Normalized defining polynomial
\( x^{8} - 24x^{6} + 150x^{4} - 72x^{2} - 108 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[6, 1]$ |
| |
| Discriminant: |
\(-29109824520192\)
\(\medspace = -\,2^{22}\cdot 3^{5}\cdot 13^{4}\)
|
| |
| Root discriminant: | \(48.20\) |
| |
| Galois root discriminant: | $2^{55/16}3^{3/4}13^{1/2}\approx 89.04370605008683$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}$, $\frac{1}{6}a^{5}$, $\frac{1}{54}a^{6}-\frac{2}{9}a^{2}+\frac{1}{3}$, $\frac{1}{54}a^{7}-\frac{2}{9}a^{3}+\frac{1}{3}a$
| 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: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ |
|
Unit group
| Rank: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{6}a^{4}-2a^{2}-1$, $\frac{1}{27}a^{6}-\frac{1}{2}a^{4}+\frac{5}{9}a^{2}+\frac{2}{3}$, $\frac{1}{27}a^{7}-\frac{5}{27}a^{6}-\frac{1}{3}a^{5}+2a^{4}-\frac{4}{9}a^{3}+\frac{2}{9}a^{2}-\frac{4}{3}a-\frac{1}{3}$, $\frac{5}{27}a^{7}-\frac{5}{27}a^{6}-\frac{13}{3}a^{5}+\frac{14}{3}a^{4}+\frac{214}{9}a^{3}-\frac{250}{9}a^{2}+\frac{52}{3}a-\frac{55}{3}$, $\frac{1}{54}a^{6}+\frac{1}{2}a^{4}-\frac{92}{9}a^{2}+\frac{37}{3}$, $\frac{5}{27}a^{7}+\frac{22}{27}a^{6}-2a^{5}-\frac{34}{3}a^{4}-\frac{38}{9}a^{3}+\frac{110}{9}a^{2}-\frac{2}{3}a+\frac{35}{3}$
|
| |
| Regulator: | \( 24457.8250687 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{1}\cdot 24457.8250687 \cdot 1}{2\cdot\sqrt{29109824520192}}\cr\approx \mathstrut & 0.911439586624 \end{aligned}\]
Galois group
$C_2\wr D_4$ (as 8T35):
| A solvable group of order 128 |
| The 20 conjugacy class representatives for $C_2 \wr C_2\wr C_2$ |
| Character table for $C_2 \wr C_2\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.4.32448.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.2.3234424946688.25 |
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 | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.48 | $x^{8} + 4 x^{7} + 22 x^{6} + 56 x^{5} + 99 x^{4} + 112 x^{3} + 98 x^{2} + 60 x + 23$ | $4$ | $2$ | $22$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, 4]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.1.2.1a1.1 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.4.3a1.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ | |
|
\(13\)
| 13.2.2.2a1.2 | $x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 13.2.2.2a1.2 | $x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |