Normalized defining polynomial
\( x^{8} - 3x^{7} + 5x^{6} - 8x^{5} + 13x^{4} - 7x^{3} - 43x^{2} + 38x + 12 \)
Invariants
| Degree: | $8$ |
| |
| Signature: | $[4, 2]$ |
| |
| Discriminant: |
\(109957989152\)
\(\medspace = 2^{5}\cdot 11^{2}\cdot 73^{4}\)
|
| |
| Root discriminant: | \(24.00\) |
| |
| Galois root discriminant: | $2^{2}11^{1/2}73^{1/2}\approx 113.34901852243803$ | ||
| Ramified primes: |
\(2\), \(11\), \(73\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{3632}a^{7}-\frac{69}{454}a^{6}-\frac{219}{3632}a^{5}+\frac{367}{3632}a^{4}+\frac{53}{1816}a^{3}+\frac{1727}{3632}a^{2}-\frac{107}{1816}a-\frac{129}{908}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{99}{3632}a^{7}-\frac{21}{454}a^{6}+\frac{111}{3632}a^{5}+\frac{13}{3632}a^{4}-\frac{201}{1816}a^{3}+\frac{269}{3632}a^{2}-\frac{1513}{1816}a-\frac{967}{908}$, $\frac{99}{3632}a^{7}-\frac{21}{454}a^{6}+\frac{111}{3632}a^{5}+\frac{13}{3632}a^{4}-\frac{201}{1816}a^{3}+\frac{269}{3632}a^{2}+\frac{303}{1816}a-\frac{59}{908}$, $\frac{357}{3632}a^{7}-\frac{117}{454}a^{6}+\frac{1721}{3632}a^{5}-\frac{3365}{3632}a^{4}+\frac{2577}{1816}a^{3}-\frac{4533}{3632}a^{2}-\frac{5511}{1816}a-\frac{653}{908}$, $\frac{269}{3632}a^{7}-\frac{87}{227}a^{6}+\frac{2833}{3632}a^{5}-\frac{4789}{3632}a^{4}+\frac{2453}{1816}a^{3}-\frac{3965}{3632}a^{2}-\frac{6083}{1816}a+\frac{4343}{908}$, $\frac{155}{3632}a^{7}-\frac{13}{227}a^{6}-\frac{1257}{3632}a^{5}+\frac{4221}{3632}a^{4}-\frac{1773}{1816}a^{3}-\frac{4715}{3632}a^{2}+\frac{2483}{1816}a+\frac{889}{908}$
|
| |
| Regulator: | \( 1952.88263128 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 1952.88263128 \cdot 1}{2\cdot\sqrt{109957989152}}\cr\approx \mathstrut & 1.85999797766 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
| A solvable group of order 1152 |
| The 20 conjugacy class representatives for $S_4\wr C_2$ |
| Character table for $S_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{73}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.3a1.2 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ | |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
|
\(11\)
| 11.4.1.0a1.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(73\)
| 73.4.2.4a1.2 | $x^{8} + 32 x^{6} + 112 x^{5} + 266 x^{4} + 1792 x^{3} + 3296 x^{2} + 560 x + 98$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |