Normalized defining polynomial
\( x^{10} + 6x^{8} + 8x^{6} + 4 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-277838626816\)
\(\medspace = -\,2^{16}\cdot 29^{2}\cdot 71^{2}\)
|
| |
| Root discriminant: | \(13.94\) |
| |
| Galois root discriminant: | $2^{187/80}29^{1/2}71^{1/2}\approx 229.34315791029596$ | ||
| Ramified primes: |
\(2\), \(29\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{4}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{8}+\frac{9}{2}a^{6}+a^{4}-3a^{2}+3$, $\frac{1}{2}a^{7}+\frac{5}{2}a^{5}+2a^{3}$, $a^{9}+\frac{9}{2}a^{7}+\frac{3}{2}a^{5}-a^{3}+3a$, $\frac{1}{2}a^{9}+\frac{7}{2}a^{7}+a^{6}+7a^{5}+5a^{4}+4a^{3}+3a^{2}-3$
|
| |
| Regulator: | \( 66.6616282796 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 66.6616282796 \cdot 1}{2\cdot\sqrt{277838626816}}\cr\approx \mathstrut & 0.619225670028 \end{aligned}\]
Galois group
$C_2\wr S_5$ (as 10T39):
| A non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2 \wr S_5$ |
| Character table for $C_2 \wr S_5$ |
Intermediate fields
| 5.3.32944.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.16a1.16 | $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ | $10$ | $1$ | $16$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$$ |
|
\(29\)
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 29.1.2.1a1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 29.3.1.0a1.1 | $x^{3} + 2 x + 27$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 29.3.1.0a1.1 | $x^{3} + 2 x + 27$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(71\)
| 71.2.2.2a1.2 | $x^{4} + 138 x^{3} + 4775 x^{2} + 966 x + 120$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 71.6.1.0a1.1 | $x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |