Normalized defining polynomial
\( x^{4} - 2x^{3} - 6x^{2} - 12x - 40 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(2, 1)$ |
| |
| Discriminant: |
\(-482296\)
\(\medspace = -\,2^{3}\cdot 19^{2}\cdot 167\)
|
| |
| Root discriminant: | \(26.35\) |
| |
| Galois root discriminant: | $2^{3/2}19^{1/2}167^{1/2}\approx 159.32357013323548$ | ||
| Ramified primes: |
\(2\), \(19\), \(167\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-334}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}-\frac{1}{2}a$
| Monogenic: | Yes | |
| 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: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{2}-2a-9$, $a^{3}-3a^{2}-7a+3$
|
| |
| Regulator: | \( 47.8495263175 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 47.8495263175 \cdot 1}{2\cdot\sqrt{482296}}\cr\approx \mathstrut & 0.865826107084 \end{aligned}\]
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: | 6.2.644347456.4, deg 6 |
| 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 | ${\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }$ | R | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(19\)
| 19.2.2.2a1.1 | $x^{4} + 36 x^{3} + 328 x^{2} + 91 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
|
\(167\)
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 167.1.2.1a1.1 | $x^{2} + 167$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |