Normalized defining polynomial
\( x^{8} - 8x^{5} - 4x^{4} + 64x^{2} - 56x + 17 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(35718692864\)
\(\medspace = 2^{24}\cdot 2129\)
|
| |
| Root discriminant: | \(20.85\) |
| |
| Galois root discriminant: | $2^{3}2129^{1/2}\approx 369.12870384189847$ | ||
| Ramified primes: |
\(2\), \(2129\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2129}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{31591}a^{7}-\frac{12910}{31591}a^{6}-\frac{6016}{31591}a^{5}-\frac{15717}{31591}a^{4}-\frac{361}{4513}a^{3}-\frac{1419}{4513}a^{2}+\frac{7225}{31591}a+\frac{13417}{31591}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{108}{4513}a^{7}+\frac{237}{4513}a^{6}+\frac{144}{4513}a^{5}-\frac{548}{4513}a^{4}-\frac{2136}{4513}a^{3}-\frac{3183}{4513}a^{2}+\frac{4064}{4513}a+\frac{9389}{4513}$, $\frac{19392}{31591}a^{7}+\frac{7955}{31591}a^{6}+\frac{3291}{31591}a^{5}-\frac{152051}{31591}a^{4}-\frac{18901}{4513}a^{3}-\frac{6000}{4513}a^{2}+\frac{1233164}{31591}a-\frac{601241}{31591}$, $\frac{509}{31591}a^{7}-\frac{262}{31591}a^{6}+\frac{2183}{31591}a^{5}-\frac{7430}{31591}a^{4}+\frac{1284}{4513}a^{3}-\frac{4704}{4513}a^{2}+\frac{44560}{31591}a+\frac{5597}{31591}$
|
| |
| Regulator: | \( 98.2805561268 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 98.2805561268 \cdot 2}{2\cdot\sqrt{35718692864}}\cr\approx \mathstrut & 0.810474397123 \end{aligned}\]
Galois group
$C_2\wr C_4$ (as 8T27):
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
| Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.2048.2 |
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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.8.24c1.61 | $x^{8} + 8 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $$[2, 3, 4]$$ |
|
\(2129\)
| $\Q_{2129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{2129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{2129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{2129}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |