Normalized defining polynomial
\( x^{10} - 5x^{8} + 20x^{6} - 40x^{4} + 35x^{2} + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-6718464000000\)
\(\medspace = -\,2^{16}\cdot 3^{8}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(19.17\) |
| |
| Galois root discriminant: | $2^{187/80}3^{4/3}5^{5/6}\approx 83.61694406953937$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{10}a^{6}+\frac{3}{10}a^{4}+\frac{3}{10}a^{2}+\frac{1}{10}$, $\frac{1}{10}a^{7}-\frac{1}{5}a^{5}-\frac{1}{2}a^{4}+\frac{3}{10}a^{3}-\frac{2}{5}a-\frac{1}{2}$, $\frac{1}{10}a^{8}+\frac{2}{5}a^{4}+\frac{1}{5}a^{2}-\frac{3}{10}$, $\frac{1}{10}a^{9}-\frac{1}{10}a^{5}-\frac{1}{2}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a-\frac{1}{2}$
| 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: |
$a$, $\frac{1}{5}a^{8}-\frac{1}{2}a^{6}+\frac{23}{10}a^{4}-\frac{11}{10}a^{2}-\frac{11}{10}$, $\frac{1}{5}a^{9}-\frac{1}{10}a^{8}-a^{7}+\frac{3}{5}a^{6}+\frac{19}{5}a^{5}-\frac{8}{5}a^{4}-\frac{38}{5}a^{3}+\frac{18}{5}a^{2}+\frac{17}{5}a+\frac{9}{10}$, $\frac{4}{5}a^{9}+\frac{11}{10}a^{8}-\frac{5}{2}a^{7}-\frac{37}{10}a^{6}+\frac{51}{5}a^{5}+\frac{69}{5}a^{4}-\frac{119}{10}a^{3}-\frac{169}{10}a^{2}+\frac{3}{5}a-\frac{3}{2}$
|
| |
| Regulator: | \( 1157.79207661 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1157.79207661 \cdot 1}{2\cdot\sqrt{6718464000000}}\cr\approx \mathstrut & 2.18708127365 \end{aligned}\]
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
| 5.1.162000.2 |
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 | R | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.16a1.16 | $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ | $10$ | $1$ | $16$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$$ |
|
\(3\)
| 3.1.3.4a2.2 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ |
| 3.1.3.4a2.2 | $x^{3} + 6 x^{2} + 12$ | $3$ | $1$ | $4$ | $C_3$ | $$[2]$$ | |
| 3.4.1.0a1.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 5.1.3.2a1.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |