Normalized defining polynomial
\( x^{21} - 441 x^{19} - 504 x^{18} + 75383 x^{17} + 143990 x^{16} - 6403488 x^{15} - 15018669 x^{14} + \cdots - 64432726399 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[21, 0]$ |
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| Discriminant: |
\(98353367498957471525704156941061377491148627676882129\)
\(\medspace = 7^{38}\cdot 31^{14}\)
|
| |
| Root discriminant: | \(333.78\) |
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| Galois root discriminant: | $7^{38/21}31^{2/3}\approx 333.7844912351955$ | ||
| Ramified primes: |
\(7\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{21}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1519=7^{2}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1519}(1152,·)$, $\chi_{1519}(1,·)$, $\chi_{1519}(67,·)$, $\chi_{1519}(583,·)$, $\chi_{1519}(652,·)$, $\chi_{1519}(718,·)$, $\chi_{1519}(1234,·)$, $\chi_{1519}(149,·)$, $\chi_{1519}(1303,·)$, $\chi_{1519}(1369,·)$, $\chi_{1519}(218,·)$, $\chi_{1519}(284,·)$, $\chi_{1519}(800,·)$, $\chi_{1519}(869,·)$, $\chi_{1519}(935,·)$, $\chi_{1519}(1451,·)$, $\chi_{1519}(366,·)$, $\chi_{1519}(435,·)$, $\chi_{1519}(501,·)$, $\chi_{1519}(1017,·)$, $\chi_{1519}(1086,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19}a^{12}-\frac{7}{19}a^{11}+\frac{4}{19}a^{10}-\frac{6}{19}a^{9}-\frac{3}{19}a^{8}+\frac{2}{19}a^{7}+\frac{2}{19}a^{6}-\frac{5}{19}a^{5}-\frac{3}{19}a^{4}-\frac{2}{19}a^{3}-\frac{9}{19}a^{2}-\frac{1}{19}a+\frac{8}{19}$, $\frac{1}{19}a^{13}-\frac{7}{19}a^{11}+\frac{3}{19}a^{10}-\frac{7}{19}a^{9}-\frac{3}{19}a^{7}+\frac{9}{19}a^{6}-\frac{4}{19}a^{4}-\frac{4}{19}a^{3}-\frac{7}{19}a^{2}+\frac{1}{19}a-\frac{1}{19}$, $\frac{1}{19}a^{14}-\frac{8}{19}a^{11}+\frac{2}{19}a^{10}-\frac{4}{19}a^{9}-\frac{5}{19}a^{8}+\frac{4}{19}a^{7}-\frac{5}{19}a^{6}-\frac{1}{19}a^{5}-\frac{6}{19}a^{4}-\frac{2}{19}a^{3}-\frac{5}{19}a^{2}-\frac{8}{19}a-\frac{1}{19}$, $\frac{1}{19}a^{15}+\frac{3}{19}a^{11}+\frac{9}{19}a^{10}+\frac{4}{19}a^{9}-\frac{1}{19}a^{8}-\frac{8}{19}a^{7}-\frac{4}{19}a^{6}-\frac{8}{19}a^{5}-\frac{7}{19}a^{4}-\frac{2}{19}a^{3}-\frac{4}{19}a^{2}-\frac{9}{19}a+\frac{7}{19}$, $\frac{1}{19}a^{16}-\frac{8}{19}a^{11}-\frac{8}{19}a^{10}-\frac{2}{19}a^{9}+\frac{1}{19}a^{8}+\frac{9}{19}a^{7}+\frac{5}{19}a^{6}+\frac{8}{19}a^{5}+\frac{7}{19}a^{4}+\frac{2}{19}a^{3}-\frac{1}{19}a^{2}-\frac{9}{19}a-\frac{5}{19}$, $\frac{1}{1273}a^{17}+\frac{3}{1273}a^{16}+\frac{17}{1273}a^{15}+\frac{33}{1273}a^{14}-\frac{9}{1273}a^{13}+\frac{5}{1273}a^{12}-\frac{26}{1273}a^{11}-\frac{314}{1273}a^{10}+\frac{125}{1273}a^{9}+\frac{7}{67}a^{8}-\frac{394}{1273}a^{7}-\frac{284}{1273}a^{6}+\frac{253}{1273}a^{5}-\frac{221}{1273}a^{4}-\frac{66}{1273}a^{3}-\frac{33}{1273}a^{2}-\frac{395}{1273}a-\frac{82}{1273}$, $\frac{1}{59233963}a^{18}+\frac{21988}{59233963}a^{17}-\frac{183938}{59233963}a^{16}+\frac{44808}{59233963}a^{15}-\frac{148385}{59233963}a^{14}-\frac{1246946}{59233963}a^{13}-\frac{295518}{59233963}a^{12}-\frac{17752198}{59233963}a^{11}+\frac{10223241}{59233963}a^{10}+\frac{25853744}{59233963}a^{9}+\frac{4015711}{59233963}a^{8}+\frac{2174675}{59233963}a^{7}-\frac{25803668}{59233963}a^{6}-\frac{15365265}{59233963}a^{5}-\frac{6994309}{59233963}a^{4}+\frac{284974}{1910773}a^{3}+\frac{29286817}{59233963}a^{2}+\frac{558377}{3117577}a+\frac{12158757}{59233963}$, $\frac{1}{59233963}a^{19}-\frac{12868}{59233963}a^{17}-\frac{885057}{59233963}a^{16}+\frac{651732}{59233963}a^{15}-\frac{1311286}{59233963}a^{14}+\frac{13874}{884089}a^{13}+\frac{24158}{1910773}a^{12}-\frac{17151559}{59233963}a^{11}-\frac{23436960}{59233963}a^{10}-\frac{28474077}{59233963}a^{9}-\frac{2799355}{59233963}a^{8}+\frac{3129306}{59233963}a^{7}-\frac{3264090}{59233963}a^{6}-\frac{10531087}{59233963}a^{5}-\frac{8101235}{59233963}a^{4}-\frac{857707}{59233963}a^{3}-\frac{10625505}{59233963}a^{2}+\frac{25983499}{59233963}a+\frac{7255342}{59233963}$, $\frac{1}{19\cdots 99}a^{20}+\frac{13\cdots 60}{19\cdots 99}a^{19}-\frac{84\cdots 69}{19\cdots 99}a^{18}-\frac{26\cdots 73}{87\cdots 37}a^{17}+\frac{19\cdots 58}{19\cdots 99}a^{16}-\frac{47\cdots 95}{19\cdots 99}a^{15}+\frac{25\cdots 81}{19\cdots 99}a^{14}-\frac{36\cdots 46}{19\cdots 99}a^{13}-\frac{18\cdots 62}{19\cdots 99}a^{12}-\frac{65\cdots 05}{19\cdots 99}a^{11}+\frac{25\cdots 48}{19\cdots 99}a^{10}+\frac{69\cdots 81}{19\cdots 99}a^{9}-\frac{33\cdots 72}{19\cdots 99}a^{8}-\frac{63\cdots 68}{19\cdots 99}a^{7}+\frac{49\cdots 12}{19\cdots 99}a^{6}-\frac{57\cdots 33}{19\cdots 99}a^{5}-\frac{46\cdots 80}{19\cdots 99}a^{4}+\frac{12\cdots 10}{19\cdots 99}a^{3}-\frac{77\cdots 95}{19\cdots 99}a^{2}-\frac{81\cdots 59}{19\cdots 99}a+\frac{44\cdots 70}{19\cdots 99}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $19$ |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
| |
| Narrow class group: | $C_{3}$, which has order $3$ (assuming GRH) |
|
Unit group
| Rank: | $20$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{68\cdots 74}{10\cdots 21}a^{20}-\frac{70\cdots 34}{10\cdots 21}a^{19}-\frac{30\cdots 97}{10\cdots 21}a^{18}-\frac{13\cdots 03}{48\cdots 23}a^{17}+\frac{51\cdots 69}{10\cdots 21}a^{16}+\frac{44\cdots 60}{10\cdots 21}a^{15}-\frac{43\cdots 37}{10\cdots 21}a^{14}-\frac{55\cdots 59}{10\cdots 21}a^{13}+\frac{20\cdots 68}{10\cdots 21}a^{12}+\frac{30\cdots 39}{10\cdots 21}a^{11}-\frac{54\cdots 51}{10\cdots 21}a^{10}-\frac{89\cdots 24}{10\cdots 21}a^{9}+\frac{76\cdots 36}{10\cdots 21}a^{8}+\frac{14\cdots 47}{10\cdots 21}a^{7}-\frac{52\cdots 85}{10\cdots 21}a^{6}-\frac{12\cdots 35}{10\cdots 21}a^{5}+\frac{10\cdots 05}{10\cdots 21}a^{4}+\frac{39\cdots 40}{10\cdots 21}a^{3}+\frac{26\cdots 07}{10\cdots 21}a^{2}+\frac{41\cdots 17}{10\cdots 21}a-\frac{38\cdots 96}{10\cdots 21}$, $\frac{57\cdots 93}{33\cdots 51}a^{20}-\frac{10\cdots 55}{33\cdots 51}a^{19}-\frac{25\cdots 97}{33\cdots 51}a^{18}-\frac{10\cdots 42}{14\cdots 13}a^{17}+\frac{42\cdots 58}{33\cdots 51}a^{16}+\frac{70\cdots 19}{33\cdots 51}a^{15}-\frac{36\cdots 73}{33\cdots 51}a^{14}-\frac{71\cdots 71}{33\cdots 51}a^{13}+\frac{16\cdots 00}{33\cdots 51}a^{12}+\frac{35\cdots 09}{33\cdots 51}a^{11}-\frac{42\cdots 89}{33\cdots 51}a^{10}-\frac{96\cdots 72}{33\cdots 51}a^{9}+\frac{57\cdots 63}{33\cdots 51}a^{8}+\frac{14\cdots 00}{33\cdots 51}a^{7}-\frac{37\cdots 44}{33\cdots 51}a^{6}-\frac{11\cdots 26}{33\cdots 51}a^{5}+\frac{55\cdots 22}{33\cdots 51}a^{4}+\frac{33\cdots 17}{33\cdots 51}a^{3}+\frac{27\cdots 26}{33\cdots 51}a^{2}+\frac{47\cdots 17}{33\cdots 51}a-\frac{52\cdots 42}{33\cdots 51}$, $\frac{55\cdots 74}{33\cdots 51}a^{20}-\frac{60\cdots 08}{33\cdots 51}a^{19}-\frac{24\cdots 23}{33\cdots 51}a^{18}-\frac{45\cdots 99}{14\cdots 13}a^{17}+\frac{41\cdots 40}{33\cdots 51}a^{16}+\frac{33\cdots 44}{33\cdots 51}a^{15}-\frac{35\cdots 43}{33\cdots 51}a^{14}-\frac{43\cdots 64}{33\cdots 51}a^{13}+\frac{16\cdots 04}{33\cdots 51}a^{12}+\frac{24\cdots 03}{33\cdots 51}a^{11}-\frac{44\cdots 36}{33\cdots 51}a^{10}-\frac{72\cdots 00}{33\cdots 51}a^{9}+\frac{63\cdots 86}{33\cdots 51}a^{8}+\frac{11\cdots 21}{33\cdots 51}a^{7}-\frac{44\cdots 89}{33\cdots 51}a^{6}-\frac{10\cdots 30}{33\cdots 51}a^{5}+\frac{90\cdots 51}{33\cdots 51}a^{4}+\frac{33\cdots 82}{33\cdots 51}a^{3}+\frac{22\cdots 64}{33\cdots 51}a^{2}+\frac{33\cdots 91}{33\cdots 51}a-\frac{28\cdots 38}{33\cdots 51}$, $\frac{11\cdots 47}{33\cdots 51}a^{20}-\frac{14\cdots 62}{33\cdots 51}a^{19}-\frac{48\cdots 91}{33\cdots 51}a^{18}+\frac{24\cdots 10}{14\cdots 13}a^{17}+\frac{83\cdots 07}{33\cdots 51}a^{16}+\frac{57\cdots 67}{33\cdots 51}a^{15}-\frac{72\cdots 95}{33\cdots 51}a^{14}-\frac{82\cdots 08}{33\cdots 51}a^{13}+\frac{34\cdots 68}{33\cdots 51}a^{12}+\frac{48\cdots 99}{33\cdots 51}a^{11}-\frac{90\cdots 63}{33\cdots 51}a^{10}-\frac{15\cdots 11}{33\cdots 51}a^{9}+\frac{12\cdots 60}{33\cdots 51}a^{8}+\frac{25\cdots 51}{33\cdots 51}a^{7}-\frac{87\cdots 03}{33\cdots 51}a^{6}-\frac{21\cdots 75}{33\cdots 51}a^{5}+\frac{16\cdots 41}{33\cdots 51}a^{4}+\frac{70\cdots 40}{33\cdots 51}a^{3}+\frac{50\cdots 65}{33\cdots 51}a^{2}+\frac{83\cdots 34}{33\cdots 51}a-\frac{16\cdots 52}{33\cdots 51}$, $\frac{15\cdots 35}{33\cdots 51}a^{20}-\frac{18\cdots 16}{33\cdots 51}a^{19}-\frac{68\cdots 81}{33\cdots 51}a^{18}+\frac{85\cdots 20}{14\cdots 13}a^{17}+\frac{11\cdots 70}{33\cdots 51}a^{16}+\frac{86\cdots 19}{33\cdots 51}a^{15}-\frac{10\cdots 59}{33\cdots 51}a^{14}-\frac{11\cdots 93}{33\cdots 51}a^{13}+\frac{47\cdots 58}{33\cdots 51}a^{12}+\frac{65\cdots 93}{33\cdots 51}a^{11}-\frac{12\cdots 70}{33\cdots 51}a^{10}-\frac{19\cdots 25}{33\cdots 51}a^{9}+\frac{17\cdots 03}{33\cdots 51}a^{8}+\frac{31\cdots 73}{33\cdots 51}a^{7}-\frac{12\cdots 17}{33\cdots 51}a^{6}-\frac{26\cdots 17}{33\cdots 51}a^{5}+\frac{24\cdots 09}{33\cdots 51}a^{4}+\frac{87\cdots 12}{33\cdots 51}a^{3}+\frac{59\cdots 16}{33\cdots 51}a^{2}+\frac{96\cdots 30}{33\cdots 51}a-\frac{89\cdots 34}{33\cdots 51}$, $\frac{11\cdots 20}{33\cdots 51}a^{20}+\frac{62\cdots 29}{33\cdots 51}a^{19}-\frac{51\cdots 04}{33\cdots 51}a^{18}-\frac{40\cdots 38}{14\cdots 13}a^{17}+\frac{92\cdots 66}{33\cdots 51}a^{16}+\frac{24\cdots 70}{33\cdots 51}a^{15}-\frac{81\cdots 64}{33\cdots 51}a^{14}-\frac{26\cdots 14}{33\cdots 51}a^{13}+\frac{38\cdots 64}{33\cdots 51}a^{12}+\frac{14\cdots 02}{33\cdots 51}a^{11}-\frac{10\cdots 07}{33\cdots 51}a^{10}-\frac{43\cdots 39}{33\cdots 51}a^{9}+\frac{12\cdots 93}{33\cdots 51}a^{8}+\frac{67\cdots 15}{33\cdots 51}a^{7}-\frac{63\cdots 01}{33\cdots 51}a^{6}-\frac{52\cdots 46}{33\cdots 51}a^{5}-\frac{15\cdots 41}{33\cdots 51}a^{4}+\frac{15\cdots 99}{33\cdots 51}a^{3}+\frac{17\cdots 38}{33\cdots 51}a^{2}+\frac{35\cdots 15}{33\cdots 51}a-\frac{45\cdots 58}{33\cdots 51}$, $\frac{20\cdots 84}{36\cdots 29}a^{20}-\frac{33\cdots 80}{36\cdots 29}a^{19}-\frac{89\cdots 24}{36\cdots 29}a^{18}+\frac{18\cdots 98}{16\cdots 27}a^{17}+\frac{15\cdots 70}{36\cdots 29}a^{16}+\frac{44\cdots 92}{36\cdots 29}a^{15}-\frac{13\cdots 16}{36\cdots 29}a^{14}-\frac{94\cdots 02}{36\cdots 29}a^{13}+\frac{62\cdots 54}{36\cdots 29}a^{12}+\frac{59\cdots 48}{36\cdots 29}a^{11}-\frac{16\cdots 78}{36\cdots 29}a^{10}-\frac{19\cdots 84}{36\cdots 29}a^{9}+\frac{23\cdots 74}{36\cdots 29}a^{8}+\frac{33\cdots 59}{36\cdots 29}a^{7}-\frac{16\cdots 64}{36\cdots 29}a^{6}-\frac{29\cdots 63}{36\cdots 29}a^{5}+\frac{37\cdots 44}{36\cdots 29}a^{4}+\frac{10\cdots 36}{36\cdots 29}a^{3}+\frac{64\cdots 24}{36\cdots 29}a^{2}+\frac{10\cdots 71}{36\cdots 29}a-\frac{32\cdots 46}{11\cdots 59}$, $\frac{15\cdots 40}{36\cdots 29}a^{20}-\frac{13\cdots 52}{36\cdots 29}a^{19}-\frac{66\cdots 12}{36\cdots 29}a^{18}-\frac{73\cdots 68}{16\cdots 27}a^{17}+\frac{11\cdots 52}{36\cdots 29}a^{16}+\frac{11\cdots 84}{36\cdots 29}a^{15}-\frac{97\cdots 83}{36\cdots 29}a^{14}-\frac{13\cdots 24}{36\cdots 29}a^{13}+\frac{45\cdots 46}{36\cdots 29}a^{12}+\frac{74\cdots 64}{36\cdots 29}a^{11}-\frac{38\cdots 85}{11\cdots 59}a^{10}-\frac{21\cdots 56}{36\cdots 29}a^{9}+\frac{17\cdots 86}{36\cdots 29}a^{8}+\frac{33\cdots 54}{36\cdots 29}a^{7}-\frac{12\cdots 86}{36\cdots 29}a^{6}-\frac{27\cdots 18}{36\cdots 29}a^{5}+\frac{27\cdots 68}{36\cdots 29}a^{4}+\frac{89\cdots 44}{36\cdots 29}a^{3}+\frac{39\cdots 21}{36\cdots 29}a^{2}-\frac{93\cdots 22}{36\cdots 29}a+\frac{39\cdots 75}{36\cdots 29}$, $\frac{27\cdots 12}{19\cdots 99}a^{20}-\frac{45\cdots 33}{64\cdots 29}a^{19}-\frac{13\cdots 13}{19\cdots 99}a^{18}+\frac{60\cdots 78}{28\cdots 27}a^{17}+\frac{24\cdots 41}{19\cdots 99}a^{16}-\frac{13\cdots 88}{64\cdots 29}a^{15}-\frac{24\cdots 53}{19\cdots 99}a^{14}+\frac{86\cdots 54}{19\cdots 99}a^{13}+\frac{12\cdots 94}{19\cdots 99}a^{12}+\frac{80\cdots 33}{19\cdots 99}a^{11}-\frac{37\cdots 41}{19\cdots 99}a^{10}-\frac{51\cdots 29}{19\cdots 99}a^{9}+\frac{56\cdots 92}{19\cdots 99}a^{8}+\frac{11\cdots 32}{19\cdots 99}a^{7}-\frac{38\cdots 22}{19\cdots 99}a^{6}-\frac{11\cdots 56}{19\cdots 99}a^{5}+\frac{51\cdots 13}{19\cdots 99}a^{4}+\frac{38\cdots 92}{19\cdots 99}a^{3}+\frac{37\cdots 48}{19\cdots 99}a^{2}+\frac{10\cdots 99}{19\cdots 99}a-\frac{83\cdots 84}{19\cdots 99}$, $\frac{35\cdots 00}{10\cdots 21}a^{20}-\frac{85\cdots 33}{10\cdots 21}a^{19}-\frac{15\cdots 09}{10\cdots 21}a^{18}-\frac{60\cdots 44}{46\cdots 23}a^{17}+\frac{26\cdots 77}{10\cdots 21}a^{16}+\frac{42\cdots 84}{10\cdots 21}a^{15}-\frac{70\cdots 77}{33\cdots 91}a^{14}-\frac{44\cdots 84}{10\cdots 21}a^{13}+\frac{99\cdots 30}{10\cdots 21}a^{12}+\frac{22\cdots 49}{10\cdots 21}a^{11}-\frac{24\cdots 07}{10\cdots 21}a^{10}-\frac{60\cdots 39}{10\cdots 21}a^{9}+\frac{32\cdots 78}{10\cdots 21}a^{8}+\frac{86\cdots 02}{10\cdots 21}a^{7}-\frac{18\cdots 12}{10\cdots 21}a^{6}-\frac{60\cdots 20}{10\cdots 21}a^{5}+\frac{18\cdots 25}{10\cdots 21}a^{4}+\frac{15\cdots 18}{10\cdots 21}a^{3}+\frac{12\cdots 42}{10\cdots 21}a^{2}+\frac{19\cdots 69}{10\cdots 21}a-\frac{24\cdots 66}{10\cdots 21}$, $\frac{22\cdots 78}{19\cdots 99}a^{20}-\frac{28\cdots 98}{19\cdots 99}a^{19}-\frac{96\cdots 34}{19\cdots 99}a^{18}+\frac{53\cdots 24}{87\cdots 37}a^{17}+\frac{16\cdots 64}{19\cdots 99}a^{16}+\frac{10\cdots 24}{19\cdots 99}a^{15}-\frac{14\cdots 08}{19\cdots 99}a^{14}-\frac{14\cdots 53}{19\cdots 99}a^{13}+\frac{67\cdots 15}{19\cdots 99}a^{12}+\frac{85\cdots 64}{19\cdots 99}a^{11}-\frac{17\cdots 42}{19\cdots 99}a^{10}-\frac{25\cdots 65}{19\cdots 99}a^{9}+\frac{25\cdots 66}{19\cdots 99}a^{8}+\frac{42\cdots 81}{19\cdots 99}a^{7}-\frac{17\cdots 05}{19\cdots 99}a^{6}-\frac{35\cdots 10}{19\cdots 99}a^{5}+\frac{38\cdots 58}{19\cdots 99}a^{4}+\frac{11\cdots 35}{19\cdots 99}a^{3}+\frac{70\cdots 36}{19\cdots 99}a^{2}+\frac{81\cdots 21}{19\cdots 99}a-\frac{87\cdots 88}{19\cdots 99}$, $\frac{29\cdots 30}{19\cdots 99}a^{20}-\frac{30\cdots 67}{19\cdots 99}a^{19}-\frac{11\cdots 22}{19\cdots 99}a^{18}+\frac{51\cdots 09}{87\cdots 37}a^{17}+\frac{19\cdots 91}{19\cdots 99}a^{16}-\frac{58\cdots 96}{64\cdots 29}a^{15}-\frac{16\cdots 30}{19\cdots 99}a^{14}+\frac{14\cdots 04}{19\cdots 99}a^{13}+\frac{79\cdots 79}{19\cdots 99}a^{12}-\frac{64\cdots 70}{19\cdots 99}a^{11}-\frac{21\cdots 69}{19\cdots 99}a^{10}+\frac{15\cdots 17}{19\cdots 99}a^{9}+\frac{34\cdots 73}{19\cdots 99}a^{8}-\frac{18\cdots 74}{19\cdots 99}a^{7}-\frac{32\cdots 30}{19\cdots 99}a^{6}+\frac{10\cdots 16}{19\cdots 99}a^{5}+\frac{17\cdots 72}{19\cdots 99}a^{4}-\frac{16\cdots 87}{19\cdots 99}a^{3}-\frac{40\cdots 50}{19\cdots 99}a^{2}-\frac{16\cdots 99}{19\cdots 99}a+\frac{76\cdots 17}{19\cdots 99}$, $\frac{18\cdots 03}{19\cdots 99}a^{20}+\frac{60\cdots 69}{19\cdots 99}a^{19}-\frac{77\cdots 38}{19\cdots 99}a^{18}-\frac{49\cdots 64}{28\cdots 27}a^{17}+\frac{12\cdots 50}{19\cdots 99}a^{16}+\frac{67\cdots 31}{19\cdots 99}a^{15}-\frac{92\cdots 29}{19\cdots 99}a^{14}-\frac{57\cdots 71}{19\cdots 99}a^{13}+\frac{33\cdots 87}{19\cdots 99}a^{12}+\frac{24\cdots 59}{19\cdots 99}a^{11}-\frac{49\cdots 03}{19\cdots 99}a^{10}-\frac{53\cdots 11}{19\cdots 99}a^{9}-\frac{65\cdots 19}{19\cdots 99}a^{8}+\frac{53\cdots 50}{19\cdots 99}a^{7}+\frac{25\cdots 46}{64\cdots 29}a^{6}-\frac{14\cdots 62}{19\cdots 99}a^{5}-\frac{48\cdots 49}{19\cdots 99}a^{4}-\frac{44\cdots 88}{19\cdots 99}a^{3}-\frac{13\cdots 17}{19\cdots 99}a^{2}-\frac{13\cdots 23}{19\cdots 99}a+\frac{81\cdots 64}{19\cdots 99}$, $\frac{50\cdots 19}{19\cdots 99}a^{20}-\frac{68\cdots 31}{19\cdots 99}a^{19}-\frac{22\cdots 32}{19\cdots 99}a^{18}+\frac{21\cdots 63}{87\cdots 37}a^{17}+\frac{37\cdots 55}{19\cdots 99}a^{16}+\frac{20\cdots 86}{19\cdots 99}a^{15}-\frac{32\cdots 66}{19\cdots 99}a^{14}-\frac{30\cdots 66}{19\cdots 99}a^{13}+\frac{15\cdots 63}{19\cdots 99}a^{12}+\frac{17\cdots 37}{19\cdots 99}a^{11}-\frac{40\cdots 25}{19\cdots 99}a^{10}-\frac{54\cdots 52}{19\cdots 99}a^{9}+\frac{58\cdots 30}{19\cdots 99}a^{8}+\frac{89\cdots 19}{19\cdots 99}a^{7}-\frac{40\cdots 58}{19\cdots 99}a^{6}-\frac{76\cdots 68}{19\cdots 99}a^{5}+\frac{95\cdots 14}{19\cdots 99}a^{4}+\frac{26\cdots 60}{19\cdots 99}a^{3}+\frac{13\cdots 35}{19\cdots 99}a^{2}+\frac{89\cdots 81}{19\cdots 99}a-\frac{13\cdots 50}{19\cdots 99}$, $\frac{19\cdots 19}{19\cdots 99}a^{20}-\frac{46\cdots 97}{19\cdots 99}a^{19}-\frac{85\cdots 94}{19\cdots 99}a^{18}-\frac{33\cdots 52}{87\cdots 37}a^{17}+\frac{14\cdots 37}{19\cdots 99}a^{16}+\frac{24\cdots 50}{19\cdots 99}a^{15}-\frac{12\cdots 02}{19\cdots 99}a^{14}-\frac{26\cdots 04}{19\cdots 99}a^{13}+\frac{57\cdots 53}{19\cdots 99}a^{12}+\frac{13\cdots 45}{19\cdots 99}a^{11}-\frac{14\cdots 99}{19\cdots 99}a^{10}-\frac{38\cdots 38}{19\cdots 99}a^{9}+\frac{19\cdots 04}{19\cdots 99}a^{8}+\frac{60\cdots 47}{19\cdots 99}a^{7}-\frac{11\cdots 89}{19\cdots 99}a^{6}-\frac{47\cdots 95}{19\cdots 99}a^{5}+\frac{20\cdots 19}{19\cdots 99}a^{4}+\frac{14\cdots 55}{19\cdots 99}a^{3}+\frac{16\cdots 47}{19\cdots 99}a^{2}+\frac{72\cdots 61}{19\cdots 99}a+\frac{10\cdots 86}{19\cdots 99}$, $\frac{17\cdots 56}{20\cdots 67}a^{20}+\frac{58\cdots 74}{20\cdots 67}a^{19}-\frac{74\cdots 84}{20\cdots 67}a^{18}-\frac{14\cdots 80}{90\cdots 21}a^{17}+\frac{11\cdots 72}{20\cdots 67}a^{16}+\frac{64\cdots 42}{20\cdots 67}a^{15}-\frac{86\cdots 98}{20\cdots 67}a^{14}-\frac{53\cdots 03}{20\cdots 67}a^{13}+\frac{30\cdots 51}{20\cdots 67}a^{12}+\frac{22\cdots 94}{20\cdots 67}a^{11}-\frac{43\cdots 90}{20\cdots 67}a^{10}-\frac{48\cdots 77}{20\cdots 67}a^{9}-\frac{10\cdots 54}{20\cdots 67}a^{8}+\frac{46\cdots 47}{20\cdots 67}a^{7}+\frac{75\cdots 55}{20\cdots 67}a^{6}-\frac{11\cdots 56}{20\cdots 67}a^{5}-\frac{44\cdots 66}{20\cdots 67}a^{4}-\frac{45\cdots 91}{20\cdots 67}a^{3}-\frac{17\cdots 90}{20\cdots 67}a^{2}-\frac{72\cdots 59}{20\cdots 67}a+\frac{55\cdots 14}{20\cdots 67}$, $\frac{23\cdots 03}{19\cdots 99}a^{20}-\frac{63\cdots 70}{19\cdots 99}a^{19}-\frac{10\cdots 91}{19\cdots 99}a^{18}+\frac{68\cdots 34}{87\cdots 37}a^{17}+\frac{17\cdots 89}{19\cdots 99}a^{16}-\frac{12\cdots 49}{19\cdots 99}a^{15}-\frac{14\cdots 47}{19\cdots 99}a^{14}+\frac{40\cdots 53}{19\cdots 99}a^{13}+\frac{66\cdots 78}{19\cdots 99}a^{12}+\frac{12\cdots 88}{19\cdots 99}a^{11}-\frac{16\cdots 26}{19\cdots 99}a^{10}-\frac{52\cdots 60}{19\cdots 99}a^{9}+\frac{75\cdots 67}{64\cdots 29}a^{8}+\frac{16\cdots 97}{19\cdots 99}a^{7}-\frac{15\cdots 45}{19\cdots 99}a^{6}-\frac{19\cdots 11}{19\cdots 99}a^{5}+\frac{34\cdots 42}{19\cdots 99}a^{4}+\frac{79\cdots 61}{19\cdots 99}a^{3}+\frac{48\cdots 90}{19\cdots 99}a^{2}+\frac{72\cdots 78}{19\cdots 99}a-\frac{71\cdots 00}{19\cdots 99}$, $\frac{11\cdots 03}{19\cdots 99}a^{20}-\frac{13\cdots 33}{19\cdots 99}a^{19}-\frac{50\cdots 95}{19\cdots 99}a^{18}+\frac{49\cdots 99}{87\cdots 37}a^{17}+\frac{87\cdots 17}{19\cdots 99}a^{16}+\frac{64\cdots 40}{19\cdots 99}a^{15}-\frac{74\cdots 13}{19\cdots 99}a^{14}-\frac{86\cdots 93}{19\cdots 99}a^{13}+\frac{35\cdots 88}{19\cdots 99}a^{12}+\frac{48\cdots 48}{19\cdots 99}a^{11}-\frac{29\cdots 05}{64\cdots 29}a^{10}-\frac{14\cdots 58}{19\cdots 99}a^{9}+\frac{13\cdots 71}{19\cdots 99}a^{8}+\frac{23\cdots 36}{19\cdots 99}a^{7}-\frac{90\cdots 49}{19\cdots 99}a^{6}-\frac{19\cdots 37}{19\cdots 99}a^{5}+\frac{18\cdots 92}{19\cdots 99}a^{4}+\frac{65\cdots 01}{19\cdots 99}a^{3}+\frac{43\cdots 49}{19\cdots 99}a^{2}+\frac{65\cdots 87}{19\cdots 99}a-\frac{63\cdots 68}{19\cdots 99}$, $\frac{15\cdots 65}{19\cdots 99}a^{20}-\frac{68\cdots 08}{19\cdots 99}a^{19}-\frac{67\cdots 65}{19\cdots 99}a^{18}-\frac{20\cdots 56}{87\cdots 37}a^{17}+\frac{11\cdots 99}{19\cdots 99}a^{16}+\frac{17\cdots 98}{19\cdots 99}a^{15}-\frac{99\cdots 42}{19\cdots 99}a^{14}-\frac{18\cdots 74}{19\cdots 99}a^{13}+\frac{46\cdots 16}{19\cdots 99}a^{12}+\frac{10\cdots 69}{19\cdots 99}a^{11}-\frac{12\cdots 33}{19\cdots 99}a^{10}-\frac{28\cdots 28}{19\cdots 99}a^{9}+\frac{16\cdots 45}{19\cdots 99}a^{8}+\frac{45\cdots 10}{19\cdots 99}a^{7}-\frac{10\cdots 88}{19\cdots 99}a^{6}-\frac{36\cdots 81}{19\cdots 99}a^{5}+\frac{86\cdots 67}{19\cdots 99}a^{4}+\frac{11\cdots 75}{19\cdots 99}a^{3}+\frac{11\cdots 94}{19\cdots 99}a^{2}+\frac{31\cdots 33}{19\cdots 99}a-\frac{26\cdots 44}{19\cdots 99}$, $\frac{21\cdots 67}{19\cdots 99}a^{20}-\frac{61\cdots 50}{19\cdots 99}a^{19}-\frac{96\cdots 75}{19\cdots 99}a^{18}-\frac{37\cdots 72}{87\cdots 37}a^{17}+\frac{16\cdots 75}{19\cdots 99}a^{16}+\frac{27\cdots 09}{19\cdots 99}a^{15}-\frac{14\cdots 17}{19\cdots 99}a^{14}-\frac{30\cdots 05}{19\cdots 99}a^{13}+\frac{67\cdots 20}{19\cdots 99}a^{12}+\frac{52\cdots 21}{64\cdots 29}a^{11}-\frac{17\cdots 55}{19\cdots 99}a^{10}-\frac{47\cdots 02}{19\cdots 99}a^{9}+\frac{23\cdots 05}{19\cdots 99}a^{8}+\frac{77\cdots 87}{19\cdots 99}a^{7}-\frac{13\cdots 06}{19\cdots 99}a^{6}-\frac{63\cdots 71}{19\cdots 99}a^{5}-\frac{82\cdots 07}{19\cdots 99}a^{4}+\frac{19\cdots 29}{19\cdots 99}a^{3}+\frac{26\cdots 91}{19\cdots 99}a^{2}+\frac{13\cdots 77}{19\cdots 99}a+\frac{78\cdots 03}{64\cdots 29}$
|
| |
| Regulator: | \( 97078377381360090000 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 97078377381360090000 \cdot 3}{2\cdot\sqrt{98353367498957471525704156941061377491148627676882129}}\cr\approx \mathstrut & 0.973753562555105 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 21 |
| The 21 conjugacy class representatives for $C_{21}$ |
| Character table for $C_{21}$ |
Intermediate fields
| 3.3.47089.1, 7.7.13841287201.1 |
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 | $21$ | $21$ | $21$ | R | ${\href{/padicField/11.7.0.1}{7} }^{3}$ | $21$ | $21$ | ${\href{/padicField/19.1.0.1}{1} }^{21}$ | $21$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | R | ${\href{/padicField/37.7.0.1}{7} }^{3}$ | $21$ | $21$ | $21$ | ${\href{/padicField/53.7.0.1}{7} }^{3}$ | $21$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.21.38a4.243 | $x^{21} + 28 x^{18} + 217$ | $21$ | $1$ | $38$ | $C_{21}$ | not computed |
|
\(31\)
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 31.1.3.2a1.2 | $x^{3} + 93$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |