Normalized defining polynomial
\( x^{10} - 3x^{9} - 6x^{8} + 19x^{7} + 15x^{6} - 40x^{5} - 20x^{4} + 29x^{3} + 11x^{2} - 4x - 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[10, 0]$ |
| |
| Discriminant: |
\(782566688557\)
\(\medspace = 17^{2}\cdot 157\cdot 4153^{2}\)
|
| |
| Root discriminant: | \(15.47\) |
| |
| Galois root discriminant: | $17^{1/2}157^{1/2}4153^{1/2}\approx 3329.31779798805$ | ||
| Ramified primes: |
\(17\), \(157\), \(4153\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{157}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{9}-4a^{8}-2a^{7}+20a^{6}-3a^{5}-32a^{4}+4a^{3}+17a^{2}+2a-2$, $a^{8}-4a^{7}-a^{6}+17a^{5}-6a^{4}-21a^{3}+8a^{2}+6a-1$, $a^{9}-4a^{8}-2a^{7}+21a^{6}-5a^{5}-37a^{4}+12a^{3}+24a^{2}-5a-3$, $a^{8}-4a^{7}-a^{6}+17a^{5}-5a^{4}-23a^{3}+5a^{2}+10a+1$, $a^{9}-3a^{8}-6a^{7}+20a^{6}+12a^{5}-43a^{4}-9a^{3}+32a^{2}+2a-5$, $a^{4}-2a^{3}-3a^{2}+4a+2$, $a$, $a^{9}-4a^{8}-a^{7}+17a^{6}-6a^{5}-20a^{4}+6a^{3}+4a^{2}+2a+1$, $a^{9}-3a^{8}-5a^{7}+17a^{6}+9a^{5}-31a^{4}-7a^{3}+19a^{2}+2a-2$
|
| |
| Regulator: | \( 301.03717021 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 301.03717021 \cdot 1}{2\cdot\sqrt{782566688557}}\cr\approx \mathstrut & 0.17423259507 \end{aligned}\]
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
| 5.5.70601.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | R | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.3.1.0a1.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 17.3.1.0a1.1 | $x^{3} + x + 14$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 17.2.2.2a1.2 | $x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(157\)
| 157.1.2.1a1.2 | $x^{2} + 785$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 157.2.1.0a1.1 | $x^{2} + 152 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 157.3.1.0a1.1 | $x^{3} + x + 152$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 157.3.1.0a1.1 | $x^{3} + x + 152$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(4153\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |