Normalized defining polynomial
\( x^{8} - 13x^{6} + 52x^{4} - 52x^{3} - 78x^{2} + 208x + 104 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(4, 2)$ |
| |
| Discriminant: |
\(1285089628160\)
\(\medspace = 2^{12}\cdot 5\cdot 13^{7}\)
|
| |
| Root discriminant: | \(32.63\) |
| |
| Galois root discriminant: | $2^{2}5^{1/2}13^{7/8}\approx 84.38114744244693$ | ||
| Ramified primes: |
\(2\), \(5\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{65}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{185348}a^{7}+\frac{3594}{46337}a^{6}+\frac{6343}{185348}a^{5}-\frac{1062}{46337}a^{4}-\frac{22426}{46337}a^{3}+\frac{16657}{46337}a^{2}-\frac{17207}{92674}a-\frac{10411}{46337}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{2}\times C_{8}$, which has order $16$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1721}{185348}a^{7}-\frac{1431}{92674}a^{6}-\frac{19229}{185348}a^{5}+\frac{5219}{92674}a^{4}+\frac{3575}{46337}a^{3}-\frac{15906}{46337}a^{2}+\frac{135107}{92674}a+\frac{15088}{46337}$, $\frac{2005}{46337}a^{7}+\frac{2266}{46337}a^{6}-\frac{24960}{46337}a^{5}-\frac{37569}{46337}a^{4}+\frac{70051}{46337}a^{3}+\frac{92243}{46337}a^{2}+\frac{42060}{46337}a+\frac{142065}{46337}$, $\frac{27265}{92674}a^{7}-\frac{24690}{46337}a^{6}-\frac{265811}{92674}a^{5}+\frac{242075}{46337}a^{4}+\frac{268009}{46337}a^{3}-\frac{1196426}{46337}a^{2}+\frac{1125358}{46337}a+\frac{796823}{46337}$, $\frac{1399}{92674}a^{7}+\frac{883}{46337}a^{6}-\frac{22847}{92674}a^{5}-\frac{5908}{46337}a^{4}+\frac{38687}{46337}a^{3}-\frac{8736}{46337}a^{2}+\frac{22647}{46337}a+\frac{15995}{46337}$, $\frac{3907}{92674}a^{7}+\frac{3294}{46337}a^{6}-\frac{54531}{92674}a^{5}-\frac{50482}{46337}a^{4}+\frac{102444}{46337}a^{3}+\frac{182513}{46337}a^{2}-\frac{85436}{46337}a-\frac{122793}{46337}$
|
| |
| Regulator: | \( 3559.39572808 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 3559.39572808 \cdot 4}{2\cdot\sqrt{1285089628160}}\cr\approx \mathstrut & 3.96660797476 \end{aligned}\]
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
| 4.2.70304.1 |
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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.1.4.8b1.1 | $x^{4} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $C_2^2$ | $$[2, 3]$$ | |
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.6.1.0a1.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(13\)
| 13.1.8.7a1.4 | $x^{8} + 104$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |