Normalized defining polynomial
\( x^{8} + 20x^{4} + 40x^{2} + 20 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(20480000000\)
\(\medspace = 2^{18}\cdot 5^{7}\)
|
| |
| Root discriminant: | \(19.45\) |
| |
| Galois root discriminant: | $2^{9/4}5^{7/8}\approx 19.449849449321462$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-5}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{12}a^{6}+\frac{1}{12}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}$, $\frac{1}{12}a^{7}+\frac{1}{12}a^{5}-\frac{1}{2}a^{3}-\frac{1}{6}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{12}a^{6}-\frac{1}{6}a^{4}+\frac{3}{2}a^{2}+\frac{4}{3}$, $\frac{9}{4}a^{7}+\frac{3}{4}a^{6}-2a^{5}-a^{4}+\frac{93}{2}a^{3}+\frac{33}{2}a^{2}+49a+7$, $\frac{1}{3}a^{7}-\frac{1}{4}a^{6}-\frac{1}{6}a^{5}+\frac{1}{2}a^{4}+7a^{3}-\frac{13}{2}a^{2}+\frac{16}{3}a-8$
|
| |
| Regulator: | \( 144.110858989 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 144.110858989 \cdot 4}{2\cdot\sqrt{20480000000}}\cr\approx \mathstrut & 3.13892688494 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 8T17):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), 4.0.8000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 16 siblings: | 16.4.419430400000000000000.4, 16.0.419430400000000000000.2 |
| Minimal sibling: | 8.0.20480000000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18b1.10 | $x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $C_4\wr C_2$ | $$[2, 2, 3]^{4}$$ |
|
\(5\)
| 5.1.8.7a1.4 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |