Normalized defining polynomial
\( x^{8} - 2x^{7} + 3x^{6} - 2x^{5} + 6x^{4} - 6x^{3} + 5x^{2} - 2x + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(203689984\)
\(\medspace = 2^{12}\cdot 223^{2}\)
|
| |
| Root discriminant: | \(10.93\) |
| |
| Galois root discriminant: | $2^{3/2}223^{2/3}\approx 104.01155011920754$ | ||
| Ramified primes: |
\(2\), \(223\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{7}-2a^{6}+3a^{5}-2a^{4}+6a^{3}-6a^{2}+5a-2$, $\frac{3}{2}a^{7}-2a^{6}+3a^{5}-a^{4}+8a^{3}-4a^{2}+\frac{7}{2}a-1$, $\frac{3}{2}a^{7}-\frac{5}{2}a^{6}+3a^{5}-a^{4}+7a^{3}-6a^{2}+\frac{5}{2}a+\frac{1}{2}$
|
| |
| Regulator: | \( 9.42900393812 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 9.42900393812 \cdot 1}{2\cdot\sqrt{203689984}}\cr\approx \mathstrut & 0.514837837987 \end{aligned}\]
Galois group
$A_4\wr C_2$ (as 8T42):
| A solvable group of order 288 |
| The 14 conjugacy class representatives for $A_4\wr C_2$ |
| Character table for $A_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ |
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ | |
|
\(223\)
| $\Q_{223}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $3$ | $3$ | $1$ | $2$ |