Normalized defining polynomial
\( x^{8} - 2x^{7} - 4x^{6} + 4x^{5} + 8x^{4} + 4x^{3} - 4x^{2} - 4x - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(2, 3)$ |
| |
| Discriminant: |
\(-754040832\)
\(\medspace = -\,2^{14}\cdot 3\cdot 23^{2}\cdot 29\)
|
| |
| Root discriminant: | \(12.87\) |
| |
| Galois root discriminant: | $2^{7/4}3^{1/2}23^{1/2}29^{1/2}\approx 150.46172505480757$ | ||
| Ramified primes: |
\(2\), \(3\), \(23\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-87}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$
| Monogenic: | Yes | |
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{7}-2a^{6}-4a^{5}+5a^{4}+7a^{3}+a^{2}-4a-3$, $a^{5}-a^{4}-3a^{3}+a+1$, $a^{7}-3a^{6}-2a^{5}+8a^{4}+3a^{3}-3a^{2}-4a+1$, $a^{4}-2a^{3}-a^{2}+a+1$
|
| |
| Regulator: | \( 37.0927109312 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 37.0927109312 \cdot 1}{2\cdot\sqrt{754040832}}\cr\approx \mathstrut & 0.67013259718 \end{aligned}\]
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
| 4.2.1472.1 |
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 | R | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.14a1.2 | $x^{8} + 2 x^{7} + 6$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.6.1.0a1.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(23\)
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.4.1.0a1.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(29\)
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 29.6.1.0a1.1 | $x^{6} + x^{4} + 25 x^{3} + 17 x^{2} + 13 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |