Normalized defining polynomial
\( x^{4} - 3x^{2} - 12x + 15 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[2, 1]$ |
| |
| Discriminant: |
\(-34128\)
\(\medspace = -\,2^{4}\cdot 3^{3}\cdot 79\)
|
| |
| Root discriminant: | \(13.59\) |
| |
| Galois root discriminant: | $2^{3/2}3^{3/4}79^{1/2}\approx 57.30591881429519$ | ||
| Ramified primes: |
\(2\), \(3\), \(79\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-237}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$
| Monogenic: | No | |
| 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: |
$a-1$, $\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{4}{5}a-1$
|
| |
| Regulator: | \( 11.5611438529 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 11.5611438529 \cdot 1}{2\cdot\sqrt{34128}}\cr\approx \mathstrut & 0.786421267945 \end{aligned}\]
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: | 6.2.32353344.1, 6.0.3407885568.3 |
| Degree 8 sibling: | 8.0.116304318664704.6 |
| 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 | ${\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a2.1 | $x^{4} + 4 x^{3} + 5 x^{2} + 4 x + 3$ | $2$ | $2$ | $4$ | $D_{4}$ | $$[2, 2]^{2}$$ |
|
\(3\)
| 3.1.4.3a1.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
|
\(79\)
| 79.2.1.0a1.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 79.1.2.1a1.1 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *24 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.948.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 79 $ | \(\Q(\sqrt{-237}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.948.3t2.a.a | $2$ | $ 2^{2} \cdot 3 \cdot 79 $ | 3.1.948.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
| 3.3594816.6t8.a.a | $3$ | $ 2^{6} \cdot 3^{2} \cdot 79^{2}$ | 4.2.34128.2 | $S_4$ (as 4T5) | $1$ | $-1$ | |
| *24 | 3.34128.4t5.a.a | $3$ | $ 2^{4} \cdot 3^{3} \cdot 79 $ | 4.2.34128.2 | $S_4$ (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 *.