Normalized defining polynomial
\( x^{4} - x^{3} + 2x^{2} + 8x + 16 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(24648\)
\(\medspace = 2^{3}\cdot 3\cdot 13\cdot 79\)
|
| |
| Root discriminant: | \(12.53\) |
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}13^{1/2}79^{1/2}\approx 156.9968152543229$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(79\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{6162}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$
| Monogenic: | No | |
| Index: | $2$ | |
| Inessential primes: | $2$ |
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: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{5527}{4}a^{3}+\frac{43869}{4}a^{2}+\frac{41613}{2}a+18207$
|
| |
| Regulator: | \( 23.510135379 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 23.510135379 \cdot 1}{2\cdot\sqrt{24648}}\cr\approx \mathstrut & 2.9559292044 \end{aligned}\]
Galois group
| A solvable group of order 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | deg 24 |
| Degree 6 siblings: | deg 6, 6.2.607523904.3 |
| Degree 8 sibling: | deg 8 |
| Degree 12 siblings: | deg 12, deg 12 |
| 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 | R | ${\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }$ | R | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(79\)
| 79.1.2.1a1.2 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 79.2.1.0a1.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |