Normalized defining polynomial
\( x^{8} + x^{6} - 63x^{5} + 325x^{4} + 2688x^{3} - 6842x^{2} - 16737x + 39951 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(8158247197093\)
\(\medspace = 7^{6}\cdot 37^{5}\)
|
| |
| Root discriminant: | \(41.11\) |
| |
| Galois root discriminant: | $7^{3/4}37^{3/4}\approx 64.56168002238364$ | ||
| Ramified primes: |
\(7\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-7}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{27032628045414}a^{7}+\frac{1270799880615}{9010876015138}a^{6}-\frac{6400812846844}{13516314022707}a^{5}+\frac{3166915288567}{9010876015138}a^{4}-\frac{3466611569422}{13516314022707}a^{3}-\frac{1290326722223}{4505438007569}a^{2}+\frac{1857546580139}{13516314022707}a+\frac{3079426146449}{9010876015138}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1160400106}{13516314022707}a^{7}+\frac{434798813}{4505438007569}a^{6}+\frac{11996469277}{13516314022707}a^{5}-\frac{16241048201}{4505438007569}a^{4}+\frac{204224705722}{13516314022707}a^{3}+\frac{494908284719}{4505438007569}a^{2}-\frac{2189304579497}{13516314022707}a-\frac{1020672727164}{4505438007569}$, $\frac{4073117788129}{27032628045414}a^{7}-\frac{4597610161557}{9010876015138}a^{6}+\frac{25247828615093}{13516314022707}a^{5}-\frac{141903471573913}{9010876015138}a^{4}+\frac{13\cdots 01}{13516314022707}a^{3}+\frac{279418308236060}{4505438007569}a^{2}-\frac{16\cdots 03}{13516314022707}a+\frac{15\cdots 71}{9010876015138}$, $\frac{111277873453}{27032628045414}a^{7}-\frac{153932128339}{9010876015138}a^{6}-\frac{559927008883}{13516314022707}a^{5}+\frac{4408144039191}{9010876015138}a^{4}-\frac{1291241377768}{13516314022707}a^{3}-\frac{14485551360635}{4505438007569}a^{2}+\frac{19596956028116}{13516314022707}a+\frac{8118907644725}{9010876015138}$
|
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| Regulator: | \( 1240.65632061 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1240.65632061 \cdot 2}{2\cdot\sqrt{8158247197093}}\cr\approx \mathstrut & 0.676974851487 \end{aligned}\]
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
| \(\Q(\sqrt{-7}) \), 4.0.1813.1 |
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: | 16.0.66556997328875790787650649.1, 16.4.91116529343230957588293738481.1 |
| Minimal sibling: | 8.0.5959274797.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.8.0.1}{8} }$ | R | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }$ | R | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.4.6a1.3 | $x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |
|
\(37\)
| 37.2.2.2a1.1 | $x^{4} + 66 x^{3} + 1093 x^{2} + 169 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 37.1.4.3a1.2 | $x^{4} + 74$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |