Normalized defining polynomial
\( x^{8} - 6x^{6} + 49x^{4} - 250x^{2} - 155 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-101580800000\)
\(\medspace = -\,2^{20}\cdot 5^{5}\cdot 31\)
|
| |
| Root discriminant: | \(23.76\) |
| |
| Galois root discriminant: | $2^{3}5^{3/4}31^{1/2}\approx 148.9355375214321$ | ||
| Ramified primes: |
\(2\), \(5\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-155}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{8327}a^{6}+\frac{2632}{8327}a^{4}-\frac{1453}{8327}a^{2}-\frac{2844}{8327}$, $\frac{1}{8327}a^{7}+\frac{2632}{8327}a^{5}-\frac{1453}{8327}a^{3}-\frac{2844}{8327}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{757}a^{6}-\frac{35}{757}a^{4}+\frac{122}{757}a^{2}-\frac{1146}{757}$, $\frac{38}{8327}a^{6}+\frac{92}{8327}a^{4}+\frac{3075}{8327}a^{2}+\frac{179}{8327}$, $\frac{4}{757}a^{7}+\frac{6}{8327}a^{6}-\frac{70}{757}a^{5}-\frac{862}{8327}a^{4}+\frac{244}{757}a^{3}+\frac{7936}{8327}a^{2}-\frac{778}{757}a+\frac{7917}{8327}$, $\frac{117}{8327}a^{7}+\frac{152}{8327}a^{6}+\frac{155}{8327}a^{5}+\frac{368}{8327}a^{4}-\frac{4866}{8327}a^{3}+\frac{3973}{8327}a^{2}+\frac{7995}{8327}a+\frac{25697}{8327}$
|
| |
| Regulator: | \( 150.90430257848288 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 150.90430257848288 \cdot 2}{2\cdot\sqrt{101580800000}}\cr\approx \mathstrut & 0.469780952967685 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), 4.2.1600.1 |
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: | 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.20a1.7 | $x^{8} + 4 x^{7} + 10 x^{6} + 20 x^{5} + 27 x^{4} + 28 x^{3} + 18 x^{2} + 8 x + 3$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, 3, \frac{7}{2}]^{4}$$ |
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |