Normalized defining polynomial
\( x^{9} - 3x^{7} - x^{6} + 3x^{5} + 2x^{4} + 316x^{3} - x^{2} - 317x + 7 \)
Invariants
| Degree: | $9$ |
| |
| Signature: | $(3, 3)$ |
| |
| Discriminant: |
\(-376367048000000\)
\(\medspace = -\,2^{9}\cdot 5^{6}\cdot 19^{6}\)
|
| |
| Root discriminant: | \(41.64\) |
| |
| Galois root discriminant: | $2^{3/2}5^{2/3}19^{2/3}\approx 58.888080312522455$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{42}a^{6}-\frac{1}{21}a^{4}-\frac{3}{7}a^{3}+\frac{1}{42}a^{2}+\frac{3}{7}a-\frac{1}{6}$, $\frac{1}{42}a^{7}-\frac{1}{21}a^{5}-\frac{3}{7}a^{4}+\frac{1}{42}a^{3}+\frac{3}{7}a^{2}-\frac{1}{6}a$, $\frac{1}{718158}a^{8}-\frac{7579}{718158}a^{7}+\frac{1899}{239386}a^{6}+\frac{7255}{359079}a^{5}-\frac{42437}{239386}a^{4}-\frac{109093}{718158}a^{3}-\frac{88486}{359079}a^{2}+\frac{162019}{718158}a-\frac{45659}{102594}$
| 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: |
$\frac{1108}{119693}a^{8}+\frac{5561}{718158}a^{7}-\frac{8864}{359079}a^{6}-\frac{5039}{359079}a^{5}+\frac{1387}{359079}a^{4}+\frac{3821}{718158}a^{3}+\frac{1079206}{359079}a^{2}+\frac{1387123}{718158}a-\frac{16891}{51297}$, $\frac{1111}{239386}a^{8}-\frac{2792}{359079}a^{7}-\frac{4444}{359079}a^{6}+\frac{2929}{359079}a^{5}+\frac{35419}{718158}a^{4}+\frac{1633}{359079}a^{3}+\frac{871477}{718158}a^{2}-\frac{852046}{359079}a+\frac{42574}{51297}$, $\frac{1847}{239386}a^{8}-\frac{95}{718158}a^{7}-\frac{7388}{359079}a^{6}+\frac{206}{359079}a^{5}+\frac{74489}{718158}a^{4}+\frac{238921}{718158}a^{3}+\frac{323681}{102594}a^{2}+\frac{836333}{718158}a-\frac{8407}{51297}$, $\frac{1285}{119693}a^{8}-\frac{2269}{239386}a^{7}-\frac{3461}{718158}a^{6}+\frac{7440}{119693}a^{5}+\frac{42890}{359079}a^{4}+\frac{37491}{239386}a^{3}+\frac{2388643}{718158}a^{2}-\frac{980661}{239386}a-\frac{13777}{102594}$, $\frac{2557103}{718158}a^{8}-\frac{2216047}{359079}a^{7}+\frac{7214708}{359079}a^{6}-\frac{12909817}{359079}a^{5}+\frac{87076105}{718158}a^{4}+\frac{14273264}{359079}a^{3}-\frac{34943235}{239386}a^{2}+\frac{316903}{51297}a-\frac{3370}{51297}$
|
| |
| Regulator: | \( 9581.34517932 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 9581.34517932 \cdot 1}{2\cdot\sqrt{376367048000000}}\cr\approx \mathstrut & 0.490027312082 \end{aligned}\]
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, 3.1.72200.1 |
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: | data not computed |
| Minimal sibling: | 6.0.115520000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ | |
|
\(5\)
| 5.3.3.6a1.1 | $x^{9} + 9 x^{7} + 9 x^{6} + 27 x^{5} + 54 x^{4} + 54 x^{3} + 81 x^{2} + 81 x + 32$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |
|
\(19\)
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *18 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| *18 | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.152.6t1.c.a | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.152.6t1.c.b | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| *18 | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *18 | 2.72200.3t2.b.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19^{2}$ | 3.1.72200.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *18 | 2.3800.6t5.a.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19 $ | 9.3.376367048000000.3 | $S_3\times C_3$ (as 9T4) | $0$ | $0$ |
| *18 | 2.3800.6t5.a.b | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19 $ | 9.3.376367048000000.3 | $S_3\times C_3$ (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 *.