Normalized defining polynomial
\( x^{8} - 4x^{7} + 8x^{5} + 2x^{4} + 20x^{3} - 56x^{2} + 48x - 10 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(4091805696\)
\(\medspace = 2^{18}\cdot 3\cdot 11^{2}\cdot 43\)
|
| |
| Root discriminant: | \(15.90\) |
| |
| Galois root discriminant: | $2^{7/3}3^{1/2}11^{1/2}43^{1/2}\approx 189.84297059381998$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{129}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{6}+\frac{2}{9}a^{4}+\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{2}{9}a-\frac{1}{9}$, $\frac{1}{81}a^{7}+\frac{1}{81}a^{6}+\frac{5}{81}a^{5}+\frac{11}{27}a^{4}+\frac{5}{81}a^{3}-\frac{4}{9}a^{2}+\frac{7}{81}a+\frac{2}{81}$
| 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: |
$\frac{13}{81}a^{7}-\frac{41}{81}a^{6}-\frac{16}{81}a^{5}+\frac{26}{27}a^{4}+\frac{11}{81}a^{3}+\frac{23}{9}a^{2}-\frac{584}{81}a+\frac{323}{81}$, $\frac{4}{81}a^{7}+\frac{13}{81}a^{6}-\frac{34}{81}a^{5}-\frac{22}{27}a^{4}-\frac{52}{81}a^{3}+\frac{181}{81}a-\frac{217}{81}$, $\frac{1}{81}a^{7}-\frac{8}{81}a^{6}+\frac{5}{81}a^{5}+\frac{5}{27}a^{4}-\frac{4}{81}a^{3}+\frac{10}{9}a^{2}-\frac{11}{81}a+\frac{11}{81}$, $\frac{23}{81}a^{7}-\frac{85}{81}a^{6}-\frac{20}{81}a^{5}+\frac{55}{27}a^{4}+\frac{88}{81}a^{3}+\frac{52}{9}a^{2}-\frac{1054}{81}a+\frac{667}{81}$, $\frac{26}{81}a^{7}-\frac{37}{81}a^{6}-\frac{86}{81}a^{5}-\frac{17}{27}a^{4}-\frac{14}{81}a^{3}+\frac{17}{3}a^{2}-\frac{619}{81}a+\frac{223}{81}$
|
| |
| Regulator: | \( 160.407953195 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 160.407953195 \cdot 1}{2\cdot\sqrt{4091805696}}\cr\approx \mathstrut & 0.791987121678 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
| A solvable group of order 384 |
| The 20 conjugacy class representatives for $C_2 \wr S_4$ |
| Character table for $C_2 \wr S_4$ |
Intermediate fields
| 4.2.2816.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | R | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.18c1.4 | $x^{8} + 2 x^{4} + 4 x^{3} + 6$ | $8$ | $1$ | $18$ | $S_4\times C_2$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.3.1.0a1.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 3.3.1.0a1.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(11\)
| 11.4.1.0a1.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(43\)
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 43.1.2.1a1.2 | $x^{2} + 129$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |