Normalized defining polynomial
\( x^{8} - 4x^{6} - 8x^{5} + 20x^{3} + 28x^{2} + 20x + 5 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-1744568320\)
\(\medspace = -\,2^{18}\cdot 5\cdot 11^{3}\)
|
| |
| Root discriminant: | \(14.30\) |
| |
| Galois root discriminant: | $2^{7/3}5^{1/2}11^{3/4}\approx 68.06640903889887$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-55}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{61}a^{7}+\frac{12}{61}a^{6}+\frac{18}{61}a^{5}+\frac{25}{61}a^{4}-\frac{5}{61}a^{3}+\frac{21}{61}a^{2}-\frac{25}{61}a+\frac{25}{61}$
| Monogenic: | Not computed | |
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{16}{61}a^{7}+\frac{9}{61}a^{6}-\frac{78}{61}a^{5}-\frac{149}{61}a^{4}-\frac{19}{61}a^{3}+\frac{397}{61}a^{2}+\frac{515}{61}a+\frac{278}{61}$, $\frac{9}{61}a^{7}-\frac{14}{61}a^{6}-\frac{21}{61}a^{5}-\frac{19}{61}a^{4}+\frac{77}{61}a^{3}+\frac{67}{61}a^{2}+\frac{19}{61}a-\frac{19}{61}$, $\frac{2}{61}a^{7}-\frac{37}{61}a^{6}+\frac{36}{61}a^{5}+\frac{111}{61}a^{4}+\frac{112}{61}a^{3}-\frac{202}{61}a^{2}-\frac{477}{61}a-\frac{194}{61}$, $\frac{34}{61}a^{7}-\frac{19}{61}a^{6}-\frac{120}{61}a^{5}-\frac{187}{61}a^{4}+\frac{13}{61}a^{3}+\frac{653}{61}a^{2}+\frac{675}{61}a+\frac{484}{61}$
|
| |
| Regulator: | \( 48.3981188033 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 48.3981188033 \cdot 1}{2\cdot\sqrt{1744568320}}\cr\approx \mathstrut & 0.574849411574 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
| A solvable group of order 384 |
| The 20 conjugacy class representatives for $C_2 \wr S_4$ |
| Character table for $C_2 \wr S_4$ |
Intermediate fields
| 4.2.2816.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18c1.4 | $x^{8} + 2 x^{4} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $S_4\times C_2$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.6.1.0a1.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(11\)
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 11.2.1.0a1.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 11.1.4.3a1.2 | $x^{4} + 22$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |