Normalized defining polynomial
\( x^{8} - 2x^{7} + 4x^{6} - 16x^{4} + 4x^{3} + 12x^{2} - 2 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(4, 2)$ |
| |
| Discriminant: |
\(702038016\)
\(\medspace = 2^{14}\cdot 3^{4}\cdot 23^{2}\)
|
| |
| Root discriminant: | \(12.76\) |
| |
| Galois root discriminant: | $2^{7/4}3^{1/2}23^{1/2}\approx 27.940040916755088$ | ||
| Ramified primes: |
\(2\), \(3\), \(23\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{157}a^{7}-\frac{48}{157}a^{6}+\frac{14}{157}a^{5}-\frac{16}{157}a^{4}-\frac{65}{157}a^{3}+\frac{11}{157}a^{2}-\frac{23}{157}a-\frac{41}{157}$
| 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: |
$\frac{120}{157}a^{7}-\frac{265}{157}a^{6}+\frac{581}{157}a^{5}-\frac{193}{157}a^{4}-\frac{1677}{157}a^{3}+\frac{849}{157}a^{2}+\frac{694}{157}a-\frac{53}{157}$, $\frac{6}{157}a^{7}+\frac{26}{157}a^{6}-\frac{73}{157}a^{5}+\frac{218}{157}a^{4}-\frac{233}{157}a^{3}-\frac{405}{157}a^{2}+\frac{333}{157}a+\frac{225}{157}$, $\frac{25}{157}a^{7}-\frac{101}{157}a^{6}+\frac{193}{157}a^{5}-\frac{243}{157}a^{4}-\frac{369}{157}a^{3}+\frac{746}{157}a^{2}+\frac{210}{157}a-\frac{83}{157}$, $\frac{139}{157}a^{7}-\frac{392}{157}a^{6}+\frac{847}{157}a^{5}-\frac{654}{157}a^{4}-\frac{1813}{157}a^{3}+\frac{2000}{157}a^{2}+\frac{414}{157}a-\frac{361}{157}$, $\frac{13}{157}a^{7}+\frac{4}{157}a^{6}+\frac{25}{157}a^{5}+\frac{106}{157}a^{4}-\frac{60}{157}a^{3}-\frac{328}{157}a^{2}+\frac{15}{157}a+\frac{95}{157}$
|
| |
| Regulator: | \( 48.2532110622 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 48.2532110622 \cdot 1}{2\cdot\sqrt{702038016}}\cr\approx \mathstrut & 0.57516920805 \end{aligned}\]
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
| \(\Q(\sqrt{3}) \), 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: | 6.0.21024576.3, 6.2.914112.2 |
| Degree 8 sibling: | 8.0.371378110464.1 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| Minimal sibling: | 6.2.914112.2 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.14a1.2 | $x^{8} + 2 x^{7} + 6$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.3.2.3a1.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(23\)
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 23.1.2.1a1.2 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |