Normalized defining polynomial
\( x^{20} - 2 x^{19} + 2 x^{18} + 40 x^{17} - 140 x^{16} + 258 x^{15} - 382 x^{14} - 94 x^{13} + 830 x^{12} + \cdots + 1 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(4, 8)$ |
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| Discriminant: |
\(7604915305565615172624892035072\)
\(\medspace = 2^{20}\cdot 3^{15}\cdot 11^{17}\)
|
| |
| Root discriminant: | \(35.00\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{43}a^{18}+\frac{14}{43}a^{17}+\frac{15}{43}a^{15}+\frac{14}{43}a^{14}+\frac{16}{43}a^{13}+\frac{21}{43}a^{12}-\frac{20}{43}a^{11}+\frac{21}{43}a^{10}+\frac{16}{43}a^{9}-\frac{6}{43}a^{8}+\frac{15}{43}a^{7}+\frac{5}{43}a^{6}-\frac{4}{43}a^{5}+\frac{17}{43}a^{4}-\frac{6}{43}a^{3}+\frac{7}{43}a^{2}+\frac{17}{43}a+\frac{4}{43}$, $\frac{1}{16\cdots 81}a^{19}-\frac{14\cdots 28}{16\cdots 81}a^{18}-\frac{21\cdots 27}{16\cdots 81}a^{17}+\frac{73\cdots 66}{16\cdots 81}a^{16}-\frac{79\cdots 86}{16\cdots 81}a^{15}-\frac{42\cdots 51}{16\cdots 81}a^{14}+\frac{48\cdots 00}{16\cdots 81}a^{13}-\frac{57\cdots 64}{16\cdots 81}a^{12}+\frac{32\cdots 50}{16\cdots 81}a^{11}-\frac{40\cdots 85}{16\cdots 81}a^{10}+\frac{41\cdots 30}{16\cdots 81}a^{9}-\frac{62\cdots 79}{16\cdots 81}a^{8}-\frac{65\cdots 18}{16\cdots 81}a^{7}-\frac{28\cdots 82}{16\cdots 81}a^{6}-\frac{12\cdots 04}{16\cdots 81}a^{5}-\frac{73\cdots 43}{16\cdots 81}a^{4}+\frac{69\cdots 00}{16\cdots 81}a^{3}+\frac{69\cdots 45}{16\cdots 81}a^{2}+\frac{19\cdots 02}{16\cdots 81}a-\frac{31\cdots 82}{16\cdots 81}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{90\cdots 92}{70\cdots 47}a^{19}-\frac{42\cdots 53}{70\cdots 47}a^{18}+\frac{63\cdots 20}{70\cdots 47}a^{17}+\frac{31\cdots 05}{70\cdots 47}a^{16}-\frac{22\cdots 76}{70\cdots 47}a^{15}+\frac{56\cdots 84}{70\cdots 47}a^{14}-\frac{94\cdots 42}{70\cdots 47}a^{13}+\frac{78\cdots 28}{70\cdots 47}a^{12}+\frac{11\cdots 64}{70\cdots 47}a^{11}-\frac{33\cdots 86}{70\cdots 47}a^{10}+\frac{45\cdots 88}{70\cdots 47}a^{9}+\frac{63\cdots 29}{70\cdots 47}a^{8}-\frac{15\cdots 56}{70\cdots 47}a^{7}+\frac{32\cdots 90}{70\cdots 47}a^{6}-\frac{47\cdots 62}{70\cdots 47}a^{5}+\frac{46\cdots 82}{70\cdots 47}a^{4}-\frac{30\cdots 92}{70\cdots 47}a^{3}+\frac{14\cdots 25}{70\cdots 47}a^{2}-\frac{39\cdots 26}{70\cdots 47}a-\frac{15\cdots 17}{70\cdots 47}$, $\frac{12\cdots 42}{70\cdots 47}a^{19}+\frac{13\cdots 41}{70\cdots 47}a^{18}-\frac{30\cdots 16}{70\cdots 47}a^{17}+\frac{77\cdots 27}{70\cdots 47}a^{16}+\frac{45\cdots 14}{70\cdots 47}a^{15}-\frac{19\cdots 82}{70\cdots 47}a^{14}+\frac{35\cdots 40}{70\cdots 47}a^{13}-\frac{59\cdots 86}{70\cdots 47}a^{12}-\frac{38\cdots 08}{70\cdots 47}a^{11}+\frac{12\cdots 14}{70\cdots 47}a^{10}-\frac{22\cdots 04}{70\cdots 47}a^{9}+\frac{19\cdots 18}{70\cdots 47}a^{8}+\frac{36\cdots 84}{70\cdots 47}a^{7}-\frac{12\cdots 23}{70\cdots 47}a^{6}+\frac{20\cdots 90}{70\cdots 47}a^{5}-\frac{23\cdots 26}{70\cdots 47}a^{4}+\frac{16\cdots 18}{70\cdots 47}a^{3}-\frac{79\cdots 29}{70\cdots 47}a^{2}+\frac{21\cdots 44}{70\cdots 47}a+\frac{13\cdots 17}{70\cdots 47}$, $\frac{16\cdots 88}{70\cdots 47}a^{19}-\frac{55\cdots 82}{70\cdots 47}a^{18}+\frac{70\cdots 48}{70\cdots 47}a^{17}+\frac{60\cdots 13}{70\cdots 47}a^{16}-\frac{31\cdots 04}{70\cdots 47}a^{15}+\frac{70\cdots 90}{70\cdots 47}a^{14}-\frac{11\cdots 96}{70\cdots 47}a^{13}+\frac{70\cdots 07}{70\cdots 47}a^{12}+\frac{15\cdots 20}{70\cdots 47}a^{11}-\frac{40\cdots 10}{70\cdots 47}a^{10}+\frac{50\cdots 48}{70\cdots 47}a^{9}+\frac{22\cdots 14}{70\cdots 47}a^{8}-\frac{20\cdots 16}{70\cdots 47}a^{7}+\frac{40\cdots 08}{70\cdots 47}a^{6}-\frac{57\cdots 76}{70\cdots 47}a^{5}+\frac{55\cdots 84}{70\cdots 47}a^{4}-\frac{37\cdots 56}{70\cdots 47}a^{3}+\frac{17\cdots 12}{70\cdots 47}a^{2}-\frac{48\cdots 60}{70\cdots 47}a-\frac{48\cdots 21}{70\cdots 47}$, $\frac{12\cdots 72}{70\cdots 47}a^{19}-\frac{91\cdots 30}{70\cdots 47}a^{18}+\frac{15\cdots 60}{70\cdots 47}a^{17}+\frac{39\cdots 99}{70\cdots 47}a^{16}-\frac{44\cdots 54}{70\cdots 47}a^{15}+\frac{12\cdots 30}{70\cdots 47}a^{14}-\frac{21\cdots 24}{70\cdots 47}a^{13}+\frac{22\cdots 57}{70\cdots 47}a^{12}+\frac{19\cdots 20}{70\cdots 47}a^{11}-\frac{74\cdots 85}{70\cdots 47}a^{10}+\frac{11\cdots 72}{70\cdots 47}a^{9}-\frac{18\cdots 80}{70\cdots 47}a^{8}-\frac{31\cdots 66}{70\cdots 47}a^{7}+\frac{73\cdots 96}{70\cdots 47}a^{6}-\frac{10\cdots 46}{70\cdots 47}a^{5}+\frac{11\cdots 96}{70\cdots 47}a^{4}-\frac{75\cdots 18}{70\cdots 47}a^{3}+\frac{35\cdots 21}{70\cdots 47}a^{2}-\frac{97\cdots 18}{70\cdots 47}a-\frac{47\cdots 50}{70\cdots 47}$, $\frac{10\cdots 17}{70\cdots 47}a^{19}-\frac{19\cdots 88}{70\cdots 47}a^{18}+\frac{19\cdots 35}{70\cdots 47}a^{17}+\frac{41\cdots 52}{70\cdots 47}a^{16}-\frac{14\cdots 58}{70\cdots 47}a^{15}+\frac{25\cdots 84}{70\cdots 47}a^{14}-\frac{37\cdots 42}{70\cdots 47}a^{13}-\frac{12\cdots 08}{70\cdots 47}a^{12}+\frac{83\cdots 13}{70\cdots 47}a^{11}-\frac{15\cdots 32}{70\cdots 47}a^{10}+\frac{97\cdots 12}{70\cdots 47}a^{9}+\frac{31\cdots 92}{70\cdots 47}a^{8}-\frac{86\cdots 27}{70\cdots 47}a^{7}+\frac{14\cdots 14}{70\cdots 47}a^{6}-\frac{36\cdots 10}{16\cdots 29}a^{5}+\frac{11\cdots 38}{70\cdots 47}a^{4}-\frac{60\cdots 98}{70\cdots 47}a^{3}+\frac{24\cdots 98}{70\cdots 47}a^{2}+\frac{22\cdots 80}{70\cdots 47}a+\frac{27\cdots 18}{70\cdots 47}$, $\frac{42\cdots 11}{16\cdots 81}a^{19}-\frac{90\cdots 04}{16\cdots 81}a^{18}+\frac{94\cdots 91}{16\cdots 81}a^{17}+\frac{16\cdots 43}{16\cdots 81}a^{16}-\frac{61\cdots 41}{16\cdots 81}a^{15}+\frac{11\cdots 52}{16\cdots 81}a^{14}-\frac{17\cdots 98}{16\cdots 81}a^{13}-\frac{24\cdots 65}{16\cdots 81}a^{12}+\frac{36\cdots 82}{16\cdots 81}a^{11}-\frac{70\cdots 33}{16\cdots 81}a^{10}+\frac{50\cdots 14}{16\cdots 81}a^{9}+\frac{12\cdots 86}{16\cdots 81}a^{8}-\frac{38\cdots 84}{16\cdots 81}a^{7}+\frac{64\cdots 87}{16\cdots 81}a^{6}-\frac{74\cdots 74}{16\cdots 81}a^{5}+\frac{55\cdots 07}{16\cdots 81}a^{4}-\frac{30\cdots 82}{16\cdots 81}a^{3}+\frac{37\cdots 46}{16\cdots 81}a^{2}+\frac{79\cdots 91}{16\cdots 81}a+\frac{36\cdots 03}{16\cdots 81}$, $\frac{83\cdots 14}{16\cdots 81}a^{19}-\frac{14\cdots 62}{16\cdots 81}a^{18}+\frac{13\cdots 23}{16\cdots 81}a^{17}+\frac{33\cdots 77}{16\cdots 81}a^{16}-\frac{10\cdots 70}{16\cdots 81}a^{15}+\frac{19\cdots 21}{16\cdots 81}a^{14}-\frac{27\cdots 34}{16\cdots 81}a^{13}-\frac{13\cdots 17}{16\cdots 81}a^{12}+\frac{65\cdots 28}{16\cdots 81}a^{11}-\frac{11\cdots 38}{16\cdots 81}a^{10}+\frac{63\cdots 43}{16\cdots 81}a^{9}+\frac{26\cdots 66}{16\cdots 81}a^{8}-\frac{66\cdots 66}{16\cdots 81}a^{7}+\frac{10\cdots 49}{16\cdots 81}a^{6}-\frac{11\cdots 41}{16\cdots 81}a^{5}+\frac{74\cdots 86}{16\cdots 81}a^{4}-\frac{37\cdots 07}{16\cdots 81}a^{3}-\frac{36\cdots 50}{16\cdots 81}a^{2}+\frac{96\cdots 59}{16\cdots 81}a-\frac{66\cdots 61}{16\cdots 81}$, $\frac{94\cdots 52}{16\cdots 81}a^{19}-\frac{15\cdots 65}{16\cdots 81}a^{18}+\frac{13\cdots 74}{16\cdots 81}a^{17}+\frac{38\cdots 36}{16\cdots 81}a^{16}-\frac{11\cdots 07}{16\cdots 81}a^{15}+\frac{20\cdots 48}{16\cdots 81}a^{14}-\frac{29\cdots 68}{16\cdots 81}a^{13}-\frac{17\cdots 70}{16\cdots 81}a^{12}+\frac{70\cdots 74}{16\cdots 81}a^{11}-\frac{12\cdots 14}{16\cdots 81}a^{10}+\frac{58\cdots 30}{16\cdots 81}a^{9}+\frac{29\cdots 51}{16\cdots 81}a^{8}-\frac{70\cdots 73}{16\cdots 81}a^{7}+\frac{11\cdots 68}{16\cdots 81}a^{6}-\frac{11\cdots 14}{16\cdots 81}a^{5}+\frac{74\cdots 08}{16\cdots 81}a^{4}-\frac{38\cdots 41}{16\cdots 81}a^{3}-\frac{54\cdots 45}{16\cdots 81}a^{2}-\frac{44\cdots 68}{16\cdots 81}a-\frac{50\cdots 68}{16\cdots 81}$, $\frac{12\cdots 04}{16\cdots 81}a^{19}-\frac{26\cdots 28}{16\cdots 81}a^{18}+\frac{29\cdots 49}{16\cdots 81}a^{17}+\frac{47\cdots 52}{16\cdots 81}a^{16}-\frac{17\cdots 08}{16\cdots 81}a^{15}+\frac{34\cdots 93}{16\cdots 81}a^{14}-\frac{52\cdots 81}{16\cdots 81}a^{13}-\frac{14\cdots 23}{16\cdots 81}a^{12}+\frac{99\cdots 66}{16\cdots 81}a^{11}-\frac{20\cdots 14}{16\cdots 81}a^{10}+\frac{16\cdots 84}{16\cdots 81}a^{9}+\frac{32\cdots 98}{16\cdots 81}a^{8}-\frac{25\cdots 49}{37\cdots 67}a^{7}+\frac{19\cdots 08}{16\cdots 81}a^{6}-\frac{23\cdots 14}{16\cdots 81}a^{5}+\frac{18\cdots 07}{16\cdots 81}a^{4}-\frac{11\cdots 64}{16\cdots 81}a^{3}+\frac{25\cdots 33}{16\cdots 81}a^{2}-\frac{15\cdots 16}{16\cdots 81}a+\frac{64\cdots 44}{16\cdots 81}$, $\frac{21\cdots 42}{16\cdots 81}a^{19}-\frac{45\cdots 95}{16\cdots 81}a^{18}+\frac{50\cdots 75}{16\cdots 81}a^{17}+\frac{83\cdots 10}{16\cdots 81}a^{16}-\frac{31\cdots 53}{16\cdots 81}a^{15}+\frac{59\cdots 33}{16\cdots 81}a^{14}-\frac{90\cdots 28}{16\cdots 81}a^{13}-\frac{42\cdots 59}{16\cdots 81}a^{12}+\frac{17\cdots 20}{16\cdots 81}a^{11}-\frac{36\cdots 54}{16\cdots 81}a^{10}+\frac{28\cdots 35}{16\cdots 81}a^{9}+\frac{58\cdots 45}{16\cdots 81}a^{8}-\frac{19\cdots 37}{16\cdots 81}a^{7}+\frac{33\cdots 82}{16\cdots 81}a^{6}-\frac{39\cdots 57}{16\cdots 81}a^{5}+\frac{31\cdots 13}{16\cdots 81}a^{4}-\frac{18\cdots 28}{16\cdots 81}a^{3}+\frac{40\cdots 64}{16\cdots 81}a^{2}-\frac{94\cdots 19}{16\cdots 81}a+\frac{11\cdots 41}{16\cdots 81}$, $\frac{18\cdots 46}{70\cdots 47}a^{19}-\frac{24\cdots 04}{70\cdots 47}a^{18}+\frac{18\cdots 68}{70\cdots 47}a^{17}+\frac{74\cdots 47}{70\cdots 47}a^{16}-\frac{20\cdots 73}{70\cdots 47}a^{15}+\frac{32\cdots 72}{70\cdots 47}a^{14}-\frac{45\cdots 04}{70\cdots 47}a^{13}-\frac{52\cdots 63}{70\cdots 47}a^{12}+\frac{12\cdots 56}{70\cdots 47}a^{11}-\frac{20\cdots 16}{70\cdots 47}a^{10}+\frac{39\cdots 60}{70\cdots 47}a^{9}+\frac{60\cdots 45}{70\cdots 47}a^{8}-\frac{12\cdots 05}{70\cdots 47}a^{7}+\frac{17\cdots 35}{70\cdots 47}a^{6}-\frac{16\cdots 34}{70\cdots 47}a^{5}+\frac{18\cdots 60}{16\cdots 29}a^{4}-\frac{32\cdots 11}{70\cdots 47}a^{3}-\frac{30\cdots 38}{70\cdots 47}a^{2}-\frac{67\cdots 20}{70\cdots 47}a-\frac{48\cdots 04}{70\cdots 47}$
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| Regulator: | \( 35301761.581 \) (assuming GRH) |
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| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 35301761.581 \cdot 1}{2\cdot\sqrt{7604915305565615172624892035072}}\cr\approx \mathstrut & 0.24875860946 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.C_{10}$ (as 20T427):
| A solvable group of order 10240 |
| The 136 conjugacy class representatives for $C_2^{10}.C_{10}$ |
| Character table for $C_2^{10}.C_{10}$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.53339349076992.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.8.81693426134005631737181457408.3 |
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 | $20$ | $20$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{6}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.2.20a83.2 | $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{16} + 4 x^{15} + 4 x^{14} + 6 x^{13} + 7 x^{12} + 10 x^{11} + 9 x^{10} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 11 x^{6} + 10 x^{5} + 9 x^{4} + 16 x^{3} + 9 x^{2} + 6 x + 5$ | $2$ | $10$ | $20$ | not computed | not computed |
|
\(3\)
| 3.5.4.15a1.1 | $x^{20} + 8 x^{16} + 4 x^{15} + 24 x^{12} + 24 x^{11} + 6 x^{10} + 32 x^{8} + 48 x^{7} + 24 x^{6} + 4 x^{5} + 16 x^{4} + 32 x^{3} + 24 x^{2} + 11 x + 1$ | $4$ | $5$ | $15$ | 20T12 | $$[\ ]_{4}^{10}$$ |
|
\(11\)
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |
| 11.1.10.9a1.1 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |