Normalized defining polynomial
\( x^{6} - 2x^{5} + 7x^{4} + 15x^{2} - 2x - 9 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(15177728\)
\(\medspace = 2^{11}\cdot 7411\)
|
| |
| Root discriminant: | \(15.74\) |
| |
| Galois root discriminant: | $2^{31/12}7411^{1/2}\approx 515.9400349659487$ | ||
| Ramified primes: |
\(2\), \(7411\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{14822}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{191}a^{5}-\frac{32}{191}a^{4}+\frac{12}{191}a^{3}+\frac{22}{191}a^{2}-\frac{72}{191}a+\frac{57}{191}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{31}{191}a^{5}-\frac{37}{191}a^{4}+\frac{181}{191}a^{3}+\frac{109}{191}a^{2}+\frac{633}{191}a+\frac{430}{191}$, $\frac{37}{191}a^{5}-\frac{38}{191}a^{4}+\frac{253}{191}a^{3}+\frac{50}{191}a^{2}+\frac{1156}{191}a+\frac{199}{191}$, $\frac{18}{191}a^{5}-\frac{3}{191}a^{4}+\frac{25}{191}a^{3}+\frac{205}{191}a^{2}+\frac{232}{191}a+\frac{644}{191}$
|
| |
| Regulator: | \( 65.5681790962 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 65.5681790962 \cdot 1}{2\cdot\sqrt{15177728}}\cr\approx \mathstrut & 1.32886074551 \end{aligned}\]
Galois group
| A non-solvable group of order 720 |
| The 11 conjugacy class representatives for $S_6$ |
| Character table for $S_6$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
| Twin sextic algebra: | 6.2.833605155903488.1 |
| Degree 6 sibling: | 6.2.833605155903488.1 |
| Degree 10 sibling: | deg 10 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 15 siblings: | deg 15, deg 15 |
| Degree 20 siblings: | deg 20, deg 20, deg 20 |
| Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
| Degree 36 sibling: | deg 36 |
| Degree 40 siblings: | deg 40, deg 40, deg 40 |
| Degree 45 sibling: | deg 45 |
| 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 | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.6.11a1.14 | $x^{6} + 4 x^{3} + 4 x + 10$ | $6$ | $1$ | $11$ | $S_4\times C_2$ | $$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$$ |
|
\(7411\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |