Normalized defining polynomial
\( x^{8} + 4x^{6} + 10x^{4} - 36x^{2} + 9 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(13271040000\)
\(\medspace = 2^{18}\cdot 3^{4}\cdot 5^{4}\)
|
| |
| Root discriminant: | \(18.42\) |
| |
| Galois root discriminant: | $2^{7/3}3^{1/2}5^{2/3}\approx 25.523718929317052$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{3}{8}a+\frac{1}{8}$, $\frac{1}{24}a^{6}+\frac{1}{24}a^{4}-\frac{1}{2}a^{3}+\frac{1}{24}a^{2}-\frac{1}{2}a-\frac{1}{8}$, $\frac{1}{24}a^{7}+\frac{1}{24}a^{5}+\frac{1}{24}a^{3}-\frac{1}{2}a^{2}-\frac{1}{8}a-\frac{1}{2}$
| 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: |
$\frac{1}{24}a^{7}-\frac{1}{24}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{4}+\frac{7}{24}a^{3}-\frac{19}{24}a^{2}-\frac{9}{4}a+\frac{3}{4}$, $\frac{1}{12}a^{7}+\frac{1}{3}a^{5}+\frac{7}{12}a^{3}-\frac{7}{2}a-2$, $\frac{7}{24}a^{7}-\frac{1}{6}a^{6}+\frac{31}{24}a^{5}-\frac{2}{3}a^{4}+\frac{79}{24}a^{3}-\frac{5}{3}a^{2}-\frac{71}{8}a+\frac{9}{2}$, $\frac{7}{12}a^{7}+\frac{5}{24}a^{6}+\frac{31}{12}a^{5}+\frac{23}{24}a^{4}+\frac{85}{12}a^{3}+\frac{65}{24}a^{2}-\frac{69}{4}a-\frac{43}{8}$, $\frac{7}{24}a^{7}+\frac{1}{12}a^{6}+\frac{31}{24}a^{5}+\frac{1}{3}a^{4}+\frac{79}{24}a^{3}+\frac{13}{12}a^{2}-\frac{79}{8}a-3$
|
| |
| Regulator: | \( 348.394041572 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 348.394041572 \cdot 1}{2\cdot\sqrt{13271040000}}\cr\approx \mathstrut & 0.955142046115 \end{aligned}\]
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
| \(\Q(\sqrt{3}) \), 4.2.19200.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.0.5760000.1, 6.2.17280000.1 |
| Degree 8 sibling: | 8.0.1474560000.7 |
| 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: | 6.0.5760000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18c1.4 | $x^{8} + 2 x^{4} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $S_4\times C_2$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |