Normalized defining polynomial
\( x^{8} - 11x^{6} + 121x^{4} - 605x^{2} + 1815 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[0, 4]$ |
| |
| Discriminant: |
\(170069856000\)
\(\medspace = 2^{8}\cdot 3\cdot 5^{3}\cdot 11^{6}\)
|
| |
| Root discriminant: | \(25.34\) |
| |
| Galois root discriminant: | $2^{15/8}3^{1/2}5^{3/4}11^{3/4}\approx 128.310999012895$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{15}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-11}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11}a^{4}$, $\frac{1}{11}a^{5}$, $\frac{1}{1441}a^{6}-\frac{64}{1441}a^{4}-\frac{26}{131}a^{2}+\frac{13}{131}$, $\frac{1}{1441}a^{7}-\frac{64}{1441}a^{5}-\frac{26}{131}a^{3}+\frac{13}{131}a$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{4}{1441}a^{6}+\frac{6}{1441}a^{4}+\frac{27}{131}a^{2}+\frac{52}{131}$, $\frac{14}{1441}a^{7}-\frac{58}{1441}a^{6}+\frac{152}{1441}a^{5}+\frac{4}{131}a^{4}-\frac{102}{131}a^{3}+\frac{198}{131}a^{2}+\frac{444}{131}a-\frac{1409}{131}$, $\frac{210}{1441}a^{7}+\frac{784}{1441}a^{6}-\frac{78}{1441}a^{5}-\frac{298}{131}a^{4}+\frac{1090}{131}a^{3}+\frac{5030}{131}a^{2}+\frac{4040}{131}a-\frac{3301}{131}$
|
| |
| Regulator: | \( 145.788814924 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 145.788814924 \cdot 2}{2\cdot\sqrt{170069856000}}\cr\approx \mathstrut & 0.550972606768 \end{aligned}\]
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
| \(\Q(\sqrt{-11}) \), 4.0.605.1 |
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 | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.2.8a2.2 | $x^{8} + 4 x^{5} + 2 x^{4} + 3 x^{2} + 4 x + 7$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $$[2, 2, 2, 2]^{4}$$ |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 3.2.1.0a1.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.1.4.3a1.3 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(11\)
| 11.2.4.6a1.3 | $x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3601 x^{4} + 3080 x^{3} + 1208 x^{2} + 268 x + 115$ | $4$ | $2$ | $6$ | $Q_8$ | $$[\ ]_{4}^{2}$$ |