Normalized defining polynomial
\( x^{6} + 84x^{4} - 35x^{3} + 1764x^{2} - 1470x + 19159 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-108049559499\)
\(\medspace = -\,3^{8}\cdot 7^{4}\cdot 19^{3}\)
|
| |
| Root discriminant: | \(69.01\) |
| |
| Galois root discriminant: | $3^{4/3}7^{2/3}19^{1/2}\approx 69.01399479708432$ | ||
| Ramified primes: |
\(3\), \(7\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-19}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{63}a^{3}-\frac{1}{3}a+\frac{2}{9}$, $\frac{1}{63}a^{4}-\frac{1}{3}a^{2}+\frac{2}{9}a$, $\frac{1}{3843}a^{5}-\frac{13}{3843}a^{4}+\frac{1}{427}a^{3}-\frac{13}{549}a^{2}+\frac{55}{549}a+\frac{17}{183}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}\times C_{3}$, which has order $9$ |
| |
| Narrow class group: | $C_{3}\times C_{3}$, which has order $9$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{23}{3843}a^{5}+\frac{2}{1281}a^{4}+\frac{817}{3843}a^{3}-\frac{116}{549}a^{2}+\frac{127}{61}a-\frac{3646}{549}$, $\frac{31}{3843}a^{5}+\frac{23}{427}a^{4}+\frac{2963}{3843}a^{3}+\frac{2159}{549}a^{2}+\frac{3049}{183}a+\frac{25066}{549}$
|
| |
| Regulator: | \( 140.834439953 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 140.834439953 \cdot 9}{2\cdot\sqrt{108049559499}}\cr\approx \mathstrut & 0.478243955382 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 6T5):
| 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
| \(\Q(\sqrt{-19}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | 18.0.1261446981901370189638661902928499.3 |
| Twin sextic algebra: | 3.1.1539.1 $\times$ \(\Q(\zeta_{7})^+\) |
| Degree 9 sibling: | 9.3.428848701651531.5 |
| 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 | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.3.8a1.2 | $x^{6} + 6 x^{5} + 21 x^{4} + 44 x^{3} + 60 x^{2} + 57 x + 23$ | $3$ | $2$ | $8$ | $S_3\times C_3$ | $$[2]^{6}$$ |
|
\(7\)
| 7.1.3.2a1.2 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.1.3.2a1.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
|
\(19\)
| 19.3.2.3a1.2 | $x^{6} + 8 x^{4} + 34 x^{3} + 16 x^{2} + 136 x + 308$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |