Normalized defining polynomial
\( x^{8} + 7644x^{4} + 146016x^{2} + 711828 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(1135283156287488\)
\(\medspace = 2^{22}\cdot 3^{6}\cdot 13^{5}\)
|
| |
| Root discriminant: | \(76.19\) |
| |
| Galois root discriminant: | $2^{11/4}3^{3/4}13^{3/4}\approx 104.98589220070072$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{312}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{312}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{1447992}a^{6}+\frac{11}{18564}a^{4}+\frac{4895}{18564}a^{2}+\frac{81}{238}$, $\frac{1}{4343976}a^{7}+\frac{47}{37128}a^{5}-\frac{22951}{55692}a^{3}+\frac{173}{476}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{20}$, which has order $80$ |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{20}$, which has order $80$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -\frac{1}{15912} a^{6} + \frac{1}{1768} a^{4} - \frac{101}{204} a^{2} - \frac{287}{68} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{1}{40222}a^{6}-\frac{41}{37128}a^{4}-\frac{23}{3094}a^{2}+\frac{1}{476}$, $\frac{361}{361998}a^{7}-\frac{61}{18564}a^{6}-\frac{181}{18564}a^{5}+\frac{5}{357}a^{4}+\frac{36002}{4641}a^{3}-\frac{5855}{238}a^{2}+\frac{14029}{238}a-\frac{33239}{119}$, $\frac{31}{13923}a^{7}-\frac{529}{111384}a^{6}-\frac{355}{18564}a^{5}+\frac{2335}{37128}a^{4}+\frac{6109}{357}a^{3}-\frac{52613}{1428}a^{2}+\frac{43255}{238}a-\frac{126731}{476}$
|
| |
| Regulator: | \( 2477.18723452 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2477.18723452 \cdot 80}{6\cdot\sqrt{1135283156287488}}\cr\approx \mathstrut & 1.52779451368 \end{aligned}\]
Galois group
$C_4\wr C_2$ (as 8T17):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.7488.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 sibling: | data not computed |
| Degree 16 siblings: | data not computed |
| Minimal sibling: | 8.0.6717651812352.36 |
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 | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 | $x^{8} + 12 x^{7} + 42 x^{6} + 88 x^{5} + 127 x^{4} + 128 x^{3} + 102 x^{2} + 76 x + 23$ | $4$ | $2$ | $22$ | $C_8$ | $$[3, 4]^{2}$$ |
|
\(3\)
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(13\)
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.1.4.3a1.1 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |