group: | $C_{9}$, which has order $9$ |
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Unit group
| Rank: | $4$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{1685}{1250482}a^{8}-\frac{5280}{625241}a^{7}-\frac{29353}{1250482}a^{6}+\frac{100861}{625241}a^{5}-\frac{74272}{625241}a^{4}-\frac{794536}{625241}a^{3}+\frac{771182}{625241}a^{2}+\frac{1244348}{625241}a-\frac{1348611}{625241}$, $\frac{4783}{3126205}a^{8}-\frac{32121}{6252410}a^{7}-\frac{61428}{3126205}a^{6}+\frac{341113}{6252410}a^{5}+\frac{836992}{3126205}a^{4}-\frac{3725532}{3126205}a^{3}-\frac{451446}{625241}a^{2}+\frac{23061616}{3126205}a-\frac{28980883}{3126205}$, $\frac{80649}{6252410}a^{8}-\frac{169227}{3126205}a^{7}+\frac{358371}{6252410}a^{6}+\frac{664546}{3126205}a^{5}-\frac{1880267}{3126205}a^{4}+\frac{320177}{3126205}a^{3}+\frac{970384}{625241}a^{2}-\frac{6020576}{3126205}a+\frac{1828773}{3126205}$, $\frac{952139}{6252410}a^{8}-\frac{1360157}{3126205}a^{7}-\frac{6559569}{6252410}a^{6}+\frac{22759396}{3126205}a^{5}-\frac{3835312}{3126205}a^{4}-\frac{97221593}{3126205}a^{3}+\frac{18970969}{625241}a^{2}+\frac{180525789}{3126205}a-\frac{240184677}{3126205}$
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| Regulator: | \( 6509.20238259 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 6509.20238259 \cdot 9}{2\cdot\sqrt{159385218048000}}\cr\approx \mathstrut & 7.23212189053 \end{aligned}\]
Galois group
$C_3^3:D_6$ (as 9T24):
| A solvable group of order 324 |
| The 17 conjugacy class representatives for $((C_3^3:C_3):C_2):C_2$ |
| Character table for $((C_3^3:C_3):C_2):C_2$ |
Intermediate fields
| 3.1.980.1 |
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 |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 9.1.159385218048000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.4.8b1.5 | $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 6$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ | |
|
\(3\)
| 3.3.1.0a1.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 3.3.2.3a1.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(7\)
| 7.1.9.8a1.2 | $x^{9} + 14$ | $9$ | $1$ | $8$ | $C_9:C_3$ | $$[\ ]_{9}^{3}$$ |