Normalized defining polynomial
\( x^{10} - x^{9} + x^{8} + 703 x^{7} - 2749 x^{6} + 5521 x^{5} + 108307 x^{4} - 1082247 x^{3} + \cdots + 9240419 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-18828036006813507909611\)
\(\medspace = -\,11^{9}\cdot 41^{8}\)
|
| |
| Root discriminant: | \(168.84\) |
| |
| Galois root discriminant: | $11^{9/10}41^{4/5}\approx 168.84200548797392$ | ||
| Ramified primes: |
\(11\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{10}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(451=11\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{451}(1,·)$, $\chi_{451}(262,·)$, $\chi_{451}(329,·)$, $\chi_{451}(51,·)$, $\chi_{451}(182,·)$, $\chi_{451}(201,·)$, $\chi_{451}(57,·)$, $\chi_{451}(346,·)$, $\chi_{451}(283,·)$, $\chi_{451}(92,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | \(\Q(\sqrt{-11}) \), 10.0.18828036006813507909611.3$^{15}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{56\cdots 27}a^{9}+\frac{22\cdots 19}{56\cdots 27}a^{8}+\frac{22\cdots 54}{56\cdots 27}a^{7}-\frac{81\cdots 97}{56\cdots 27}a^{6}+\frac{28\cdots 41}{56\cdots 27}a^{5}-\frac{80\cdots 99}{56\cdots 27}a^{4}-\frac{46\cdots 43}{56\cdots 27}a^{3}-\frac{10\cdots 07}{56\cdots 27}a^{2}-\frac{12\cdots 00}{56\cdots 27}a+\frac{55\cdots 87}{20\cdots 83}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{605}$, which has order $605$ (assuming GRH) |
| |
| Narrow class group: | $C_{605}$, which has order $605$ (assuming GRH) |
| |
| Relative class number: | $121$ (assuming GRH) |
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{41\cdots 40}{20\cdots 83}a^{9}-\frac{27\cdots 26}{20\cdots 83}a^{8}-\frac{39\cdots 65}{20\cdots 83}a^{7}+\frac{29\cdots 70}{20\cdots 83}a^{6}-\frac{90\cdots 14}{20\cdots 83}a^{5}-\frac{12\cdots 15}{20\cdots 83}a^{4}+\frac{53\cdots 90}{20\cdots 83}a^{3}-\frac{35\cdots 71}{20\cdots 83}a^{2}+\frac{77\cdots 21}{20\cdots 83}a-\frac{37\cdots 98}{20\cdots 83}$, $\frac{27\cdots 65}{20\cdots 83}a^{9}+\frac{14\cdots 88}{20\cdots 83}a^{8}+\frac{26\cdots 45}{20\cdots 83}a^{7}+\frac{18\cdots 10}{20\cdots 83}a^{6}-\frac{47\cdots 52}{20\cdots 83}a^{5}+\frac{68\cdots 42}{20\cdots 83}a^{4}+\frac{24\cdots 57}{20\cdots 83}a^{3}-\frac{22\cdots 15}{20\cdots 83}a^{2}+\frac{49\cdots 86}{20\cdots 83}a-\frac{10\cdots 07}{20\cdots 83}$, $\frac{30\cdots 24}{20\cdots 83}a^{9}+\frac{16\cdots 09}{20\cdots 83}a^{8}+\frac{43\cdots 54}{20\cdots 83}a^{7}+\frac{23\cdots 18}{20\cdots 83}a^{6}-\frac{56\cdots 79}{20\cdots 83}a^{5}+\frac{40\cdots 82}{20\cdots 83}a^{4}+\frac{35\cdots 46}{20\cdots 83}a^{3}-\frac{30\cdots 89}{20\cdots 83}a^{2}+\frac{66\cdots 69}{20\cdots 83}a-\frac{13\cdots 94}{20\cdots 83}$, $\frac{14\cdots 15}{20\cdots 83}a^{9}+\frac{61\cdots 09}{20\cdots 83}a^{8}-\frac{42\cdots 68}{20\cdots 83}a^{7}+\frac{10\cdots 37}{20\cdots 83}a^{6}-\frac{25\cdots 50}{20\cdots 83}a^{5}-\frac{61\cdots 53}{20\cdots 83}a^{4}+\frac{17\cdots 02}{20\cdots 83}a^{3}-\frac{13\cdots 37}{20\cdots 83}a^{2}+\frac{29\cdots 45}{20\cdots 83}a-\frac{11\cdots 11}{20\cdots 83}$
|
| |
| Regulator: | \( 5741.1404574 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 5741.1404574 \cdot 605}{2\cdot\sqrt{18828036006813507909611}}\cr\approx \mathstrut & 0.12394253734 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 10 |
| The 10 conjugacy class representatives for $C_{10}$ |
| Character table for $C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), 5.5.41371966801.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }$ | R | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.1.10.9a1.3 | $x^{10} + 33$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |
|
\(41\)
| 41.2.5.8a1.1 | $x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |