Normalized defining polynomial
\( x^{8} - 4x^{7} - 6x^{6} + 16x^{5} + 41x^{4} + 24x^{3} - 26x^{2} + 44x - 9 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-51597803520000\)
\(\medspace = -\,2^{22}\cdot 3^{9}\cdot 5^{4}\)
|
| |
| Root discriminant: | \(51.77\) |
| |
| Galois root discriminant: | $2^{11/4}3^{3/2}5^{1/2}\approx 78.16266749191567$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{4}-\frac{1}{3}a^{3}-\frac{1}{9}a^{2}+\frac{4}{9}a$, $\frac{1}{9}a^{6}-\frac{1}{9}a^{4}-\frac{4}{9}a^{3}+\frac{1}{3}a^{2}+\frac{1}{9}a$, $\frac{1}{27}a^{7}+\frac{1}{27}a^{4}-\frac{1}{3}a^{2}-\frac{2}{27}a+\frac{1}{3}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}$, which has order $4$ |
| |
| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{27}a^{7}-\frac{2}{9}a^{6}+\frac{1}{9}a^{5}+\frac{22}{27}a^{4}+\frac{5}{9}a^{3}-\frac{19}{9}a^{2}-\frac{14}{27}a-\frac{2}{3}$, $\frac{1}{9}a^{7}-\frac{1}{3}a^{6}-\frac{10}{9}a^{5}+\frac{8}{9}a^{4}+\frac{23}{3}a^{3}+\frac{55}{9}a^{2}-\frac{13}{3}a+7$, $\frac{2}{27}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{47}{27}a^{4}+\frac{4}{3}a^{3}+\frac{2}{3}a^{2}+\frac{32}{27}a-\frac{1}{3}$, $\frac{52}{27}a^{7}-\frac{109}{9}a^{6}+\frac{152}{9}a^{5}-\frac{284}{27}a^{4}+\frac{763}{9}a^{3}-\frac{707}{9}a^{2}+\frac{1654}{27}a-\frac{35}{3}$
|
| |
| Regulator: | \( 6077.06743939 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 6077.06743939 \cdot 4}{2\cdot\sqrt{51597803520000}}\cr\approx \mathstrut & 1.67883510063 \end{aligned}\]
Galois group
$\PGL(2,7)$ (as 8T43):
| A non-solvable group of order 336 |
| The 9 conjugacy class representatives for $\PGL(2,7)$ |
| Character table for $\PGL(2,7)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 14 sibling: | deg 14 |
| Degree 16 sibling: | deg 16 |
| Degree 21 sibling: | deg 21 |
| Degree 24 sibling: | deg 24 |
| Degree 28 siblings: | deg 28, deg 28 |
| Degree 42 siblings: | deg 42, deg 42, some 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 | R | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.72 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.1.6.9a1.1 | $x^{6} + 6 x^{4} + 3$ | $6$ | $1$ | $9$ | $C_6$ | $$[2]_{2}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |