Normalized defining polynomial
\( x^{4} - x^{3} + 391x^{2} + 8073x + 73008 \)
Invariants
| Degree: | $4$ |
| |
| Signature: | $[0, 2]$ |
| |
| Discriminant: |
\(17698520889\)
\(\medspace = 3^{2}\cdot 7\cdot 167\cdot 1297^{2}\)
|
| |
| Root discriminant: | \(364.74\) |
| |
| Galois root discriminant: | $3^{1/2}7^{1/2}167^{1/2}1297^{1/2}\approx 2132.7397872220604$ | ||
| Ramified primes: |
\(3\), \(7\), \(167\), \(1297\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{1169}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | 4.0.15951866553.1$^{2}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{1560}a^{3}+\frac{34}{195}a^{2}-\frac{233}{1560}a+\frac{2}{5}$
| Monogenic: | No | |
| Index: | $4$ | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{22}\times C_{264}$, which has order $11616$ |
| |
| Narrow class group: | $C_{2}\times C_{22}\times C_{264}$, which has order $11616$ |
| |
| Relative class number: | $1056$ |
Unit group
| Rank: | $1$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental unit: |
$\frac{1}{260}a^{3}+\frac{3}{65}a^{2}+\frac{27}{260}a+\frac{347}{5}$
|
| |
| Regulator: | \( 8.553717928916416 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 8.553717928916416 \cdot 11616}{2\cdot\sqrt{17698520889}}\cr\approx \mathstrut & 14.7425558834434 \end{aligned}\]
Galois group
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $D_{4}$ |
| Character table for $D_{4}$ |
Intermediate fields
| \(\Q(\sqrt{1297}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(167\)
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 167.1.2.1a1.2 | $x^{2} + 835$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(1297\)
| Deg $4$ | $2$ | $2$ | $2$ |