Normalized defining polynomial
\( x^{8} - 2x^{7} - 8x^{5} + 20x^{4} - 4x^{3} + 8x^{2} - 36x + 22 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(2108178432\)
\(\medspace = 2^{14}\cdot 3^{2}\cdot 17\cdot 29^{2}\)
|
| |
| Root discriminant: | \(14.64\) |
| |
| Galois root discriminant: | $2^{7/4}3^{1/2}17^{1/2}29^{1/2}\approx 129.35600051794452$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{587}a^{7}+\frac{123}{587}a^{6}+\frac{113}{587}a^{5}+\frac{29}{587}a^{4}+\frac{123}{587}a^{3}+\frac{109}{587}a^{2}+\frac{132}{587}a+\frac{28}{587}$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{57}{587}a^{7}-\frac{33}{587}a^{6}-\frac{16}{587}a^{5}-\frac{695}{587}a^{4}+\frac{554}{587}a^{3}+\frac{343}{587}a^{2}+\frac{1654}{587}a-\frac{1339}{587}$, $a-1$, $\frac{62}{587}a^{7}-\frac{5}{587}a^{6}-\frac{38}{587}a^{5}-\frac{550}{587}a^{4}-\frac{5}{587}a^{3}+\frac{301}{587}a^{2}+\frac{1140}{587}a-\frac{25}{587}$
|
| |
| Regulator: | \( 19.9002997912 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 19.9002997912 \cdot 2}{2\cdot\sqrt{2108178432}}\cr\approx \mathstrut & 0.67550008722 \end{aligned}\]
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
| 4.2.5568.1 |
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 | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | R | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.14a1.2 | $x^{8} + 2 x^{7} + 6$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(3\)
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(17\)
| 17.1.2.1a1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 17.6.1.0a1.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 29.2.1.0a1.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 29.2.2.2a1.2 | $x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |