Normalized defining polynomial
\( x^{8} - 2x^{7} + 4x^{6} + 6x^{5} - 16x^{4} + 4x^{3} + 12x^{2} - 12x + 4 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(366799104\)
\(\medspace = 2^{8}\cdot 3^{4}\cdot 7^{2}\cdot 19^{2}\)
|
| |
| Root discriminant: | \(11.76\) |
| |
| Galois root discriminant: | $2^{31/24}3^{1/2}7^{2/3}19^{1/2}\approx 67.63422722739293$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(19\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{12}a^{7}-\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{5}{12} a^{7} + \frac{1}{2} a^{6} - \frac{7}{6} a^{5} - \frac{23}{6} a^{4} + \frac{9}{2} a^{3} + \frac{4}{3} a^{2} - \frac{16}{3} a + \frac{10}{3} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{12}a^{7}-\frac{1}{6}a^{5}+\frac{5}{3}a^{4}-\frac{3}{2}a^{3}-\frac{5}{3}a^{2}+\frac{11}{3}a-\frac{5}{3}$, $\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}+\frac{4}{3}a^{4}-7a^{3}-\frac{7}{3}a^{2}+\frac{16}{3}a-\frac{13}{3}$, $\frac{11}{12}a^{7}-a^{6}+\frac{8}{3}a^{5}+\frac{22}{3}a^{4}-\frac{13}{2}a^{3}-\frac{22}{3}a^{2}+\frac{16}{3}a+\frac{2}{3}$
|
| |
| Regulator: | \( 45.9624058449 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 45.9624058449 \cdot 1}{6\cdot\sqrt{366799104}}\cr\approx \mathstrut & 0.623385710817 \end{aligned}\]
Galois group
$A_4\wr C_2$ (as 8T42):
| A solvable group of order 288 |
| The 14 conjugacy class representatives for $A_4\wr C_2$ |
| Character table for $A_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 36 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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.8a2.1 | $x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 18 x^{3} + 12 x^{2} + 6 x + 3$ | $4$ | $2$ | $8$ | $A_4\wr C_2$ | $$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{6}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.1.3.2a1.2 | $x^{3} + 14$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 19.3.1.0a1.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |