Normalized defining polynomial
\( x^{8} - 2x^{7} + 9x^{6} + 77x^{5} + 100x^{4} + 881x^{3} + 4914x^{2} + 9916x + 8152 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(28822563882225\)
\(\medspace = 3^{2}\cdot 5^{2}\cdot 71^{6}\)
|
| |
| Root discriminant: | \(48.14\) |
| |
| Galois root discriminant: | $3^{1/2}5^{1/2}71^{3/4}\approx 94.73040435062453$ | ||
| Ramified primes: |
\(3\), \(5\), \(71\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-71}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{12}a^{5}-\frac{1}{12}a^{4}-\frac{1}{4}a^{2}+\frac{5}{12}a-\frac{1}{6}$, $\frac{1}{240}a^{6}-\frac{3}{80}a^{5}-\frac{11}{120}a^{4}+\frac{39}{80}a^{3}-\frac{91}{240}a^{2}-\frac{1}{2}a-\frac{23}{60}$, $\frac{1}{15430320}a^{7}-\frac{5579}{3086064}a^{6}-\frac{31807}{3857580}a^{5}+\frac{2111629}{15430320}a^{4}-\frac{5479393}{15430320}a^{3}+\frac{2009663}{7715160}a^{2}+\frac{47728}{964395}a+\frac{311912}{964395}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{21}$, which has order $21$ |
| |
| Narrow class group: | $C_{21}$, which has order $21$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{535}{3086064}a^{7}+\frac{3131}{1928790}a^{6}-\frac{160637}{15430320}a^{5}+\frac{981869}{15430320}a^{4}+\frac{617423}{7715160}a^{3}-\frac{2657353}{15430320}a^{2}+\frac{2162305}{771516}a+\frac{26818951}{3857580}$, $\frac{2933}{342896}a^{7}+\frac{2023}{214310}a^{6}-\frac{37191}{1714480}a^{5}+\frac{1430247}{1714480}a^{4}+\frac{4329269}{857240}a^{3}+\frac{20188581}{1714480}a^{2}+\frac{1131571}{85724}a+\frac{2463373}{428620}$, $\frac{1014800467}{3857580}a^{7}-\frac{3157410013}{3857580}a^{6}-\frac{31344648949}{3857580}a^{5}-\frac{58247649463}{1928790}a^{4}-\frac{738989129167}{3857580}a^{3}-\frac{592587826999}{771516}a^{2}-\frac{5324203178159}{3857580}a-\frac{366112679431}{385758}$
|
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| Regulator: | \( 4688.50890616 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 4688.50890616 \cdot 21}{2\cdot\sqrt{28822563882225}}\cr\approx \mathstrut & 14.2914802215 \end{aligned}\]
Galois group
$C_2^2\wr C_2$ (as 8T18):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2 \wr C_2$ |
| Character table for $C_2^2 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-71}) \), 4.0.5368665.2, 4.0.25205.1, 4.0.1073733.1 |
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 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Minimal sibling: | 8.0.28822563882225.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(71\)
| 71.2.4.6a1.2 | $x^{8} + 276 x^{7} + 28594 x^{6} + 1319832 x^{5} + 23067339 x^{4} + 9238824 x^{3} + 1401106 x^{2} + 94668 x + 2472$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |