Normalized defining polynomial
\( x^{8} - 2x^{7} - 4x^{6} + 12x^{5} - 8x^{4} - 16x^{3} + 16x^{2} + 12x + 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(4584914944\)
\(\medspace = 2^{14}\cdot 23^{4}\)
|
| |
| Root discriminant: | \(16.13\) |
| |
| Galois root discriminant: | $2^{7/4}23^{1/2}\approx 16.131190144457708$ | ||
| Ramified primes: |
\(2\), \(23\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{307}a^{7}+\frac{67}{307}a^{6}+\frac{14}{307}a^{5}+\frac{57}{307}a^{4}-\frac{66}{307}a^{3}+\frac{35}{307}a^{2}-\frac{25}{307}a+\frac{129}{307}$
| 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: |
$\frac{146}{307}a^{7}-\frac{349}{307}a^{6}-\frac{412}{307}a^{5}+\frac{1875}{307}a^{4}-\frac{1961}{307}a^{3}-\frac{1337}{307}a^{2}+\frac{2490}{307}a+\frac{721}{307}$, $\frac{36}{307}a^{7}-\frac{44}{307}a^{6}-\frac{110}{307}a^{5}+\frac{210}{307}a^{4}-\frac{227}{307}a^{3}-\frac{275}{307}a^{2}-\frac{286}{307}a+\frac{39}{307}$, $\frac{18}{307}a^{7}-\frac{22}{307}a^{6}-\frac{55}{307}a^{5}+\frac{105}{307}a^{4}-\frac{267}{307}a^{3}+\frac{16}{307}a^{2}+\frac{471}{307}a+\frac{173}{307}$, $\frac{79}{307}a^{7}-\frac{233}{307}a^{6}-\frac{122}{307}a^{5}+\frac{1126}{307}a^{4}-\frac{1530}{307}a^{3}-\frac{305}{307}a^{2}+\frac{1402}{307}a+\frac{367}{307}$, $\frac{207}{307}a^{7}-\frac{560}{307}a^{6}-\frac{479}{307}a^{5}+\frac{2896}{307}a^{4}-\frac{3531}{307}a^{3}-\frac{1351}{307}a^{2}+\frac{4649}{307}a-\frac{313}{307}$
|
| |
| Regulator: | \( 114.686019531 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 114.686019531 \cdot 1}{2\cdot\sqrt{4584914944}}\cr\approx \mathstrut & 0.534927052508 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{23}) \), 4.2.1472.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.2.778688.4, 6.0.33856.1 |
| Degree 8 sibling: | 8.0.8667136.1 |
| Degree 12 siblings: | 12.2.1687248699392.2, 12.0.73358639104.1, 12.0.606355001344.2, 12.4.38806720086016.2, 12.0.38806720086016.1, 12.0.38806720086016.3 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| Minimal sibling: | 6.0.33856.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.14a1.1 | $x^{8} + 2 x^{7} + 2$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(23\)
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.2.2.2a1.2 | $x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |