Normalized defining polynomial
\( x^{8} + 2x^{6} - 63x^{4} + 202x^{2} - 171 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[6, 1]$ |
| |
| Discriminant: |
\(-990748082176\)
\(\medspace = -\,2^{20}\cdot 19\cdot 223^{2}\)
|
| |
| Root discriminant: | \(31.59\) |
| |
| Galois root discriminant: | $2^{3}19^{1/2}223^{1/2}\approx 520.7379379303951$ | ||
| Ramified primes: |
\(2\), \(19\), \(223\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13}a^{6}-\frac{2}{13}a^{4}-\frac{3}{13}a^{2}+\frac{6}{13}$, $\frac{1}{39}a^{7}+\frac{11}{39}a^{5}-\frac{1}{13}a^{3}+\frac{19}{39}a$
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{7}{13}a^{6}+\frac{38}{13}a^{4}-\frac{307}{13}a^{2}+\frac{341}{13}$, $\frac{2}{13}a^{6}+\frac{9}{13}a^{4}-\frac{97}{13}a^{2}+\frac{142}{13}$, $\frac{5}{13}a^{6}+\frac{29}{13}a^{4}-\frac{210}{13}a^{2}+\frac{199}{13}$, $\frac{1}{39}a^{7}-\frac{5}{13}a^{6}+\frac{11}{39}a^{5}-\frac{29}{13}a^{4}-\frac{1}{13}a^{3}+\frac{210}{13}a^{2}-\frac{137}{39}a-\frac{186}{13}$, $\frac{30}{13}a^{7}-\frac{55}{13}a^{6}+\frac{161}{13}a^{5}-\frac{293}{13}a^{4}-\frac{1351}{13}a^{3}+\frac{2492}{13}a^{2}+\frac{1493}{13}a-\frac{2813}{13}$, $\frac{22}{13}a^{7}-\frac{21}{13}a^{6}+\frac{99}{13}a^{5}-\frac{88}{13}a^{4}-\frac{1119}{13}a^{3}+\frac{1116}{13}a^{2}+\frac{1783}{13}a-\frac{1894}{13}$
|
| |
| Regulator: | \( 3223.64404143 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 3223.64404143 \cdot 1}{2\cdot\sqrt{990748082176}}\cr\approx \mathstrut & 0.651171383909 \end{aligned}\]
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
| 4.4.14272.1 |
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: | 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.20a1.7 | $x^{8} + 4 x^{7} + 10 x^{6} + 20 x^{5} + 27 x^{4} + 28 x^{3} + 18 x^{2} + 8 x + 3$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, 3, \frac{7}{2}]^{4}$$ |
|
\(19\)
| 19.1.2.1a1.1 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 19.2.1.0a1.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 19.4.1.0a1.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(223\)
| $\Q_{223}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{223}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |