Normalized defining polynomial
\( x^{8} + 16x^{6} + 96x^{4} + 384x^{2} + 960 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(412782428160\)
\(\medspace = 2^{22}\cdot 3^{9}\cdot 5\)
|
| |
| Root discriminant: | \(28.31\) |
| |
| Galois root discriminant: | $2^{3}3^{3/2}5^{1/2}\approx 92.951600308978$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{15}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{16}a^{4}-\frac{1}{2}$, $\frac{1}{16}a^{5}-\frac{1}{2}a$, $\frac{1}{32}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{64}a^{7}-\frac{1}{8}a^{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{32}a^{6}+\frac{5}{16}a^{4}+\frac{5}{4}a^{2}+\frac{11}{2}$, $\frac{1}{16}a^{6}+\frac{11}{16}a^{4}+\frac{5}{2}a^{2}+\frac{19}{2}$, $\frac{15}{32}a^{7}+\frac{19}{16}a^{6}+\frac{17}{2}a^{5}+\frac{65}{4}a^{4}+\frac{197}{4}a^{3}+\frac{103}{2}a^{2}+92a-11$
|
| |
| Regulator: | \( 208.556086735 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 208.556086735 \cdot 2}{2\cdot\sqrt{412782428160}}\cr\approx \mathstrut & 0.505919876256 \end{aligned}\]
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
| 4.2.5184.1 |
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: | 8.4.412782428160.1 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.76 | $x^{8} + 12 x^{7} + 38 x^{6} + 80 x^{5} + 111 x^{4} + 120 x^{3} + 86 x^{2} + 44 x + 11$ | $4$ | $2$ | $22$ | $D_{8}$ | $$[2, 3, 4]^{2}$$ |
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.1.6.9a2.3 | $x^{6} + 6 x^{4} + 6$ | $6$ | $1$ | $9$ | $D_{6}$ | $$[2]_{2}^{2}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.3.1.0a1.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 5.3.1.0a1.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |