Normalized defining polynomial
\( x^{10} - x^{9} - 3x^{8} - x^{7} + 13x^{6} - 19x^{5} + 13x^{4} - x^{3} - 3x^{2} - x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(2, 4)$ |
| |
| Discriminant: |
\(657216138005\)
\(\medspace = 5\cdot 7^{8}\cdot 151^{2}\)
|
| |
| Root discriminant: | \(15.20\) |
| |
| Galois root discriminant: | $5^{1/2}7^{4/5}151^{1/2}\approx 130.33225451034468$ | ||
| Ramified primes: |
\(5\), \(7\), \(151\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}-\frac{5}{11}a^{7}+\frac{5}{11}a^{6}-\frac{5}{11}a^{5}-\frac{5}{11}a^{4}-\frac{5}{11}a^{3}+\frac{5}{11}a^{2}-\frac{5}{11}a+\frac{1}{11}$, $\frac{1}{33}a^{9}+\frac{1}{33}a^{8}+\frac{8}{33}a^{7}+\frac{1}{11}a^{6}-\frac{2}{33}a^{5}-\frac{2}{33}a^{4}-\frac{1}{11}a^{3}+\frac{14}{33}a^{2}+\frac{4}{33}a-\frac{5}{33}$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $a^{9}-a^{8}-3a^{7}-a^{6}+13a^{5}-19a^{4}+13a^{3}-a^{2}-3a$, $\frac{9}{11}a^{9}-\frac{1}{11}a^{8}-\frac{32}{11}a^{7}-\frac{34}{11}a^{6}+\frac{98}{11}a^{5}-\frac{78}{11}a^{4}+\frac{1}{11}a^{3}+\frac{65}{11}a^{2}-\frac{24}{11}a-2$, $\frac{17}{33}a^{9}+\frac{5}{33}a^{8}-\frac{68}{33}a^{7}-\frac{25}{11}a^{6}+\frac{191}{33}a^{5}-\frac{73}{33}a^{4}-\frac{63}{11}a^{3}+\frac{277}{33}a^{2}-\frac{70}{33}a-\frac{64}{33}$, $\frac{167}{33}a^{9}-\frac{85}{33}a^{8}-\frac{49}{3}a^{7}-13a^{6}+\frac{1949}{33}a^{5}-\frac{2242}{33}a^{4}+33a^{3}+\frac{35}{3}a^{2}-\frac{316}{33}a-\frac{328}{33}$
|
| |
| Regulator: | \( 164.131547041 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 164.131547041 \cdot 1}{2\cdot\sqrt{657216138005}}\cr\approx \mathstrut & 0.631084367683 \end{aligned}\]
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
| 5.3.362551.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(7\)
| 7.2.5.8a1.1 | $x^{10} + 30 x^{9} + 375 x^{8} + 2520 x^{7} + 9810 x^{6} + 22356 x^{5} + 29430 x^{4} + 22680 x^{3} + 10125 x^{2} + 2430 x + 250$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
|
\(151\)
| 151.3.1.0a1.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 151.3.1.0a1.1 | $x^{3} + x + 145$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 151.2.2.2a1.2 | $x^{4} + 298 x^{3} + 22213 x^{2} + 1788 x + 187$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |