Normalized defining polynomial
\( x^{8} - 2x^{7} - 2x^{6} + 14x^{5} - 12x^{4} - 14x^{3} + 34x^{2} - 10x - 11 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(4, 2)$ |
| |
| Discriminant: |
\(1275346944\)
\(\medspace = 2^{14}\cdot 3^{4}\cdot 31^{2}\)
|
| |
| Root discriminant: | \(13.75\) |
| |
| Galois root discriminant: | $2^{7/4}3^{1/2}31^{1/2}\approx 32.43724541951093$ | ||
| Ramified primes: |
\(2\), \(3\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{341}a^{7}+\frac{49}{341}a^{6}+\frac{10}{31}a^{5}+\frac{168}{341}a^{4}+\frac{1}{11}a^{3}-\frac{138}{341}a^{2}+\frac{157}{341}a+\frac{14}{31}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{42}{341}a^{7}+\frac{12}{341}a^{6}-\frac{14}{31}a^{5}+\frac{236}{341}a^{4}+\frac{9}{11}a^{3}-\frac{340}{341}a^{2}+\frac{115}{341}a-\frac{1}{31}$, $\frac{34}{341}a^{7}-\frac{39}{341}a^{6}-\frac{1}{31}a^{5}+\frac{256}{341}a^{4}-\frac{10}{11}a^{3}+\frac{82}{341}a^{2}+\frac{564}{341}a-\frac{51}{31}$, $\frac{34}{341}a^{7}-\frac{39}{341}a^{6}-\frac{1}{31}a^{5}+\frac{256}{341}a^{4}-\frac{10}{11}a^{3}+\frac{82}{341}a^{2}+\frac{223}{341}a-\frac{20}{31}$, $\frac{19}{341}a^{7}-\frac{92}{341}a^{6}+\frac{4}{31}a^{5}+\frac{464}{341}a^{4}-\frac{25}{11}a^{3}-\frac{235}{341}a^{2}+\frac{1278}{341}a-\frac{44}{31}$, $\frac{84}{341}a^{7}+\frac{24}{341}a^{6}-\frac{28}{31}a^{5}+\frac{472}{341}a^{4}+\frac{18}{11}a^{3}-\frac{1021}{341}a^{2}+\frac{230}{341}a+\frac{60}{31}$
|
| |
| Regulator: | \( 54.3294214763 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 54.3294214763 \cdot 1}{2\cdot\sqrt{1275346944}}\cr\approx \mathstrut & 0.48047481838 \end{aligned}\]
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
| \(\Q(\sqrt{3}) \), 4.2.1984.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 siblings: | 6.2.1660608.2, 6.0.51478848.4 |
| Degree 8 sibling: | 8.0.1225608413184.12 |
| 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: | 6.2.1660608.2 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.14a1.2 | $x^{8} + 2 x^{7} + 6$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(3\)
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |
|
\(31\)
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.2.2a1.2 | $x^{4} + 58 x^{3} + 847 x^{2} + 174 x + 40$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |