Normalized defining polynomial
\( x^{8} - 16x^{6} - 8x^{5} + 54x^{4} + 80x^{3} + 16x^{2} - 200x - 127 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[6, 1]$ |
| |
| Discriminant: |
\(-150307078144\)
\(\medspace = -\,2^{24}\cdot 17^{2}\cdot 31\)
|
| |
| Root discriminant: | \(24.95\) |
| |
| Galois root discriminant: | $2^{3}17^{1/2}31^{1/2}\approx 183.65184453198395$ | ||
| Ramified primes: |
\(2\), \(17\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{10}a^{5}-\frac{1}{10}a^{4}-\frac{1}{5}a^{3}-\frac{1}{10}a+\frac{3}{10}$, $\frac{1}{60}a^{6}+\frac{7}{60}a^{4}-\frac{1}{30}a^{3}+\frac{29}{60}a^{2}+\frac{11}{30}a+\frac{23}{60}$, $\frac{1}{300}a^{7}+\frac{1}{300}a^{6}-\frac{1}{60}a^{5}+\frac{47}{300}a^{4}-\frac{13}{100}a^{3}-\frac{33}{100}a^{2}-\frac{31}{100}a-\frac{133}{300}$
| 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: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{50}a^{7}+\frac{1}{50}a^{6}-\frac{3}{10}a^{5}-\frac{9}{25}a^{4}+\frac{31}{50}a^{3}+\frac{51}{50}a^{2}+\frac{67}{50}a-\frac{44}{25}$, $\frac{3}{100}a^{7}-\frac{11}{300}a^{6}-\frac{9}{20}a^{5}+\frac{73}{300}a^{4}+\frac{469}{300}a^{3}+\frac{329}{300}a^{2}-\frac{287}{300}a-\frac{1627}{300}$, $\frac{4}{75}a^{7}+\frac{4}{75}a^{6}-\frac{2}{3}a^{5}-\frac{82}{75}a^{4}+\frac{11}{50}a^{3}+\frac{161}{50}a^{2}+\frac{247}{50}a+\frac{181}{150}$, $\frac{1}{75}a^{7}+\frac{1}{75}a^{6}-\frac{4}{15}a^{5}-\frac{13}{75}a^{4}+\frac{69}{50}a^{3}+\frac{9}{50}a^{2}-\frac{77}{50}a-\frac{131}{150}$, $\frac{13}{300}a^{7}+\frac{2}{75}a^{6}-\frac{37}{60}a^{5}-\frac{17}{25}a^{4}+\frac{343}{300}a^{3}+\frac{242}{75}a^{2}+\frac{901}{300}a-\frac{326}{75}$, $\frac{1}{60}a^{7}+\frac{1}{30}a^{6}-\frac{11}{60}a^{5}-\frac{1}{2}a^{4}-\frac{29}{60}a^{3}+\frac{5}{6}a^{2}+\frac{35}{12}a+\frac{11}{30}$
|
| |
| Regulator: | \( 645.464274636 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 645.464274636 \cdot 2}{2\cdot\sqrt{150307078144}}\cr\approx \mathstrut & 0.669487289060 \end{aligned}\]
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
| \(\Q(\sqrt{2}) \), 4.4.34816.1 |
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: | 8.4.8565293056.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.61 | $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $$[2, 3, 4]$$ |
|
\(17\)
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 17.4.1.0a1.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(31\)
| 31.1.2.1a1.2 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |