Normalized defining polynomial
\( x^{8} + 16x^{6} - 8x^{4} - 576x^{2} - 704 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[2, 3]$ |
| |
| Discriminant: |
\(-3489136640000\)
\(\medspace = -\,2^{22}\cdot 5^{4}\cdot 11^{3}\)
|
| |
| Root discriminant: | \(36.97\) |
| |
| Galois root discriminant: | $2^{55/16}5^{1/2}11^{1/2}\approx 80.34742350667435$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{4}a^{3}$, $\frac{1}{40}a^{4}+\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{40}a^{5}-\frac{1}{20}a^{3}-\frac{2}{5}a$, $\frac{1}{80}a^{6}-\frac{2}{5}$, $\frac{1}{160}a^{7}-\frac{1}{5}a$
| 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: |
$\frac{1}{40}a^{4}+\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{80}a^{6}+\frac{1}{8}a^{4}-a^{2}-\frac{7}{5}$, $\frac{1}{80}a^{6}+\frac{3}{20}a^{4}-\frac{3}{10}a^{2}+\frac{1}{5}$, $\frac{231953}{40}a^{7}-\frac{1120571}{80}a^{6}+\frac{633078}{5}a^{5}-\frac{2446733}{8}a^{4}+\frac{13847471}{20}a^{3}-1672437a^{2}+699671a-\frac{8450318}{5}$
|
| |
| Regulator: | \( 894.637518749 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 894.637518749 \cdot 2}{2\cdot\sqrt{3489136640000}}\cr\approx \mathstrut & 0.475212346003 \end{aligned}\]
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
| \(\Q(\sqrt{5}) \), 4.4.17600.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.4.22a1.48 | $x^{8} + 4 x^{7} + 22 x^{6} + 56 x^{5} + 99 x^{4} + 112 x^{3} + 98 x^{2} + 60 x + 23$ | $4$ | $2$ | $22$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 2, 3, \frac{7}{2}, 4]^{2}$$ |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(11\)
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 11.1.2.1a1.1 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |