Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{5} + 12x^{4} - 16x^{2} - 16x - 8 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-2252079104\)
\(\medspace = -\,2^{18}\cdot 11^{2}\cdot 71\)
|
| |
| Root discriminant: | \(14.76\) |
| |
| Galois root discriminant: | $2^{7/3}11^{1/2}71^{1/2}\approx 140.84091573614046$ | ||
| Ramified primes: |
\(2\), \(11\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-71}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$
| 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: |
$\frac{1}{4}a^{7}-\frac{5}{4}a^{6}+a^{5}+2a^{4}+a^{3}-3a^{2}-3a-1$, $\frac{1}{4}a^{7}-\frac{5}{4}a^{6}+a^{5}+2a^{4}+\frac{1}{2}a^{3}-2a^{2}-3a-1$, $\frac{1}{2}a^{6}-\frac{5}{2}a^{5}+2a^{4}+\frac{9}{2}a^{3}-5a-5$, $\frac{1}{4}a^{7}-\frac{5}{4}a^{6}+a^{5}+2a^{4}+\frac{1}{2}a^{3}-2a^{2}-2a-1$
|
| |
| Regulator: | \( 51.0755845182 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 51.0755845182 \cdot 1}{2\cdot\sqrt{2252079104}}\cr\approx \mathstrut & 0.533938363084 \end{aligned}\]
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $x^{8} + 2 x^{4} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $S_4\times C_2$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
|
\(11\)
| 11.1.2.1a1.2 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 11.1.2.1a1.2 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 11.4.1.0a1.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(71\)
| 71.1.2.1a1.1 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 71.3.1.0a1.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 71.3.1.0a1.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |