Normalized defining polynomial
\( x^{8} - 40x^{6} + 440x^{4} - 6080x^{2} + 115520 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(118292480000000\)
\(\medspace = 2^{22}\cdot 5^{7}\cdot 19^{2}\)
|
| |
| Root discriminant: | \(57.43\) |
| |
| Galois root discriminant: | $2^{11/4}5^{7/8}19^{1/2}\approx 119.89692430103565$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 4.0.8000.2 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{8}a^{4}$, $\frac{1}{8}a^{5}$, $\frac{1}{42256}a^{6}-\frac{457}{21128}a^{4}+\frac{436}{2641}a^{2}-\frac{60}{139}$, $\frac{1}{84512}a^{7}+\frac{273}{5282}a^{5}+\frac{218}{2641}a^{3}-\frac{30}{139}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
$\frac{3}{21128}a^{6}-\frac{101}{21128}a^{4}-\frac{25}{2641}a^{2}-\frac{82}{139}$, $\frac{791}{42256}a^{7}+\frac{1451}{21128}a^{6}-\frac{5117}{10564}a^{5}-\frac{8031}{5282}a^{4}+\frac{21579}{5282}a^{3}+\frac{37207}{2641}a^{2}-\frac{13822}{139}a-\frac{72511}{139}$, $\frac{34336453}{42256}a^{7}+\frac{37209853}{10564}a^{6}-\frac{144666729}{10564}a^{5}-\frac{263772845}{5282}a^{4}+\frac{80217872}{2641}a^{3}+\frac{228393485}{2641}a^{2}-\frac{455920070}{139}a-\frac{2541998251}{139}$
|
| |
| Regulator: | \( 1459.94767878 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1459.94767878 \cdot 4}{2\cdot\sqrt{118292480000000}}\cr\approx \mathstrut & 0.418415912553 \end{aligned}\]
Galois group
$\OD_{16}:C_2$ (as 8T16):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $(C_8:C_2):C_2$ |
| Character table for $(C_8:C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.8000.2 |
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: | 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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | R | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(5\)
| 5.1.8.7a1.4 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.1.2.1a1.2 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |