Normalized defining polynomial
\( x^{8} + 4x^{6} + 16x^{4} - 72x^{2} + 36 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(326488817664\)
\(\medspace = 2^{22}\cdot 3^{4}\cdot 31^{2}\)
|
| |
| Root discriminant: | \(27.49\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}31^{1/2}\approx 64.87449083902186$ | ||
| Ramified primes: |
\(2\), \(3\), \(31\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a$, $\frac{1}{24}a^{6}+\frac{1}{24}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}-\frac{1}{4}$, $\frac{1}{48}a^{7}-\frac{1}{24}a^{5}+\frac{11}{24}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
| 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: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{12}a^{7}-\frac{1}{8}a^{6}+\frac{1}{3}a^{5}-\frac{5}{8}a^{4}+\frac{4}{3}a^{3}-\frac{5}{2}a^{2}-6a+\frac{23}{4}$, $\frac{1}{48}a^{7}+\frac{1}{24}a^{6}+\frac{1}{12}a^{5}+\frac{1}{24}a^{4}+\frac{5}{24}a^{3}+\frac{2}{3}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{48}a^{7}-\frac{1}{24}a^{6}+\frac{1}{12}a^{5}-\frac{1}{24}a^{4}+\frac{5}{24}a^{3}-\frac{2}{3}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{11}{48}a^{7}+\frac{5}{24}a^{6}+\frac{25}{24}a^{5}+\frac{5}{6}a^{4}+\frac{97}{24}a^{3}+\frac{43}{12}a^{2}-\frac{27}{2}a-\frac{23}{2}$, $\frac{11}{48}a^{7}-\frac{5}{24}a^{6}+\frac{25}{24}a^{5}-\frac{5}{6}a^{4}+\frac{97}{24}a^{3}-\frac{43}{12}a^{2}-\frac{27}{2}a+\frac{23}{2}$
|
| |
| Regulator: | \( 1319.19249096 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1319.19249096 \cdot 2}{2\cdot\sqrt{326488817664}}\cr\approx \mathstrut & 1.45832303032 \end{aligned}\]
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
| \(\Q(\sqrt{6}) \), 4.2.47616.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.0.1098215424.2, 6.2.212557824.1 |
| Degree 8 sibling: | 8.0.2178859401216.5 |
| 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: | 6.2.212557824.1 |
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 | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.22d1.22 | $x^{8} + 4 x^{7} + 2 x^{4} + 4 x^{2} + 14$ | $8$ | $1$ | $22$ | $D_4$ | $$[2, 3, \frac{7}{2}]$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(31\)
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.2.2a1.2 | $x^{4} + 58 x^{3} + 847 x^{2} + 174 x + 40$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |