Normalized defining polynomial
\( x^{8} - 8x^{6} - 24x^{4} - 240x^{2} + 380 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(4, 2)$ |
| |
| Discriminant: |
\(249036800000\)
\(\medspace = 2^{22}\cdot 5^{5}\cdot 19\)
|
| |
| Root discriminant: | \(26.58\) |
| |
| Galois root discriminant: | $2^{3}5^{3/4}19^{1/2}\approx 116.59885635459175$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{95}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{3916}a^{6}-\frac{53}{3916}a^{4}+\frac{691}{1958}a^{2}+\frac{113}{1958}$, $\frac{1}{3916}a^{7}-\frac{53}{3916}a^{5}+\frac{691}{1958}a^{3}+\frac{113}{1958}a$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{178}a^{6}-\frac{17}{356}a^{4}-\frac{21}{89}a^{2}-\frac{219}{178}$, $\frac{41}{3916}a^{6}-\frac{215}{3916}a^{4}-\frac{1039}{1958}a^{2}-\frac{5157}{1958}$, $\frac{23}{3916}a^{7}-\frac{1}{178}a^{6}-\frac{60}{979}a^{5}+\frac{17}{356}a^{4}+\frac{229}{1958}a^{3}+\frac{21}{89}a^{2}-\frac{2127}{979}a-\frac{493}{178}$, $\frac{1}{178}a^{7}-\frac{5}{979}a^{6}-\frac{17}{356}a^{5}+\frac{81}{3916}a^{4}-\frac{21}{89}a^{3}-\frac{57}{979}a^{2}-\frac{219}{178}a+\frac{677}{1958}$, $\frac{261}{1958}a^{7}-\frac{35}{1958}a^{6}-\frac{2085}{1958}a^{5}-\frac{541}{979}a^{4}-\frac{3701}{979}a^{3}+\frac{7143}{979}a^{2}-\frac{19457}{979}a+\frac{21499}{979}$
|
| |
| Regulator: | \( 734.553092357 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 734.553092357 \cdot 1}{2\cdot\sqrt{249036800000}}\cr\approx \mathstrut & 0.464880309405 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.1600.1 |
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 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 | ${\href{/padicField/3.8.0.1}{8} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.76 | $x^{8} + 12 x^{7} + 38 x^{6} + 80 x^{5} + 111 x^{4} + 120 x^{3} + 86 x^{2} + 44 x + 11$ | $4$ | $2$ | $22$ | $D_{8}$ | $$[2, 3, 4]^{2}$$ |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 19.4.1.0a1.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |