Normalized defining polynomial
\( x^{10} - x^{9} + x^{8} + 2x^{7} - 3x^{6} + 2x^{5} - 2x^{3} + x^{2} - x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-330230659\)
\(\medspace = -\,11^{2}\cdot 19\cdot 379^{2}\)
|
| |
| Root discriminant: | \(7.11\) |
| |
| Galois root discriminant: | $11^{1/2}19^{1/2}379^{1/2}\approx 281.4444883098619$ | ||
| Ramified primes: |
\(11\), \(19\), \(379\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$
| Monogenic: | Yes | |
| 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: |
$a$, $a^{7}+a^{5}+a^{4}+a^{2}-a$, $a^{6}-a^{5}+a^{4}+a^{3}-2a^{2}+a-1$, $a^{9}-a^{8}+a^{7}+2a^{6}-2a^{5}+2a^{4}+a^{3}-a^{2}+a-1$
|
| |
| Regulator: | \( 0.770950046286 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 0.770950046286 \cdot 1}{2\cdot\sqrt{330230659}}\cr\approx \mathstrut & 0.2077241016176 \end{aligned}\]
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
| 5.1.4169.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 11.2.2.2a1.2 | $x^{4} + 14 x^{3} + 53 x^{2} + 28 x + 15$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(19\)
| 19.1.2.1a1.2 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 19.8.1.0a1.1 | $x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
|
\(379\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |