Normalized defining polynomial
\( x^{8} - 4x^{6} - 8x^{5} + 10x^{4} + 8x^{3} - 4x^{2} + 1 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $(4, 2)$ |
| |
| Discriminant: |
\(355352576\)
\(\medspace = 2^{14}\cdot 23^{2}\cdot 41\)
|
| |
| Root discriminant: | \(11.72\) |
| |
| Galois root discriminant: | $2^{7/4}23^{1/2}41^{1/2}\approx 103.29001459261505$ | ||
| Ramified primes: |
\(2\), \(23\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$
| 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: |
$a$, $\frac{3}{4}a^{7}-3a^{5}-\frac{25}{4}a^{4}+\frac{29}{4}a^{3}+\frac{13}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{3}{4}a^{7}-\frac{11}{4}a^{5}-6a^{4}+\frac{27}{4}a^{3}+4a^{2}-\frac{3}{4}a$, $\frac{3}{2}a^{7}-\frac{1}{2}a^{6}-\frac{23}{4}a^{5}-\frac{41}{4}a^{4}+18a^{3}+6a^{2}-\frac{25}{4}a+\frac{9}{4}$, $\frac{3}{4}a^{7}-3a^{5}-\frac{25}{4}a^{4}+\frac{29}{4}a^{3}+\frac{13}{2}a^{2}-\frac{3}{2}a-\frac{3}{4}$
|
| |
| Regulator: | \( 25.2687845117 \) |
| |
| Unit signature rank: | \( 3 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 25.2687845117 \cdot 1}{2\cdot\sqrt{355352576}}\cr\approx \mathstrut & 0.42335465704 \end{aligned}\]
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
| 4.2.1472.1 |
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: | 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.14a1.2 | $x^{8} + 2 x^{7} + 6$ | $8$ | $1$ | $14$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(23\)
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 23.2.2.2a1.2 | $x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(41\)
| 41.1.2.1a1.2 | $x^{2} + 246$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 41.6.1.0a1.1 | $x^{6} + 4 x^{4} + 33 x^{3} + 39 x^{2} + 6 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |