Normalized defining polynomial
\( x^{4} - 2x^{3} - 6x + 3 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(2, 1)$ |
| |
| Discriminant: |
\(-3888\)
\(\medspace = -\,2^{4}\cdot 3^{5}\)
|
| |
| Root discriminant: | \(7.90\) |
| |
| Galois root discriminant: | $2^{7/6}3^{11/6}\approx 16.82379477331321$ | ||
| Ramified primes: |
\(2\), \(3\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{2}a^{2}-a+\frac{1}{2}$, $\frac{1}{2}a^{3}-a^{2}-\frac{1}{2}a-1$
|
| |
| Regulator: | \( 4.4210602631 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 4.4210602631 \cdot 1}{2\cdot\sqrt{3888}}\cr\approx \mathstrut & 0.89099069938 \end{aligned}\]
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: | 6.2.3779136.3, 6.0.11337408.4 |
| Degree 8 sibling: | 8.0.136048896.1 |
| Degree 12 siblings: | 12.2.171382426877952.3, 12.0.128536820158464.9 |
| 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 | ${\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.4a1.1 | $x^{4} + 2 x + 2$ | $4$ | $1$ | $4$ | $S_4$ | $$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.3.5a1.3 | $x^{3} + 18 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *24 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.972.3t2.d.a | $2$ | $ 2^{2} \cdot 3^{5}$ | \(\Q(\sqrt[3]{6})\) | $S_3$ (as 3T2) | $1$ | $0$ | |
| 3.11664.6t8.b.a | $3$ | $ 2^{4} \cdot 3^{6}$ | 4.2.3888.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *24 | 3.3888.4t5.b.a | $3$ | $ 2^{4} \cdot 3^{5}$ | 4.2.3888.1 | $S_4$ (as 4T5) | $1$ | $1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.