Normalized defining polynomial
\( x^{8} - 2x^{7} + 9x^{6} - 4x^{5} - 25x^{4} + 53x^{3} - 144x^{2} - 74x + 877 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(20933562253041\)
\(\medspace = 3^{4}\cdot 23^{4}\cdot 31^{4}\)
|
| |
| Root discriminant: | \(46.25\) |
| |
| Galois root discriminant: | $3^{1/2}23^{1/2}31^{1/2}\approx 46.24932431938871$ | ||
| Ramified primes: |
\(3\), \(23\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2139}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7272997}a^{7}-\frac{1501566}{7272997}a^{6}+\frac{2922260}{7272997}a^{5}-\frac{1318610}{7272997}a^{4}-\frac{1578274}{7272997}a^{3}+\frac{2440127}{7272997}a^{2}-\frac{1884118}{7272997}a+\frac{657448}{7272997}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}\times C_{4}$, which has order $16$ |
| |
| Narrow class group: | $C_{4}\times C_{4}$, which has order $16$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{50451}{7272997}a^{7}+\frac{30486}{7272997}a^{6}+\frac{17073}{7272997}a^{5}+\frac{910449}{7272997}a^{4}-\frac{730418}{7272997}a^{3}-\frac{3172942}{7272997}a^{2}+\frac{2433572}{7272997}a+\frac{4042728}{7272997}$, $\frac{60705}{7272997}a^{7}-\frac{92629}{7272997}a^{6}+\frac{123473}{7272997}a^{5}+\frac{384932}{7272997}a^{4}-\frac{1933689}{7272997}a^{3}-\frac{1220364}{7272997}a^{2}+\frac{7040629}{7272997}a+\frac{3446301}{7272997}$, $\frac{15824}{7272997}a^{7}+\frac{100815}{7272997}a^{6}+\frac{127314}{7272997}a^{5}+\frac{543753}{7272997}a^{4}+\frac{863922}{7272997}a^{3}+\frac{228575}{7272997}a^{2}+\frac{5004468}{7272997}a+\frac{10344439}{7272997}$
|
| |
| Regulator: | \( 157.11457639 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 157.11457639 \cdot 16}{2\cdot\sqrt{20933562253041}}\cr\approx \mathstrut & 0.42815830267 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 8T24):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2139}) \), 4.2.147591.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.2.66309.1, 6.0.2055579.1 |
| Degree 8 sibling: | 8.4.21783103281.1 |
| Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| Minimal sibling: | 6.2.66309.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
|
\(23\)
| 23.4.2.4a1.2 | $x^{8} + 6 x^{6} + 38 x^{5} + 19 x^{4} + 114 x^{3} + 391 x^{2} + 190 x + 48$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
|
\(31\)
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |