Normalized defining polynomial
\( x^{8} + 30x^{6} + 219x^{4} + 469x^{2} + 9 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(0, 4)$ |
| |
| Discriminant: |
\(234444145241761\)
\(\medspace = 7^{4}\cdot 13^{4}\cdot 43^{4}\)
|
| |
| Root discriminant: | \(62.55\) |
| |
| Galois root discriminant: | $7^{2/3}13^{1/2}43^{1/2}\approx 86.51762663289117$ | ||
| Ramified primes: |
\(7\), \(13\), \(43\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-559}) \), 6.0.1342159.1$^{3}$, 8.0.234444145241761.14$^{4}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a$, $\frac{1}{18}a^{6}-\frac{1}{18}a^{4}-\frac{1}{6}a^{3}+\frac{7}{18}a^{2}+\frac{1}{3}a$, $\frac{1}{18}a^{7}-\frac{1}{18}a^{5}+\frac{1}{18}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a-\frac{1}{2}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{32}$, which has order $64$ |
| |
| Narrow class group: | $C_{2}\times C_{32}$, which has order $64$ |
| |
| Relative class number: | $32$ |
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{9}a^{7}+\frac{29}{9}a^{5}+\frac{193}{9}a^{3}+\frac{118}{3}a$, $\frac{1}{3}a^{6}+10a^{4}+\frac{218}{3}a^{2}+154$, $\frac{5}{3}a^{6}+\frac{134}{3}a^{4}+233a^{2}+307$
|
| |
| Regulator: | \( 1346.29565406 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 1346.29565406 \cdot 64}{2\cdot\sqrt{234444145241761}}\cr\approx \mathstrut & 4.3852080210 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 8T13):
| A solvable group of order 24 |
| The 8 conjugacy class representatives for $A_4\times C_2$ |
| Character table for $A_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-559}) \), 4.4.15311569.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 24 |
| Degree 6 sibling: | 6.0.1342159.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Minimal sibling: | 6.0.1342159.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
|
\(13\)
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
|
\(43\)
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 43.2.2.2a1.2 | $x^{4} + 84 x^{3} + 1770 x^{2} + 252 x + 52$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |