Normalized defining polynomial
\( x^{8} - 10x^{6} + 57x^{4} - 286x^{2} + 637 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(1679412953088\)
\(\medspace = 2^{20}\cdot 3^{6}\cdot 13^{3}\)
|
| |
| Root discriminant: | \(33.74\) |
| |
| Galois root discriminant: | $2^{3}3^{3/4}13^{3/4}\approx 124.84996997998198$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{61}a^{6}+\frac{5}{61}a^{4}+\frac{10}{61}a^{2}-\frac{14}{61}$, $\frac{1}{427}a^{7}-\frac{178}{427}a^{5}+\frac{71}{427}a^{3}+\frac{169}{427}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ |
| |
| Narrow class group: | $C_{4}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( \frac{6}{61} a^{6} - \frac{31}{61} a^{4} + \frac{182}{61} a^{2} - \frac{755}{61} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{96}{427}a^{7}+\frac{48}{61}a^{6}-\frac{435}{427}a^{5}-\frac{248}{61}a^{4}+\frac{2973}{427}a^{3}+\frac{1578}{61}a^{2}-\frac{11531}{427}a-\frac{6223}{61}$, $\frac{96}{427}a^{7}-\frac{48}{61}a^{6}-\frac{435}{427}a^{5}+\frac{248}{61}a^{4}+\frac{2973}{427}a^{3}-\frac{1578}{61}a^{2}-\frac{11531}{427}a+\frac{6223}{61}$, $\frac{7538}{427}a^{7}+\frac{2367}{61}a^{6}-\frac{38987}{427}a^{5}-\frac{12260}{61}a^{4}+\frac{240995}{427}a^{3}+\frac{75703}{61}a^{2}-\frac{992167}{427}a-\frac{311969}{61}$
|
| |
| Regulator: | \( 2137.5612008 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 2137.5612008 \cdot 4}{6\cdot\sqrt{1679412953088}}\cr\approx \mathstrut & 1.7138311140 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.22464.1 |
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 32 siblings: | data not computed |
| Minimal sibling: | 8.0.139951079424.2 |
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 | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.20a1.7 | $x^{8} + 4 x^{7} + 10 x^{6} + 20 x^{5} + 27 x^{4} + 28 x^{3} + 18 x^{2} + 8 x + 3$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, 3, \frac{7}{2}]^{4}$$ |
|
\(3\)
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.1.4.3a1.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(13\)
| 13.1.4.3a1.2 | $x^{4} + 26$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 13.4.1.0a1.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |