Normalized defining polynomial
\( x^{4} - 2x^{3} - 74x^{2} + 75x + 4464 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $(0, 2)$ |
| |
| Discriminant: |
\(65416069\)
\(\medspace = 19\cdot 151^{3}\)
|
| |
| Root discriminant: | \(89.93\) |
| |
| Galois root discriminant: | $19^{1/2}151^{3/4}\approx 187.76278180736003$ | ||
| Ramified primes: |
\(19\), \(151\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2869}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-151}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{9}a^{2}-\frac{1}{9}a+\frac{1}{3}$, $\frac{1}{45}a^{3}+\frac{1}{45}a^{2}+\frac{19}{45}a-\frac{1}{15}$
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{7}$, which has order $7$ |
| |
| Narrow class group: | $C_{7}$, which has order $7$ |
|
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{35\cdots 95}{9}a^{3}+28\!\cdots\!42a^{2}-\frac{86\cdots 77}{9}a-\frac{44\cdots 29}{3}$
|
| |
| Regulator: | \( 99.3926924016 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 99.3926924016 \cdot 7}{2\cdot\sqrt{65416069}}\cr\approx \mathstrut & 1.69800910554 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{-151}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | 8.0.1544813612112006721.1 |
| Degree 4 sibling: | \(\Q(\sqrt{302 +10 \sqrt{2869}})\) |
| 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 | ${\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(151\)
| 151.1.4.3a1.1 | $x^{4} + 151$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |