Normalized defining polynomial
\( x^{6} - 4x^{4} - 11x^{3} - 3x^{2} + 8x + 4 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(4, 1)$ |
| |
| Discriminant: |
\(-5809048\)
\(\medspace = -\,2^{3}\cdot 7^{3}\cdot 29\cdot 73\)
|
| |
| Root discriminant: | \(13.41\) |
| |
| Galois root discriminant: | $2^{3/2}7^{1/2}29^{1/2}73^{1/2}\approx 344.31381035328803$ | ||
| Ramified primes: |
\(2\), \(7\), \(29\), \(73\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-29638}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{3}{8}a^{2}+\frac{3}{8}a+\frac{1}{4}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a+1$, $\frac{5}{8}a^{5}-\frac{1}{4}a^{4}-2a^{3}-\frac{47}{8}a^{2}-\frac{9}{8}a+\frac{5}{4}$, $\frac{7}{8}a^{5}-\frac{3}{4}a^{4}-3a^{3}-\frac{53}{8}a^{2}+\frac{29}{8}a+\frac{15}{4}$, $\frac{17}{8}a^{5}-\frac{5}{4}a^{4}-8a^{3}-\frac{147}{8}a^{2}+\frac{43}{8}a+\frac{57}{4}$
|
| |
| Regulator: | \( 29.264088283 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 29.264088283 \cdot 1}{2\cdot\sqrt{5809048}}\cr\approx \mathstrut & 0.61031264918 \end{aligned}\]
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: | 6.0.34004044436992.1 |
| Degree 6 sibling: | 6.0.34004044436992.1 |
| 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 | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | R | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(7\)
| 7.3.2.3a1.1 | $x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 7 x + 16$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 29.3.1.0a1.1 | $x^{3} + 2 x + 27$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 73.1.2.1a1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |