Normalized defining polynomial
\( x^{20} - 4 x^{19} + 50 x^{18} - 160 x^{17} + 1183 x^{16} - 3204 x^{15} + 16682 x^{14} - 38000 x^{13} + \cdots + 95829823 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $(0, 10)$ |
| |
| Discriminant: |
\(97756416090547441671665711234313879552\)
\(\medspace = 2^{55}\cdot 3^{10}\cdot 11^{16}\)
|
| |
| Root discriminant: | \(79.34\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}11^{4/5}\approx 79.34275267180627$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(528=2^{4}\cdot 3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{528}(1,·)$, $\chi_{528}(389,·)$, $\chi_{528}(289,·)$, $\chi_{528}(265,·)$, $\chi_{528}(269,·)$, $\chi_{528}(25,·)$, $\chi_{528}(221,·)$, $\chi_{528}(5,·)$, $\chi_{528}(97,·)$, $\chi_{528}(485,·)$, $\chi_{528}(433,·)$, $\chi_{528}(169,·)$, $\chi_{528}(317,·)$, $\chi_{528}(49,·)$, $\chi_{528}(245,·)$, $\chi_{528}(361,·)$, $\chi_{528}(313,·)$, $\chi_{528}(509,·)$, $\chi_{528}(125,·)$, $\chi_{528}(53,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{69}a^{15}+\frac{5}{69}a^{14}-\frac{3}{23}a^{13}+\frac{7}{69}a^{12}+\frac{5}{69}a^{10}-\frac{11}{23}a^{9}+\frac{25}{69}a^{8}+\frac{32}{69}a^{7}-\frac{8}{23}a^{6}+\frac{28}{69}a^{5}+\frac{34}{69}a^{4}-\frac{5}{23}a^{3}-\frac{4}{23}a^{2}-\frac{9}{23}a+\frac{2}{23}$, $\frac{1}{69}a^{16}-\frac{11}{69}a^{14}+\frac{2}{23}a^{13}+\frac{11}{69}a^{12}+\frac{5}{69}a^{11}+\frac{11}{69}a^{10}+\frac{2}{23}a^{9}-\frac{8}{23}a^{8}-\frac{13}{69}a^{6}+\frac{32}{69}a^{5}-\frac{1}{69}a^{4}-\frac{29}{69}a^{3}+\frac{11}{23}a^{2}+\frac{26}{69}a-\frac{7}{69}$, $\frac{1}{69}a^{17}-\frac{8}{69}a^{14}+\frac{4}{69}a^{13}-\frac{10}{69}a^{12}+\frac{11}{69}a^{11}-\frac{8}{69}a^{10}+\frac{4}{69}a^{9}-\frac{1}{69}a^{8}+\frac{17}{69}a^{7}-\frac{25}{69}a^{6}-\frac{5}{23}a^{5}+\frac{1}{3}a^{4}-\frac{17}{69}a^{3}-\frac{14}{69}a^{2}-\frac{28}{69}a-\frac{26}{69}$, $\frac{1}{31\cdots 81}a^{18}+\frac{76\cdots 58}{10\cdots 27}a^{17}+\frac{21\cdots 04}{31\cdots 81}a^{16}-\frac{40\cdots 76}{31\cdots 81}a^{15}-\frac{58\cdots 55}{46\cdots 49}a^{14}-\frac{26\cdots 67}{31\cdots 81}a^{13}-\frac{22\cdots 39}{31\cdots 81}a^{12}+\frac{23\cdots 34}{31\cdots 81}a^{11}+\frac{46\cdots 84}{31\cdots 81}a^{10}+\frac{99\cdots 14}{10\cdots 27}a^{9}-\frac{27\cdots 23}{10\cdots 27}a^{8}+\frac{94\cdots 30}{31\cdots 81}a^{7}+\frac{15\cdots 22}{10\cdots 27}a^{6}-\frac{13\cdots 51}{31\cdots 81}a^{5}-\frac{91\cdots 35}{31\cdots 81}a^{4}+\frac{57\cdots 28}{31\cdots 81}a^{3}-\frac{80\cdots 02}{31\cdots 81}a^{2}-\frac{16\cdots 19}{10\cdots 27}a-\frac{10\cdots 78}{31\cdots 81}$, $\frac{1}{10\cdots 07}a^{19}-\frac{12\cdots 96}{10\cdots 07}a^{18}+\frac{39\cdots 55}{10\cdots 07}a^{17}-\frac{13\cdots 18}{36\cdots 69}a^{16}+\frac{15\cdots 25}{36\cdots 69}a^{15}-\frac{17\cdots 60}{10\cdots 07}a^{14}+\frac{13\cdots 38}{10\cdots 07}a^{13}-\frac{13\cdots 21}{10\cdots 07}a^{12}+\frac{14\cdots 68}{10\cdots 07}a^{11}+\frac{25\cdots 52}{10\cdots 07}a^{10}+\frac{15\cdots 84}{10\cdots 07}a^{9}+\frac{53\cdots 72}{10\cdots 07}a^{8}+\frac{42\cdots 05}{36\cdots 69}a^{7}-\frac{22\cdots 15}{10\cdots 07}a^{6}+\frac{12\cdots 92}{36\cdots 69}a^{5}+\frac{41\cdots 49}{10\cdots 07}a^{4}+\frac{20\cdots 20}{10\cdots 07}a^{3}+\frac{29\cdots 43}{36\cdots 69}a^{2}+\frac{13\cdots 69}{36\cdots 69}a+\frac{50\cdots 08}{10\cdots 07}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{131042}$, which has order $131042$ (assuming GRH) |
| |
| Narrow class group: | $C_{131042}$, which has order $131042$ (assuming GRH) |
| |
| Relative class number: | $131042$ (assuming GRH) |
Unit group
| Rank: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{62\cdots 56}{36\cdots 69}a^{19}-\frac{42\cdots 52}{36\cdots 69}a^{18}+\frac{25\cdots 04}{36\cdots 69}a^{17}-\frac{66\cdots 69}{36\cdots 69}a^{16}+\frac{52\cdots 76}{36\cdots 69}a^{15}+\frac{48\cdots 80}{36\cdots 69}a^{14}+\frac{63\cdots 84}{36\cdots 69}a^{13}+\frac{28\cdots 60}{36\cdots 69}a^{12}+\frac{49\cdots 48}{36\cdots 69}a^{11}+\frac{36\cdots 08}{36\cdots 69}a^{10}+\frac{25\cdots 60}{36\cdots 69}a^{9}+\frac{21\cdots 55}{36\cdots 69}a^{8}+\frac{12\cdots 40}{36\cdots 69}a^{7}+\frac{80\cdots 64}{36\cdots 69}a^{6}+\frac{59\cdots 48}{36\cdots 69}a^{5}+\frac{59\cdots 32}{36\cdots 69}a^{4}+\frac{13\cdots 36}{36\cdots 69}a^{3}+\frac{86\cdots 96}{36\cdots 69}a^{2}+\frac{16\cdots 04}{36\cdots 69}a-\frac{16\cdots 61}{36\cdots 69}$, $\frac{49\cdots 32}{36\cdots 69}a^{19}+\frac{69\cdots 98}{36\cdots 69}a^{18}+\frac{16\cdots 04}{36\cdots 69}a^{17}+\frac{43\cdots 68}{36\cdots 69}a^{16}+\frac{27\cdots 68}{36\cdots 69}a^{15}+\frac{11\cdots 26}{36\cdots 69}a^{14}+\frac{24\cdots 72}{36\cdots 69}a^{13}+\frac{16\cdots 16}{36\cdots 69}a^{12}+\frac{98\cdots 24}{36\cdots 69}a^{11}+\frac{14\cdots 74}{36\cdots 69}a^{10}-\frac{15\cdots 88}{36\cdots 69}a^{9}+\frac{85\cdots 76}{36\cdots 69}a^{8}-\frac{39\cdots 64}{36\cdots 69}a^{7}+\frac{39\cdots 12}{36\cdots 69}a^{6}+\frac{34\cdots 60}{36\cdots 69}a^{5}+\frac{20\cdots 55}{36\cdots 69}a^{4}-\frac{40\cdots 36}{36\cdots 69}a^{3}+\frac{35\cdots 26}{36\cdots 69}a^{2}-\frac{10\cdots 04}{36\cdots 69}a+\frac{75\cdots 12}{36\cdots 69}$, $\frac{27\cdots 96}{36\cdots 69}a^{19}-\frac{99\cdots 16}{36\cdots 69}a^{18}+\frac{13\cdots 76}{36\cdots 69}a^{17}-\frac{41\cdots 12}{36\cdots 69}a^{16}+\frac{33\cdots 20}{36\cdots 69}a^{15}-\frac{84\cdots 48}{36\cdots 69}a^{14}+\frac{48\cdots 44}{36\cdots 69}a^{13}-\frac{10\cdots 69}{36\cdots 69}a^{12}+\frac{44\cdots 36}{36\cdots 69}a^{11}-\frac{77\cdots 74}{36\cdots 69}a^{10}+\frac{27\cdots 56}{36\cdots 69}a^{9}-\frac{39\cdots 56}{36\cdots 69}a^{8}+\frac{13\cdots 04}{36\cdots 69}a^{7}-\frac{16\cdots 82}{36\cdots 69}a^{6}+\frac{59\cdots 32}{36\cdots 69}a^{5}-\frac{59\cdots 53}{36\cdots 69}a^{4}+\frac{17\cdots 60}{36\cdots 69}a^{3}-\frac{11\cdots 94}{36\cdots 69}a^{2}+\frac{17\cdots 28}{36\cdots 69}a-\frac{13\cdots 24}{36\cdots 69}$, $\frac{27\cdots 96}{36\cdots 69}a^{19}-\frac{99\cdots 16}{36\cdots 69}a^{18}+\frac{13\cdots 76}{36\cdots 69}a^{17}-\frac{41\cdots 12}{36\cdots 69}a^{16}+\frac{33\cdots 20}{36\cdots 69}a^{15}-\frac{84\cdots 48}{36\cdots 69}a^{14}+\frac{48\cdots 44}{36\cdots 69}a^{13}-\frac{10\cdots 69}{36\cdots 69}a^{12}+\frac{44\cdots 36}{36\cdots 69}a^{11}-\frac{77\cdots 74}{36\cdots 69}a^{10}+\frac{27\cdots 56}{36\cdots 69}a^{9}-\frac{39\cdots 56}{36\cdots 69}a^{8}+\frac{13\cdots 04}{36\cdots 69}a^{7}-\frac{16\cdots 82}{36\cdots 69}a^{6}+\frac{59\cdots 32}{36\cdots 69}a^{5}-\frac{59\cdots 53}{36\cdots 69}a^{4}+\frac{17\cdots 60}{36\cdots 69}a^{3}-\frac{11\cdots 94}{36\cdots 69}a^{2}+\frac{17\cdots 28}{36\cdots 69}a+\frac{22\cdots 45}{36\cdots 69}$, $\frac{29015885365676}{73\cdots 63}a^{19}+\frac{344395780525762}{73\cdots 63}a^{18}-\frac{381019137243520}{73\cdots 63}a^{17}+\frac{13\cdots 11}{73\cdots 63}a^{16}-\frac{76\cdots 20}{24\cdots 21}a^{15}+\frac{25\cdots 14}{73\cdots 63}a^{14}-\frac{47\cdots 30}{73\cdots 63}a^{13}+\frac{28\cdots 17}{73\cdots 63}a^{12}-\frac{46\cdots 76}{73\cdots 63}a^{11}+\frac{18\cdots 56}{73\cdots 63}a^{10}-\frac{25\cdots 56}{73\cdots 63}a^{9}+\frac{29\cdots 05}{24\cdots 21}a^{8}-\frac{95\cdots 68}{73\cdots 63}a^{7}+\frac{14\cdots 50}{24\cdots 21}a^{6}-\frac{15\cdots 02}{24\cdots 21}a^{5}+\frac{12\cdots 66}{73\cdots 63}a^{4}-\frac{81\cdots 16}{73\cdots 63}a^{3}+\frac{27\cdots 21}{73\cdots 63}a^{2}-\frac{38\cdots 48}{24\cdots 21}a+\frac{33\cdots 20}{73\cdots 63}$, $\frac{14\cdots 32}{10\cdots 07}a^{19}+\frac{15\cdots 76}{10\cdots 07}a^{18}+\frac{50\cdots 92}{10\cdots 07}a^{17}+\frac{10\cdots 25}{10\cdots 07}a^{16}+\frac{28\cdots 48}{36\cdots 69}a^{15}+\frac{29\cdots 32}{10\cdots 07}a^{14}+\frac{80\cdots 86}{10\cdots 07}a^{13}+\frac{45\cdots 35}{10\cdots 07}a^{12}+\frac{36\cdots 36}{10\cdots 07}a^{11}+\frac{42\cdots 38}{10\cdots 07}a^{10}+\frac{33\cdots 20}{10\cdots 07}a^{9}+\frac{81\cdots 31}{36\cdots 69}a^{8}+\frac{23\cdots 60}{10\cdots 07}a^{7}+\frac{37\cdots 62}{36\cdots 69}a^{6}+\frac{57\cdots 38}{36\cdots 69}a^{5}+\frac{58\cdots 91}{10\cdots 07}a^{4}-\frac{11\cdots 84}{10\cdots 07}a^{3}+\frac{10\cdots 09}{10\cdots 07}a^{2}-\frac{46\cdots 32}{36\cdots 69}a+\frac{69\cdots 49}{10\cdots 07}$, $\frac{97\cdots 12}{10\cdots 07}a^{19}-\frac{14\cdots 43}{36\cdots 69}a^{18}+\frac{52\cdots 28}{10\cdots 07}a^{17}-\frac{19\cdots 32}{10\cdots 07}a^{16}+\frac{13\cdots 12}{10\cdots 07}a^{15}-\frac{42\cdots 78}{10\cdots 07}a^{14}+\frac{65\cdots 08}{36\cdots 69}a^{13}-\frac{53\cdots 80}{10\cdots 07}a^{12}+\frac{18\cdots 68}{10\cdots 07}a^{11}-\frac{41\cdots 81}{10\cdots 07}a^{10}+\frac{11\cdots 70}{10\cdots 07}a^{9}-\frac{19\cdots 59}{10\cdots 07}a^{8}+\frac{14\cdots 40}{36\cdots 69}a^{7}-\frac{62\cdots 08}{10\cdots 07}a^{6}+\frac{15\cdots 34}{10\cdots 07}a^{5}-\frac{72\cdots 13}{36\cdots 69}a^{4}+\frac{42\cdots 32}{10\cdots 07}a^{3}-\frac{34\cdots 29}{10\cdots 07}a^{2}+\frac{37\cdots 22}{10\cdots 07}a+\frac{59\cdots 22}{10\cdots 07}$, $\frac{20\cdots 16}{10\cdots 07}a^{19}-\frac{80\cdots 62}{10\cdots 07}a^{18}+\frac{10\cdots 56}{10\cdots 07}a^{17}-\frac{34\cdots 65}{10\cdots 07}a^{16}+\frac{87\cdots 72}{36\cdots 69}a^{15}-\frac{71\cdots 14}{10\cdots 07}a^{14}+\frac{38\cdots 50}{10\cdots 07}a^{13}-\frac{89\cdots 67}{10\cdots 07}a^{12}+\frac{35\cdots 56}{10\cdots 07}a^{11}-\frac{69\cdots 00}{10\cdots 07}a^{10}+\frac{21\cdots 72}{10\cdots 07}a^{9}-\frac{11\cdots 44}{36\cdots 69}a^{8}+\frac{94\cdots 00}{10\cdots 07}a^{7}-\frac{47\cdots 10}{36\cdots 69}a^{6}+\frac{12\cdots 66}{36\cdots 69}a^{5}-\frac{55\cdots 50}{10\cdots 07}a^{4}+\frac{11\cdots 16}{10\cdots 07}a^{3}-\frac{10\cdots 51}{10\cdots 07}a^{2}+\frac{36\cdots 60}{36\cdots 69}a+\frac{23\cdots 41}{10\cdots 07}$, $\frac{82\cdots 76}{36\cdots 69}a^{19}-\frac{13\cdots 64}{10\cdots 07}a^{18}+\frac{47\cdots 44}{36\cdots 69}a^{17}-\frac{18\cdots 39}{36\cdots 69}a^{16}+\frac{36\cdots 68}{10\cdots 07}a^{15}-\frac{11\cdots 28}{10\cdots 07}a^{14}+\frac{18\cdots 16}{36\cdots 69}a^{13}-\frac{14\cdots 42}{10\cdots 07}a^{12}+\frac{52\cdots 64}{10\cdots 07}a^{11}-\frac{38\cdots 79}{36\cdots 69}a^{10}+\frac{10\cdots 52}{36\cdots 69}a^{9}-\frac{19\cdots 91}{36\cdots 69}a^{8}+\frac{46\cdots 32}{36\cdots 69}a^{7}-\frac{22\cdots 47}{10\cdots 07}a^{6}+\frac{19\cdots 36}{36\cdots 69}a^{5}-\frac{28\cdots 20}{36\cdots 69}a^{4}+\frac{16\cdots 28}{10\cdots 07}a^{3}-\frac{52\cdots 85}{36\cdots 69}a^{2}+\frac{16\cdots 34}{10\cdots 07}a-\frac{10\cdots 62}{10\cdots 07}$
|
| |
| Regulator: | \( 530208.250733 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 530208.250733 \cdot 131042}{2\cdot\sqrt{97756416090547441671665711234313879552}}\cr\approx \mathstrut & 0.336940368674 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6 +3 \sqrt{2}})\), \(\Q(\zeta_{11})^+\), 10.10.7024111812608.1 |
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 | R | R | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | $20$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/23.1.0.1}{1} }^{20}$ | $20$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | $20$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.55a1.6634 | $x^{20} + 4 x^{17} + 12 x^{15} + 6 x^{14} + 36 x^{12} + 4 x^{11} + 34 x^{10} + 36 x^{9} + x^{8} + 68 x^{7} + 12 x^{6} + 36 x^{5} + 34 x^{4} + 36 x^{2} + 31$ | $4$ | $5$ | $55$ | 20T1 | not computed |
|
\(3\)
| 3.10.2.10a1.1 | $x^{20} + 4 x^{16} + 4 x^{15} + 4 x^{14} + 4 x^{12} + 10 x^{11} + 16 x^{10} + 8 x^{9} + 4 x^{8} + 4 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + x^{2} + 7 x + 4$ | $2$ | $10$ | $10$ | 20T1 | $$[\ ]_{2}^{10}$$ |
|
\(11\)
| 11.4.5.16a1.4 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1600 x^{15} + 6440 x^{14} + 19600 x^{13} + 48360 x^{12} + 122000 x^{11} + 252208 x^{10} + 442800 x^{9} + 706720 x^{8} + 904000 x^{7} + 817760 x^{6} + 498400 x^{5} + 201200 x^{4} + 52800 x^{3} + 8640 x^{2} + 800 x + 43$ | $5$ | $4$ | $16$ | 20T1 | $$[\ ]_{5}^{4}$$ |