Normalized defining polynomial
\( x^{8} - 2x^{7} + 2x^{6} + x^{5} - 5x^{4} + 8x^{3} - x^{2} - 10x + 7 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(336098889\)
\(\medspace = 3^{6}\cdot 7^{2}\cdot 97^{2}\)
|
| |
| Root discriminant: | \(11.64\) |
| |
| Galois root discriminant: | $3^{3/4}7^{2/3}97^{2/3}\approx 176.09787848408035$ | ||
| Ramified primes: |
\(3\), \(7\), \(97\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{83}a^{7}+\frac{10}{83}a^{6}+\frac{39}{83}a^{5}-\frac{29}{83}a^{4}-\frac{21}{83}a^{3}+\frac{5}{83}a^{2}-\frac{24}{83}a+\frac{34}{83}$
| 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: |
\( -\frac{22}{83} a^{7} + \frac{29}{83} a^{6} - \frac{28}{83} a^{5} - \frac{26}{83} a^{4} + \frac{47}{83} a^{3} - \frac{110}{83} a^{2} - \frac{53}{83} a + \frac{165}{83} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{7}{83}a^{7}-\frac{13}{83}a^{6}+\frac{24}{83}a^{5}-\frac{37}{83}a^{4}+\frac{19}{83}a^{3}+\frac{35}{83}a^{2}-\frac{85}{83}a+\frac{72}{83}$, $\frac{24}{83}a^{7}-\frac{9}{83}a^{6}+\frac{23}{83}a^{5}+\frac{51}{83}a^{4}-\frac{89}{83}a^{3}+\frac{37}{83}a^{2}+\frac{88}{83}a-\frac{97}{83}$, $\frac{2}{83}a^{7}+\frac{20}{83}a^{6}-\frac{5}{83}a^{5}+\frac{25}{83}a^{4}+\frac{41}{83}a^{3}-\frac{73}{83}a^{2}+\frac{118}{83}a-\frac{15}{83}$
|
| |
| Regulator: | \( 34.2133769715 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 34.2133769715 \cdot 1}{6\cdot\sqrt{336098889}}\cr\approx \mathstrut & 0.484764297823 \end{aligned}\]
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.4.6a1.2 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 64 x + 19$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 7.1.3.2a1.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
|
\(97\)
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{97}$ | $x + 92$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 97.3.1.0a1.1 | $x^{3} + 9 x + 92$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 97.1.3.2a1.2 | $x^{3} + 485$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |