Normalized defining polynomial
\( x^{8} - 12x^{6} + 204x^{4} - 56x^{2} - 588 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-22024235778048\)
\(\medspace = -\,2^{22}\cdot 3^{7}\cdot 7^{4}\)
|
| |
| Root discriminant: | \(46.54\) |
| |
| Galois root discriminant: | $2^{55/16}3^{7/6}7^{1/2}\approx 103.27164477359116$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{2}a$, $\frac{1}{8232}a^{6}+\frac{53}{2058}a^{4}-\frac{1879}{4116}a^{2}-\frac{13}{49}$, $\frac{1}{8232}a^{7}+\frac{53}{2058}a^{5}-\frac{1879}{4116}a^{3}-\frac{13}{49}a$
| 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: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{65}{8232}a^{6}-\frac{313}{4116}a^{4}+\frac{5461}{4116}a^{2}+\frac{221}{98}$, $\frac{43}{2744}a^{6}-\frac{61}{343}a^{4}+\frac{4267}{1372}a^{2}+\frac{38}{49}$, $\frac{16}{1029}a^{7}+\frac{2}{49}a^{6}-\frac{419}{2058}a^{5}-\frac{17}{49}a^{4}+\frac{3670}{1029}a^{3}+\frac{226}{49}a^{2}+\frac{247}{49}a+\frac{55}{7}$, $\frac{2507581}{686}a^{7}-\frac{212018}{49}a^{6}-\frac{13818246}{343}a^{5}+\frac{2524658}{49}a^{4}+\frac{243244963}{343}a^{3}-\frac{48462248}{49}a^{2}+\frac{57139140}{49}a-\frac{11292427}{7}$
|
| |
| Regulator: | \( 7049.85729541 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 7049.85729541 \cdot 1}{2\cdot\sqrt{22024235778048}}\cr\approx \mathstrut & 0.745245741221 \end{aligned}\]
Galois group
$C_2\wr S_4$ (as 8T44):
| A solvable group of order 384 |
| The 20 conjugacy class representatives for $C_2 \wr S_4$ |
| Character table for $C_2 \wr S_4$ |
Intermediate fields
| 4.2.12096.1 |
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 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 8.2.449474199552.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.47 | $x^{8} + 4 x^{7} + 14 x^{6} + 32 x^{5} + 51 x^{4} + 56 x^{3} + 50 x^{2} + 36 x + 15$ | $4$ | $2$ | $22$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, 4]^{2}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.2.3.6a2.1 | $x^{6} + 6 x^{5} + 18 x^{4} + 32 x^{3} + 39 x^{2} + 30 x + 17$ | $3$ | $2$ | $6$ | $D_{6}$ | $$[\frac{3}{2}]_{2}^{2}$$ | |
|
\(7\)
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 7.2.2.2a1.2 | $x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |