Normalized defining polynomial
\( x^{22} - 33 x^{20} + 385 x^{18} - 2057 x^{16} + 6754 x^{14} - 18106 x^{12} - 3800 x^{11} + 27874 x^{10} + \cdots - 13 \)
Invariants
| Degree: | $22$ |
| |
| Signature: | $[14, 4]$ |
| |
| Discriminant: |
\(1086355631527929051985063244906158383693824\)
\(\medspace = 2^{32}\cdot 7^{10}\cdot 11^{23}\)
|
| |
| Root discriminant: | \(81.42\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{11}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{3}{8}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{8}a^{18}-\frac{1}{4}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{8}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{19}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{8}a^{20}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{30\cdots 60}a^{21}+\frac{41\cdots 19}{76\cdots 90}a^{20}-\frac{69\cdots 07}{30\cdots 60}a^{19}-\frac{47\cdots 71}{15\cdots 80}a^{18}-\frac{32\cdots 77}{30\cdots 60}a^{17}-\frac{16\cdots 69}{38\cdots 95}a^{16}-\frac{19\cdots 27}{15\cdots 80}a^{15}-\frac{12\cdots 37}{15\cdots 80}a^{14}-\frac{28\cdots 51}{30\cdots 76}a^{13}+\frac{10\cdots 29}{30\cdots 76}a^{12}+\frac{57\cdots 31}{76\cdots 90}a^{11}+\frac{10\cdots 07}{15\cdots 80}a^{10}-\frac{30\cdots 51}{15\cdots 80}a^{9}+\frac{17\cdots 21}{30\cdots 76}a^{8}+\frac{11\cdots 31}{30\cdots 76}a^{7}-\frac{11\cdots 71}{15\cdots 80}a^{6}-\frac{36\cdots 03}{30\cdots 60}a^{5}+\frac{53\cdots 31}{15\cdots 80}a^{4}-\frac{93\cdots 57}{30\cdots 60}a^{3}-\frac{59\cdots 28}{38\cdots 95}a^{2}-\frac{83\cdots 61}{30\cdots 60}a-\frac{50\cdots 71}{15\cdots 80}$
| 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: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $17$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{14\cdots 07}{76\cdots 19}a^{21}+\frac{44\cdots 27}{15\cdots 38}a^{20}-\frac{46\cdots 12}{76\cdots 19}a^{19}-\frac{14\cdots 93}{15\cdots 38}a^{18}+\frac{21\cdots 21}{30\cdots 76}a^{17}+\frac{16\cdots 29}{15\cdots 38}a^{16}-\frac{11\cdots 93}{30\cdots 76}a^{15}-\frac{43\cdots 72}{76\cdots 19}a^{14}+\frac{36\cdots 25}{30\cdots 76}a^{13}+\frac{14\cdots 19}{76\cdots 19}a^{12}-\frac{98\cdots 11}{30\cdots 76}a^{11}-\frac{90\cdots 09}{76\cdots 19}a^{10}+\frac{14\cdots 49}{30\cdots 76}a^{9}-\frac{21\cdots 05}{76\cdots 19}a^{8}-\frac{24\cdots 11}{30\cdots 76}a^{7}+\frac{14\cdots 65}{15\cdots 38}a^{6}+\frac{12\cdots 23}{30\cdots 76}a^{5}-\frac{32\cdots 27}{76\cdots 19}a^{4}-\frac{21\cdots 01}{30\cdots 76}a^{3}+\frac{16\cdots 69}{76\cdots 19}a^{2}+\frac{56\cdots 61}{15\cdots 38}a+\frac{20\cdots 84}{76\cdots 19}$, $\frac{43\cdots 51}{30\cdots 76}a^{21}-\frac{13\cdots 45}{30\cdots 76}a^{20}-\frac{71\cdots 95}{15\cdots 38}a^{19}+\frac{23\cdots 81}{15\cdots 38}a^{18}+\frac{16\cdots 41}{30\cdots 76}a^{17}-\frac{54\cdots 05}{30\cdots 76}a^{16}-\frac{22\cdots 95}{76\cdots 19}a^{15}+\frac{29\cdots 75}{30\cdots 76}a^{14}+\frac{14\cdots 07}{15\cdots 38}a^{13}-\frac{99\cdots 21}{30\cdots 76}a^{12}-\frac{19\cdots 31}{76\cdots 19}a^{11}-\frac{13\cdots 09}{30\cdots 76}a^{10}+\frac{61\cdots 83}{15\cdots 38}a^{9}-\frac{86\cdots 25}{30\cdots 76}a^{8}-\frac{47\cdots 73}{76\cdots 19}a^{7}+\frac{26\cdots 65}{30\cdots 76}a^{6}+\frac{76\cdots 11}{30\cdots 76}a^{5}-\frac{71\cdots 97}{15\cdots 38}a^{4}-\frac{30\cdots 87}{15\cdots 38}a^{3}+\frac{21\cdots 47}{30\cdots 76}a^{2}-\frac{20\cdots 31}{30\cdots 76}a-\frac{34\cdots 85}{15\cdots 38}$, $\frac{13\cdots 59}{76\cdots 19}a^{21}-\frac{68\cdots 59}{61\cdots 52}a^{20}-\frac{86\cdots 65}{15\cdots 38}a^{19}+\frac{28\cdots 55}{76\cdots 19}a^{18}+\frac{10\cdots 29}{15\cdots 38}a^{17}-\frac{26\cdots 05}{61\cdots 52}a^{16}-\frac{53\cdots 27}{15\cdots 38}a^{15}+\frac{70\cdots 19}{30\cdots 76}a^{14}+\frac{87\cdots 06}{76\cdots 19}a^{13}-\frac{22\cdots 19}{30\cdots 76}a^{12}-\frac{23\cdots 60}{76\cdots 19}a^{11}-\frac{13\cdots 27}{30\cdots 76}a^{10}+\frac{71\cdots 79}{15\cdots 38}a^{9}-\frac{11\cdots 29}{30\cdots 76}a^{8}-\frac{10\cdots 49}{15\cdots 38}a^{7}+\frac{31\cdots 45}{30\cdots 76}a^{6}+\frac{17\cdots 07}{76\cdots 19}a^{5}-\frac{30\cdots 71}{61\cdots 52}a^{4}+\frac{11\cdots 97}{15\cdots 38}a^{3}+\frac{17\cdots 43}{30\cdots 76}a^{2}-\frac{81\cdots 68}{76\cdots 19}a-\frac{55\cdots 05}{61\cdots 52}$, $\frac{56\cdots 01}{15\cdots 38}a^{21}-\frac{12\cdots 29}{61\cdots 52}a^{20}-\frac{37\cdots 89}{30\cdots 76}a^{19}+\frac{21\cdots 39}{30\cdots 76}a^{18}+\frac{10\cdots 31}{76\cdots 19}a^{17}-\frac{49\cdots 87}{61\cdots 52}a^{16}-\frac{23\cdots 29}{30\cdots 76}a^{15}+\frac{65\cdots 19}{15\cdots 38}a^{14}+\frac{75\cdots 95}{30\cdots 76}a^{13}-\frac{10\cdots 95}{76\cdots 19}a^{12}-\frac{20\cdots 79}{30\cdots 76}a^{11}-\frac{79\cdots 10}{76\cdots 19}a^{10}+\frac{30\cdots 31}{30\cdots 76}a^{9}-\frac{58\cdots 50}{76\cdots 19}a^{8}-\frac{46\cdots 99}{30\cdots 76}a^{7}+\frac{17\cdots 92}{76\cdots 19}a^{6}+\frac{15\cdots 43}{30\cdots 76}a^{5}-\frac{65\cdots 87}{61\cdots 52}a^{4}-\frac{39\cdots 26}{76\cdots 19}a^{3}+\frac{36\cdots 19}{30\cdots 76}a^{2}-\frac{58\cdots 59}{30\cdots 76}a-\frac{91\cdots 01}{61\cdots 52}$, $\frac{19\cdots 54}{76\cdots 19}a^{21}+\frac{18\cdots 01}{76\cdots 19}a^{20}-\frac{25\cdots 31}{30\cdots 76}a^{19}-\frac{49\cdots 67}{61\cdots 52}a^{18}+\frac{28\cdots 51}{30\cdots 76}a^{17}+\frac{57\cdots 75}{61\cdots 52}a^{16}-\frac{37\cdots 75}{76\cdots 19}a^{15}-\frac{15\cdots 31}{30\cdots 76}a^{14}+\frac{24\cdots 43}{15\cdots 38}a^{13}+\frac{48\cdots 57}{30\cdots 76}a^{12}-\frac{31\cdots 10}{76\cdots 19}a^{11}-\frac{78\cdots 13}{15\cdots 38}a^{10}+\frac{77\cdots 63}{15\cdots 38}a^{9}+\frac{94\cdots 38}{76\cdots 19}a^{8}-\frac{10\cdots 98}{76\cdots 19}a^{7}+\frac{11\cdots 87}{30\cdots 76}a^{6}+\frac{24\cdots 41}{15\cdots 38}a^{5}-\frac{39\cdots 69}{30\cdots 76}a^{4}-\frac{16\cdots 69}{30\cdots 76}a^{3}-\frac{18\cdots 37}{61\cdots 52}a^{2}+\frac{89\cdots 63}{30\cdots 76}a+\frac{14\cdots 21}{61\cdots 52}$, $\frac{24\cdots 69}{76\cdots 19}a^{21}-\frac{18\cdots 31}{30\cdots 76}a^{20}-\frac{31\cdots 91}{30\cdots 76}a^{19}+\frac{14\cdots 70}{76\cdots 19}a^{18}+\frac{37\cdots 69}{30\cdots 76}a^{17}-\frac{69\cdots 19}{30\cdots 76}a^{16}-\frac{98\cdots 05}{15\cdots 38}a^{15}+\frac{91\cdots 16}{76\cdots 19}a^{14}+\frac{31\cdots 03}{15\cdots 38}a^{13}-\frac{11\cdots 73}{30\cdots 76}a^{12}-\frac{42\cdots 93}{76\cdots 19}a^{11}-\frac{12\cdots 30}{76\cdots 19}a^{10}+\frac{65\cdots 01}{76\cdots 19}a^{9}-\frac{23\cdots 79}{30\cdots 76}a^{8}-\frac{18\cdots 11}{15\cdots 38}a^{7}+\frac{15\cdots 50}{76\cdots 19}a^{6}+\frac{23\cdots 79}{15\cdots 38}a^{5}-\frac{69\cdots 59}{76\cdots 19}a^{4}+\frac{33\cdots 87}{30\cdots 76}a^{3}+\frac{58\cdots 20}{76\cdots 19}a^{2}-\frac{19\cdots 09}{30\cdots 76}a-\frac{69\cdots 19}{15\cdots 38}$, $\frac{49\cdots 83}{15\cdots 38}a^{21}-\frac{14\cdots 77}{30\cdots 76}a^{20}-\frac{32\cdots 05}{30\cdots 76}a^{19}+\frac{97\cdots 51}{61\cdots 52}a^{18}+\frac{38\cdots 15}{30\cdots 76}a^{17}-\frac{11\cdots 13}{61\cdots 52}a^{16}-\frac{51\cdots 37}{76\cdots 19}a^{15}+\frac{29\cdots 97}{30\cdots 76}a^{14}+\frac{34\cdots 57}{15\cdots 38}a^{13}-\frac{93\cdots 29}{30\cdots 76}a^{12}-\frac{45\cdots 62}{76\cdots 19}a^{11}+\frac{52\cdots 83}{76\cdots 19}a^{10}+\frac{17\cdots 85}{15\cdots 38}a^{9}-\frac{13\cdots 60}{76\cdots 19}a^{8}-\frac{34\cdots 34}{76\cdots 19}a^{7}+\frac{11\cdots 83}{30\cdots 76}a^{6}-\frac{14\cdots 84}{76\cdots 19}a^{5}-\frac{24\cdots 25}{15\cdots 38}a^{4}+\frac{27\cdots 53}{30\cdots 76}a^{3}+\frac{64\cdots 69}{61\cdots 52}a^{2}-\frac{13\cdots 89}{30\cdots 76}a-\frac{55\cdots 59}{61\cdots 52}$, $\frac{20\cdots 51}{30\cdots 76}a^{21}-\frac{19\cdots 75}{30\cdots 76}a^{20}-\frac{68\cdots 69}{30\cdots 76}a^{19}+\frac{12\cdots 57}{61\cdots 52}a^{18}+\frac{80\cdots 41}{30\cdots 76}a^{17}-\frac{14\cdots 89}{61\cdots 52}a^{16}-\frac{42\cdots 85}{30\cdots 76}a^{15}+\frac{19\cdots 89}{15\cdots 38}a^{14}+\frac{14\cdots 29}{30\cdots 76}a^{13}-\frac{32\cdots 45}{76\cdots 19}a^{12}-\frac{37\cdots 71}{30\cdots 76}a^{11}-\frac{45\cdots 17}{30\cdots 76}a^{10}+\frac{58\cdots 77}{30\cdots 76}a^{9}-\frac{44\cdots 49}{30\cdots 76}a^{8}-\frac{86\cdots 99}{30\cdots 76}a^{7}+\frac{32\cdots 16}{76\cdots 19}a^{6}+\frac{69\cdots 44}{76\cdots 19}a^{5}-\frac{66\cdots 67}{30\cdots 76}a^{4}+\frac{50\cdots 66}{76\cdots 19}a^{3}+\frac{18\cdots 01}{61\cdots 52}a^{2}-\frac{10\cdots 99}{15\cdots 38}a-\frac{67\cdots 65}{61\cdots 52}$, $\frac{14\cdots 27}{30\cdots 76}a^{21}+\frac{10\cdots 63}{61\cdots 52}a^{20}-\frac{12\cdots 97}{76\cdots 19}a^{19}-\frac{34\cdots 95}{61\cdots 52}a^{18}+\frac{56\cdots 89}{30\cdots 76}a^{17}+\frac{98\cdots 85}{15\cdots 38}a^{16}-\frac{15\cdots 73}{15\cdots 38}a^{15}-\frac{51\cdots 79}{15\cdots 38}a^{14}+\frac{48\cdots 09}{15\cdots 38}a^{13}+\frac{16\cdots 55}{15\cdots 38}a^{12}-\frac{64\cdots 07}{76\cdots 19}a^{11}-\frac{14\cdots 17}{30\cdots 76}a^{10}+\frac{92\cdots 01}{76\cdots 19}a^{9}-\frac{16\cdots 17}{30\cdots 76}a^{8}-\frac{35\cdots 61}{15\cdots 38}a^{7}+\frac{33\cdots 91}{15\cdots 38}a^{6}+\frac{53\cdots 67}{30\cdots 76}a^{5}-\frac{63\cdots 83}{61\cdots 52}a^{4}-\frac{28\cdots 07}{76\cdots 19}a^{3}+\frac{13\cdots 81}{61\cdots 52}a^{2}+\frac{62\cdots 99}{30\cdots 76}a+\frac{24\cdots 79}{30\cdots 76}$, $\frac{14\cdots 57}{71\cdots 32}a^{21}+\frac{12\cdots 51}{14\cdots 64}a^{20}-\frac{45\cdots 63}{71\cdots 32}a^{19}-\frac{99\cdots 45}{35\cdots 66}a^{18}+\frac{46\cdots 35}{71\cdots 32}a^{17}+\frac{45\cdots 59}{14\cdots 64}a^{16}-\frac{18\cdots 55}{71\cdots 32}a^{15}-\frac{11\cdots 73}{71\cdots 32}a^{14}+\frac{39\cdots 79}{71\cdots 32}a^{13}+\frac{35\cdots 37}{71\cdots 32}a^{12}-\frac{81\cdots 37}{71\cdots 32}a^{11}-\frac{98\cdots 09}{71\cdots 32}a^{10}-\frac{30\cdots 69}{71\cdots 32}a^{9}+\frac{84\cdots 97}{71\cdots 32}a^{8}-\frac{12\cdots 93}{71\cdots 32}a^{7}-\frac{18\cdots 93}{71\cdots 32}a^{6}+\frac{54\cdots 04}{17\cdots 33}a^{5}+\frac{24\cdots 95}{14\cdots 64}a^{4}-\frac{12\cdots 78}{17\cdots 33}a^{3}-\frac{17\cdots 87}{71\cdots 32}a^{2}+\frac{12\cdots 25}{35\cdots 66}a+\frac{41\cdots 43}{14\cdots 64}$, $\frac{55\cdots 13}{30\cdots 60}a^{21}-\frac{12\cdots 01}{15\cdots 80}a^{20}-\frac{18\cdots 01}{30\cdots 60}a^{19}+\frac{83\cdots 59}{30\cdots 60}a^{18}+\frac{21\cdots 39}{30\cdots 60}a^{17}-\frac{97\cdots 61}{30\cdots 60}a^{16}-\frac{55\cdots 01}{15\cdots 80}a^{15}+\frac{64\cdots 66}{38\cdots 95}a^{14}+\frac{17\cdots 41}{15\cdots 38}a^{13}-\frac{82\cdots 39}{15\cdots 38}a^{12}-\frac{47\cdots 99}{15\cdots 80}a^{11}+\frac{28\cdots 89}{38\cdots 95}a^{10}+\frac{75\cdots 57}{15\cdots 80}a^{9}-\frac{86\cdots 59}{15\cdots 38}a^{8}-\frac{17\cdots 13}{30\cdots 76}a^{7}+\frac{51\cdots 53}{38\cdots 95}a^{6}-\frac{69\cdots 09}{30\cdots 60}a^{5}-\frac{81\cdots 07}{15\cdots 80}a^{4}+\frac{56\cdots 19}{30\cdots 60}a^{3}+\frac{88\cdots 93}{30\cdots 60}a^{2}-\frac{29\cdots 73}{30\cdots 60}a-\frac{19\cdots 71}{30\cdots 60}$, $\frac{70\cdots 55}{61\cdots 52}a^{21}-\frac{25\cdots 51}{61\cdots 52}a^{20}-\frac{23\cdots 21}{61\cdots 52}a^{19}+\frac{21\cdots 23}{15\cdots 38}a^{18}+\frac{27\cdots 17}{61\cdots 52}a^{17}-\frac{12\cdots 46}{76\cdots 19}a^{16}-\frac{36\cdots 67}{15\cdots 38}a^{15}+\frac{28\cdots 91}{30\cdots 76}a^{14}+\frac{11\cdots 61}{15\cdots 38}a^{13}-\frac{49\cdots 87}{15\cdots 38}a^{12}-\frac{31\cdots 61}{15\cdots 38}a^{11}-\frac{26\cdots 34}{76\cdots 19}a^{10}+\frac{97\cdots 57}{30\cdots 76}a^{9}-\frac{18\cdots 67}{76\cdots 19}a^{8}-\frac{37\cdots 47}{76\cdots 19}a^{7}+\frac{21\cdots 77}{30\cdots 76}a^{6}+\frac{10\cdots 49}{61\cdots 52}a^{5}-\frac{21\cdots 33}{61\cdots 52}a^{4}-\frac{26\cdots 11}{61\cdots 52}a^{3}+\frac{72\cdots 63}{15\cdots 38}a^{2}-\frac{49\cdots 55}{61\cdots 52}a-\frac{13\cdots 36}{76\cdots 19}$, $\frac{74\cdots 57}{61\cdots 52}a^{21}-\frac{42\cdots 81}{61\cdots 52}a^{20}-\frac{24\cdots 11}{61\cdots 52}a^{19}+\frac{14\cdots 39}{61\cdots 52}a^{18}+\frac{29\cdots 21}{61\cdots 52}a^{17}-\frac{40\cdots 25}{15\cdots 38}a^{16}-\frac{20\cdots 60}{76\cdots 19}a^{15}+\frac{42\cdots 71}{30\cdots 76}a^{14}+\frac{26\cdots 05}{30\cdots 76}a^{13}-\frac{34\cdots 18}{76\cdots 19}a^{12}-\frac{18\cdots 96}{76\cdots 19}a^{11}+\frac{21\cdots 01}{30\cdots 76}a^{10}+\frac{63\cdots 49}{15\cdots 38}a^{9}-\frac{30\cdots 54}{76\cdots 19}a^{8}-\frac{35\cdots 79}{76\cdots 19}a^{7}+\frac{31\cdots 69}{30\cdots 76}a^{6}-\frac{30\cdots 83}{61\cdots 52}a^{5}-\frac{37\cdots 43}{61\cdots 52}a^{4}+\frac{45\cdots 75}{61\cdots 52}a^{3}+\frac{67\cdots 07}{61\cdots 52}a^{2}+\frac{29\cdots 99}{61\cdots 52}a-\frac{17\cdots 56}{76\cdots 19}$, $\frac{64\cdots 03}{30\cdots 60}a^{21}-\frac{13\cdots 04}{38\cdots 95}a^{20}-\frac{12\cdots 43}{15\cdots 80}a^{19}+\frac{44\cdots 38}{38\cdots 95}a^{18}+\frac{45\cdots 28}{38\cdots 95}a^{17}-\frac{39\cdots 31}{30\cdots 60}a^{16}-\frac{13\cdots 21}{15\cdots 80}a^{15}+\frac{24\cdots 26}{38\cdots 95}a^{14}+\frac{55\cdots 41}{15\cdots 38}a^{13}-\frac{14\cdots 44}{76\cdots 19}a^{12}-\frac{40\cdots 61}{38\cdots 95}a^{11}+\frac{61\cdots 01}{15\cdots 80}a^{10}+\frac{19\cdots 01}{76\cdots 90}a^{9}-\frac{40\cdots 69}{76\cdots 19}a^{8}-\frac{80\cdots 33}{30\cdots 76}a^{7}+\frac{33\cdots 01}{76\cdots 90}a^{6}+\frac{83\cdots 81}{30\cdots 60}a^{5}-\frac{46\cdots 71}{76\cdots 90}a^{4}-\frac{16\cdots 03}{15\cdots 80}a^{3}+\frac{15\cdots 29}{15\cdots 80}a^{2}+\frac{20\cdots 24}{38\cdots 95}a+\frac{35\cdots 99}{30\cdots 60}$, $\frac{56\cdots 47}{30\cdots 60}a^{21}+\frac{36\cdots 13}{76\cdots 90}a^{20}-\frac{18\cdots 89}{30\cdots 60}a^{19}-\frac{23\cdots 77}{15\cdots 80}a^{18}+\frac{21\cdots 81}{30\cdots 60}a^{17}+\frac{27\cdots 13}{15\cdots 80}a^{16}-\frac{57\cdots 29}{15\cdots 80}a^{15}-\frac{14\cdots 19}{15\cdots 80}a^{14}+\frac{94\cdots 27}{76\cdots 19}a^{13}+\frac{95\cdots 05}{30\cdots 76}a^{12}-\frac{50\cdots 11}{15\cdots 80}a^{11}-\frac{59\cdots 24}{38\cdots 95}a^{10}+\frac{73\cdots 23}{15\cdots 80}a^{9}-\frac{17\cdots 35}{76\cdots 19}a^{8}-\frac{26\cdots 63}{30\cdots 76}a^{7}+\frac{13\cdots 23}{15\cdots 80}a^{6}+\frac{17\cdots 29}{30\cdots 60}a^{5}-\frac{65\cdots 13}{15\cdots 80}a^{4}-\frac{48\cdots 49}{30\cdots 60}a^{3}+\frac{64\cdots 81}{15\cdots 80}a^{2}+\frac{48\cdots 13}{30\cdots 60}a+\frac{13\cdots 53}{15\cdots 80}$, $\frac{50\cdots 83}{30\cdots 60}a^{21}+\frac{27\cdots 76}{38\cdots 95}a^{20}-\frac{16\cdots 11}{30\cdots 60}a^{19}-\frac{90\cdots 22}{38\cdots 95}a^{18}+\frac{96\cdots 57}{15\cdots 80}a^{17}+\frac{42\cdots 17}{15\cdots 80}a^{16}-\frac{12\cdots 69}{38\cdots 95}a^{15}-\frac{22\cdots 71}{15\cdots 80}a^{14}+\frac{16\cdots 33}{15\cdots 38}a^{13}+\frac{36\cdots 14}{76\cdots 19}a^{12}-\frac{11\cdots 51}{38\cdots 95}a^{11}-\frac{28\cdots 56}{38\cdots 95}a^{10}+\frac{16\cdots 53}{38\cdots 95}a^{9}-\frac{90\cdots 19}{30\cdots 76}a^{8}-\frac{10\cdots 33}{15\cdots 38}a^{7}+\frac{13\cdots 87}{15\cdots 80}a^{6}+\frac{93\cdots 41}{30\cdots 60}a^{5}-\frac{16\cdots 78}{38\cdots 95}a^{4}-\frac{13\cdots 41}{30\cdots 60}a^{3}+\frac{16\cdots 56}{38\cdots 95}a^{2}+\frac{57\cdots 41}{15\cdots 80}a+\frac{33\cdots 01}{76\cdots 90}$, $\frac{28\cdots 49}{30\cdots 60}a^{21}-\frac{31\cdots 91}{30\cdots 60}a^{20}-\frac{92\cdots 63}{30\cdots 60}a^{19}+\frac{10\cdots 67}{30\cdots 60}a^{18}+\frac{13\cdots 99}{38\cdots 95}a^{17}-\frac{15\cdots 26}{38\cdots 95}a^{16}-\frac{14\cdots 09}{76\cdots 90}a^{15}+\frac{16\cdots 71}{76\cdots 90}a^{14}+\frac{94\cdots 99}{15\cdots 38}a^{13}-\frac{52\cdots 12}{76\cdots 19}a^{12}-\frac{25\cdots 57}{15\cdots 80}a^{11}-\frac{25\cdots 37}{15\cdots 80}a^{10}+\frac{39\cdots 51}{15\cdots 80}a^{9}-\frac{61\cdots 89}{30\cdots 76}a^{8}-\frac{57\cdots 31}{15\cdots 38}a^{7}+\frac{22\cdots 09}{38\cdots 95}a^{6}+\frac{33\cdots 63}{30\cdots 60}a^{5}-\frac{89\cdots 37}{30\cdots 60}a^{4}+\frac{22\cdots 17}{30\cdots 60}a^{3}+\frac{12\cdots 79}{30\cdots 60}a^{2}-\frac{23\cdots 87}{15\cdots 80}a-\frac{22\cdots 89}{15\cdots 80}$
|
| |
| Regulator: | \( 67225391241000 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 67225391241000 \cdot 1}{2\cdot\sqrt{1086355631527929051985063244906158383693824}}\cr\approx \mathstrut & 0.823486996052983 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_2^{10}.F_{11}$ (as 22T37):
| A solvable group of order 225280 |
| The 88 conjugacy class representatives for $C_2\times C_2^{10}.F_{11}$ |
| Character table for $C_2\times C_2^{10}.F_{11}$ |
Intermediate fields
| 11.11.4910318845910094848.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.14.691317220063227578535949337667555335077888.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{9}$ | $20{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.22.32a1.38 | $x^{22} + 2 x^{19} + 2 x^{13} + 2 x^{11} + 6$ | $22$ | $1$ | $32$ | not computed | not computed |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.5.2.5a1.1 | $x^{10} + 2 x^{6} + 8 x^{5} + x^{2} + 15 x + 16$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ | |
| 7.5.2.5a1.1 | $x^{10} + 2 x^{6} + 8 x^{5} + x^{2} + 15 x + 16$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ | |
|
\(11\)
| 11.1.22.23a2.6 | $x^{22} + 11 x^{2} + 22$ | $22$ | $1$ | $23$ | not computed | not computed |