Normalized defining polynomial
\( x^{20} - 264 x^{18} - 976 x^{17} + 26874 x^{16} + 202704 x^{15} - 909396 x^{14} - 15075000 x^{13} + \cdots + 31046369877 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[12, 4]$ |
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| Discriminant: |
\(41041796771414445392922107616798059242325016576\)
\(\medspace = 2^{30}\cdot 3^{20}\cdot 13^{4}\cdot 347^{4}\cdot 5145233^{2}\)
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| Root discriminant: | \(214.12\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(347\), \(5145233\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{28}a^{15}+\frac{1}{7}a^{14}+\frac{1}{28}a^{13}+\frac{3}{14}a^{12}-\frac{3}{28}a^{11}+\frac{1}{28}a^{10}-\frac{13}{28}a^{9}-\frac{3}{14}a^{8}-\frac{1}{7}a^{7}-\frac{5}{28}a^{6}-\frac{9}{28}a^{5}-\frac{3}{7}a^{4}+\frac{13}{28}a^{3}+\frac{1}{7}a^{2}+\frac{3}{14}a-\frac{11}{28}$, $\frac{1}{252}a^{16}-\frac{19}{84}a^{14}-\frac{10}{63}a^{13}-\frac{3}{28}a^{12}-\frac{19}{84}a^{11}-\frac{5}{28}a^{10}-\frac{3}{7}a^{9}-\frac{10}{21}a^{8}+\frac{9}{28}a^{7}+\frac{9}{28}a^{6}-\frac{5}{21}a^{5}+\frac{47}{252}a^{4}-\frac{5}{14}a^{3}+\frac{17}{42}a^{2}-\frac{7}{36}a-\frac{3}{14}$, $\frac{1}{2268}a^{17}-\frac{1}{108}a^{15}+\frac{241}{1134}a^{14}+\frac{43}{252}a^{13}+\frac{95}{756}a^{12}+\frac{53}{252}a^{11}+\frac{31}{126}a^{10}-\frac{17}{54}a^{9}-\frac{1}{252}a^{8}+\frac{13}{36}a^{7}-\frac{62}{189}a^{6}+\frac{731}{2268}a^{5}+\frac{1}{21}a^{4}-\frac{68}{189}a^{3}-\frac{787}{2268}a^{2}+\frac{5}{21}a-\frac{2}{7}$, $\frac{1}{20412}a^{18}-\frac{1}{972}a^{16}-\frac{247}{20412}a^{15}+\frac{475}{2268}a^{14}-\frac{641}{3402}a^{13}+\frac{449}{2268}a^{12}-\frac{325}{2268}a^{11}+\frac{653}{6804}a^{10}-\frac{293}{1134}a^{9}+\frac{703}{2268}a^{8}-\frac{583}{3402}a^{7}+\frac{1621}{10206}a^{6}-\frac{89}{756}a^{5}+\frac{661}{1701}a^{4}-\frac{2297}{10206}a^{3}+\frac{103}{378}a^{2}-\frac{5}{63}a-\frac{3}{28}$, $\frac{1}{20\cdots 48}a^{19}+\frac{35\cdots 11}{33\cdots 96}a^{18}-\frac{17\cdots 21}{34\cdots 58}a^{17}+\frac{14\cdots 79}{10\cdots 74}a^{16}+\frac{71\cdots 99}{77\cdots 24}a^{15}+\frac{28\cdots 27}{69\cdots 16}a^{14}+\frac{66\cdots 80}{58\cdots 93}a^{13}-\frac{49\cdots 39}{23\cdots 72}a^{12}-\frac{20\cdots 55}{99\cdots 88}a^{11}-\frac{53\cdots 33}{23\cdots 72}a^{10}+\frac{56\cdots 59}{23\cdots 72}a^{9}-\frac{59\cdots 13}{34\cdots 58}a^{8}-\frac{14\cdots 39}{20\cdots 48}a^{7}-\frac{11\cdots 53}{23\cdots 72}a^{6}+\frac{67\cdots 17}{34\cdots 58}a^{5}-\frac{52\cdots 31}{52\cdots 37}a^{4}+\frac{13\cdots 23}{58\cdots 93}a^{3}+\frac{11\cdots 21}{25\cdots 08}a^{2}+\frac{15\cdots 33}{31\cdots 68}a-\frac{28\cdots 93}{31\cdots 68}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $15$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{12\cdots 84}{58\cdots 93}a^{19}-\frac{26\cdots 46}{58\cdots 93}a^{18}-\frac{11\cdots 28}{19\cdots 31}a^{17}-\frac{54\cdots 56}{58\cdots 93}a^{16}+\frac{35\cdots 04}{58\cdots 93}a^{15}+\frac{87\cdots 28}{27\cdots 33}a^{14}-\frac{52\cdots 56}{19\cdots 31}a^{13}-\frac{59\cdots 05}{21\cdots 59}a^{12}-\frac{13\cdots 24}{19\cdots 31}a^{11}+\frac{16\cdots 98}{19\cdots 31}a^{10}+\frac{25\cdots 40}{64\cdots 77}a^{9}-\frac{88\cdots 68}{19\cdots 31}a^{8}-\frac{39\cdots 24}{58\cdots 93}a^{7}-\frac{24\cdots 28}{83\cdots 99}a^{6}-\frac{12\cdots 72}{19\cdots 31}a^{5}-\frac{49\cdots 73}{58\cdots 93}a^{4}-\frac{46\cdots 36}{83\cdots 99}a^{3}-\frac{22\cdots 46}{23\cdots 51}a^{2}+\frac{58\cdots 80}{71\cdots 53}a+\frac{23\cdots 36}{79\cdots 17}$, $\frac{69\cdots 32}{52\cdots 37}a^{19}-\frac{49\cdots 94}{58\cdots 93}a^{18}-\frac{53\cdots 88}{17\cdots 79}a^{17}+\frac{56\cdots 47}{74\cdots 91}a^{16}+\frac{19\cdots 52}{58\cdots 93}a^{15}+\frac{81\cdots 84}{17\cdots 79}a^{14}-\frac{10\cdots 28}{58\cdots 93}a^{13}-\frac{51\cdots 73}{58\cdots 93}a^{12}+\frac{54\cdots 24}{17\cdots 79}a^{11}+\frac{21\cdots 94}{58\cdots 93}a^{10}+\frac{48\cdots 88}{58\cdots 93}a^{9}-\frac{58\cdots 56}{17\cdots 79}a^{8}-\frac{13\cdots 92}{52\cdots 37}a^{7}-\frac{43\cdots 60}{58\cdots 93}a^{6}-\frac{20\cdots 20}{17\cdots 79}a^{5}-\frac{55\cdots 27}{52\cdots 37}a^{4}-\frac{25\cdots 76}{58\cdots 93}a^{3}+\frac{17\cdots 82}{64\cdots 77}a^{2}+\frac{55\cdots 80}{71\cdots 53}a+\frac{13\cdots 28}{79\cdots 17}$, $\frac{18\cdots 40}{52\cdots 37}a^{19}-\frac{68\cdots 88}{58\cdots 93}a^{18}-\frac{15\cdots 52}{17\cdots 79}a^{17}-\frac{21\cdots 54}{52\cdots 37}a^{16}+\frac{18\cdots 80}{19\cdots 31}a^{15}+\frac{66\cdots 68}{17\cdots 79}a^{14}-\frac{25\cdots 52}{58\cdots 93}a^{13}-\frac{21\cdots 38}{58\cdots 93}a^{12}+\frac{32\cdots 16}{17\cdots 79}a^{11}+\frac{72\cdots 56}{58\cdots 93}a^{10}+\frac{28\cdots 40}{58\cdots 93}a^{9}-\frac{59\cdots 91}{17\cdots 79}a^{8}-\frac{70\cdots 64}{74\cdots 91}a^{7}-\frac{22\cdots 88}{58\cdots 93}a^{6}-\frac{13\cdots 92}{17\cdots 79}a^{5}-\frac{72\cdots 18}{74\cdots 91}a^{4}-\frac{35\cdots 04}{58\cdots 93}a^{3}-\frac{57\cdots 12}{64\cdots 77}a^{2}+\frac{65\cdots 28}{71\cdots 53}a+\frac{24\cdots 61}{79\cdots 17}$, $\frac{35\cdots 79}{20\cdots 48}a^{19}-\frac{10\cdots 37}{23\cdots 72}a^{18}-\frac{30\cdots 79}{69\cdots 16}a^{17}-\frac{11\cdots 59}{20\cdots 48}a^{16}+\frac{12\cdots 54}{27\cdots 33}a^{15}+\frac{77\cdots 31}{34\cdots 58}a^{14}-\frac{47\cdots 17}{23\cdots 72}a^{13}-\frac{46\cdots 11}{23\cdots 72}a^{12}-\frac{27\cdots 19}{69\cdots 16}a^{11}+\frac{36\cdots 51}{58\cdots 93}a^{10}+\frac{67\cdots 47}{23\cdots 72}a^{9}-\frac{91\cdots 11}{69\cdots 16}a^{8}-\frac{10\cdots 75}{20\cdots 48}a^{7}-\frac{12\cdots 51}{58\cdots 93}a^{6}-\frac{52\cdots 33}{99\cdots 88}a^{5}-\frac{15\cdots 67}{20\cdots 48}a^{4}-\frac{64\cdots 29}{11\cdots 86}a^{3}-\frac{10\cdots 12}{64\cdots 77}a^{2}+\frac{50\cdots 24}{79\cdots 17}a+\frac{13\cdots 99}{31\cdots 68}$, $\frac{87\cdots 73}{20\cdots 48}a^{19}-\frac{29\cdots 35}{58\cdots 93}a^{18}-\frac{38\cdots 97}{34\cdots 58}a^{17}-\frac{27\cdots 29}{10\cdots 74}a^{16}+\frac{14\cdots 71}{12\cdots 48}a^{15}+\frac{48\cdots 19}{69\cdots 16}a^{14}-\frac{11\cdots 55}{23\cdots 72}a^{13}-\frac{13\cdots 53}{23\cdots 72}a^{12}-\frac{13\cdots 11}{34\cdots 58}a^{11}+\frac{10\cdots 56}{58\cdots 93}a^{10}+\frac{10\cdots 19}{11\cdots 86}a^{9}+\frac{10\cdots 17}{69\cdots 16}a^{8}-\frac{20\cdots 65}{14\cdots 82}a^{7}-\frac{14\cdots 13}{23\cdots 72}a^{6}-\frac{51\cdots 31}{34\cdots 58}a^{5}-\frac{39\cdots 89}{20\cdots 48}a^{4}-\frac{29\cdots 25}{23\cdots 72}a^{3}-\frac{70\cdots 25}{31\cdots 68}a^{2}+\frac{13\cdots 59}{71\cdots 53}a+\frac{21\cdots 97}{31\cdots 68}$, $\frac{12\cdots 50}{52\cdots 37}a^{19}-\frac{27\cdots 55}{25\cdots 08}a^{18}-\frac{98\cdots 23}{17\cdots 79}a^{17}+\frac{81\cdots 65}{20\cdots 48}a^{16}+\frac{71\cdots 29}{11\cdots 86}a^{15}+\frac{12\cdots 93}{69\cdots 16}a^{14}-\frac{71\cdots 61}{23\cdots 72}a^{13}-\frac{48\cdots 49}{23\cdots 72}a^{12}+\frac{53\cdots 07}{17\cdots 79}a^{11}+\frac{89\cdots 23}{11\cdots 86}a^{10}+\frac{59\cdots 41}{23\cdots 72}a^{9}-\frac{26\cdots 75}{69\cdots 16}a^{8}-\frac{58\cdots 33}{10\cdots 74}a^{7}-\frac{15\cdots 65}{77\cdots 24}a^{6}-\frac{69\cdots 98}{17\cdots 79}a^{5}-\frac{23\cdots 75}{52\cdots 37}a^{4}-\frac{20\cdots 19}{77\cdots 24}a^{3}-\frac{67\cdots 67}{21\cdots 59}a^{2}+\frac{58\cdots 83}{14\cdots 06}a+\frac{10\cdots 96}{79\cdots 17}$, $\frac{26\cdots 43}{23\cdots 72}a^{19}-\frac{28\cdots 89}{25\cdots 08}a^{18}-\frac{11\cdots 09}{38\cdots 62}a^{17}-\frac{47\cdots 29}{58\cdots 93}a^{16}+\frac{41\cdots 25}{12\cdots 54}a^{15}+\frac{15\cdots 75}{77\cdots 24}a^{14}-\frac{32\cdots 61}{25\cdots 08}a^{13}-\frac{41\cdots 15}{25\cdots 08}a^{12}-\frac{70\cdots 65}{38\cdots 62}a^{11}+\frac{30\cdots 63}{64\cdots 77}a^{10}+\frac{33\cdots 69}{12\cdots 54}a^{9}+\frac{59\cdots 73}{38\cdots 62}a^{8}-\frac{88\cdots 61}{23\cdots 72}a^{7}-\frac{24\cdots 05}{12\cdots 54}a^{6}-\frac{36\cdots 11}{77\cdots 24}a^{5}-\frac{77\cdots 59}{11\cdots 86}a^{4}-\frac{12\cdots 69}{25\cdots 08}a^{3}-\frac{27\cdots 15}{25\cdots 08}a^{2}+\frac{28\cdots 99}{41\cdots 16}a+\frac{43\cdots 07}{15\cdots 34}$, $\frac{35\cdots 17}{20\cdots 48}a^{19}-\frac{83\cdots 13}{23\cdots 72}a^{18}-\frac{15\cdots 69}{34\cdots 58}a^{17}-\frac{14\cdots 07}{20\cdots 48}a^{16}+\frac{10\cdots 55}{23\cdots 72}a^{15}+\frac{83\cdots 53}{34\cdots 58}a^{14}-\frac{11\cdots 71}{58\cdots 93}a^{13}-\frac{24\cdots 59}{11\cdots 86}a^{12}-\frac{13\cdots 72}{17\cdots 79}a^{11}+\frac{37\cdots 00}{58\cdots 93}a^{10}+\frac{71\cdots 27}{23\cdots 72}a^{9}+\frac{39\cdots 07}{34\cdots 58}a^{8}-\frac{26\cdots 29}{52\cdots 37}a^{7}-\frac{13\cdots 56}{58\cdots 93}a^{6}-\frac{92\cdots 32}{17\cdots 79}a^{5}-\frac{14\cdots 13}{20\cdots 48}a^{4}-\frac{56\cdots 81}{11\cdots 86}a^{3}-\frac{11\cdots 23}{12\cdots 48}a^{2}+\frac{99\cdots 27}{14\cdots 06}a+\frac{11\cdots 73}{45\cdots 24}$, $\frac{12\cdots 92}{24\cdots 97}a^{19}-\frac{13\cdots 27}{11\cdots 86}a^{18}-\frac{15\cdots 69}{11\cdots 86}a^{17}-\frac{32\cdots 52}{17\cdots 79}a^{16}+\frac{81\cdots 52}{58\cdots 93}a^{15}+\frac{80\cdots 43}{11\cdots 86}a^{14}-\frac{40\cdots 70}{64\cdots 77}a^{13}-\frac{11\cdots 58}{19\cdots 31}a^{12}-\frac{69\cdots 63}{58\cdots 93}a^{11}+\frac{12\cdots 66}{64\cdots 77}a^{10}+\frac{34\cdots 67}{38\cdots 62}a^{9}-\frac{13\cdots 73}{11\cdots 86}a^{8}-\frac{53\cdots 73}{34\cdots 58}a^{7}-\frac{38\cdots 27}{58\cdots 93}a^{6}-\frac{86\cdots 27}{58\cdots 93}a^{5}-\frac{33\cdots 69}{17\cdots 79}a^{4}-\frac{14\cdots 21}{11\cdots 86}a^{3}-\frac{19\cdots 06}{92\cdots 11}a^{2}+\frac{13\cdots 45}{71\cdots 53}a+\frac{53\cdots 13}{79\cdots 17}$, $\frac{15\cdots 99}{34\cdots 58}a^{19}-\frac{13\cdots 99}{83\cdots 99}a^{18}-\frac{12\cdots 21}{11\cdots 86}a^{17}-\frac{39\cdots 61}{17\cdots 79}a^{16}+\frac{13\cdots 01}{11\cdots 86}a^{15}+\frac{26\cdots 77}{58\cdots 93}a^{14}-\frac{22\cdots 29}{38\cdots 62}a^{13}-\frac{17\cdots 79}{38\cdots 62}a^{12}+\frac{36\cdots 09}{11\cdots 86}a^{11}+\frac{60\cdots 97}{38\cdots 62}a^{10}+\frac{23\cdots 57}{38\cdots 62}a^{9}-\frac{57\cdots 65}{11\cdots 86}a^{8}-\frac{20\cdots 87}{17\cdots 79}a^{7}-\frac{77\cdots 71}{16\cdots 98}a^{6}-\frac{56\cdots 15}{58\cdots 93}a^{5}-\frac{29\cdots 43}{24\cdots 97}a^{4}-\frac{42\cdots 83}{58\cdots 93}a^{3}-\frac{45\cdots 91}{43\cdots 18}a^{2}+\frac{52\cdots 77}{47\cdots 02}a+\frac{84\cdots 25}{22\cdots 62}$, $\frac{17\cdots 25}{52\cdots 37}a^{19}-\frac{41\cdots 53}{33\cdots 96}a^{18}-\frac{58\cdots 83}{69\cdots 16}a^{17}-\frac{82\cdots 38}{52\cdots 37}a^{16}+\frac{52\cdots 82}{58\cdots 93}a^{15}+\frac{59\cdots 77}{17\cdots 79}a^{14}-\frac{99\cdots 79}{23\cdots 72}a^{13}-\frac{79\cdots 41}{23\cdots 72}a^{12}+\frac{23\cdots 07}{99\cdots 88}a^{11}+\frac{68\cdots 49}{58\cdots 93}a^{10}+\frac{52\cdots 17}{11\cdots 86}a^{9}-\frac{13\cdots 67}{34\cdots 58}a^{8}-\frac{46\cdots 55}{52\cdots 37}a^{7}-\frac{80\cdots 73}{23\cdots 72}a^{6}-\frac{12\cdots 86}{17\cdots 79}a^{5}-\frac{26\cdots 07}{29\cdots 64}a^{4}-\frac{63\cdots 15}{11\cdots 86}a^{3}-\frac{76\cdots 97}{95\cdots 04}a^{2}+\frac{23\cdots 19}{28\cdots 12}a+\frac{88\cdots 23}{31\cdots 68}$, $\frac{48\cdots 63}{20\cdots 48}a^{19}-\frac{33\cdots 67}{58\cdots 93}a^{18}-\frac{20\cdots 91}{34\cdots 58}a^{17}-\frac{80\cdots 21}{10\cdots 74}a^{16}+\frac{41\cdots 72}{64\cdots 77}a^{15}+\frac{30\cdots 47}{99\cdots 88}a^{14}-\frac{16\cdots 71}{58\cdots 93}a^{13}-\frac{92\cdots 89}{33\cdots 96}a^{12}-\frac{12\cdots 35}{69\cdots 16}a^{11}+\frac{20\cdots 29}{23\cdots 72}a^{10}+\frac{92\cdots 77}{23\cdots 72}a^{9}-\frac{66\cdots 25}{69\cdots 16}a^{8}-\frac{36\cdots 79}{52\cdots 37}a^{7}-\frac{17\cdots 81}{58\cdots 93}a^{6}-\frac{65\cdots 85}{99\cdots 88}a^{5}-\frac{17\cdots 83}{20\cdots 48}a^{4}-\frac{63\cdots 47}{11\cdots 86}a^{3}-\frac{22\cdots 13}{25\cdots 08}a^{2}+\frac{58\cdots 34}{71\cdots 53}a+\frac{22\cdots 52}{79\cdots 17}$, $\frac{36\cdots 79}{10\cdots 74}a^{19}-\frac{92\cdots 91}{11\cdots 86}a^{18}-\frac{63\cdots 25}{69\cdots 16}a^{17}-\frac{14\cdots 31}{10\cdots 74}a^{16}+\frac{22\cdots 67}{23\cdots 72}a^{15}+\frac{17\cdots 95}{34\cdots 58}a^{14}-\frac{10\cdots 99}{23\cdots 72}a^{13}-\frac{14\cdots 63}{33\cdots 96}a^{12}-\frac{68\cdots 43}{69\cdots 16}a^{11}+\frac{78\cdots 24}{58\cdots 93}a^{10}+\frac{72\cdots 41}{11\cdots 86}a^{9}-\frac{46\cdots 57}{69\cdots 16}a^{8}-\frac{22\cdots 77}{20\cdots 48}a^{7}-\frac{54\cdots 93}{11\cdots 86}a^{6}-\frac{73\cdots 41}{69\cdots 16}a^{5}-\frac{14\cdots 17}{10\cdots 74}a^{4}-\frac{14\cdots 99}{16\cdots 98}a^{3}-\frac{56\cdots 57}{36\cdots 44}a^{2}+\frac{31\cdots 95}{23\cdots 51}a+\frac{37\cdots 84}{79\cdots 17}$, $\frac{67\cdots 41}{10\cdots 74}a^{19}-\frac{73\cdots 51}{25\cdots 08}a^{18}-\frac{15\cdots 41}{99\cdots 88}a^{17}+\frac{86\cdots 39}{10\cdots 74}a^{16}+\frac{39\cdots 07}{23\cdots 72}a^{15}+\frac{93\cdots 75}{17\cdots 79}a^{14}-\frac{70\cdots 71}{83\cdots 99}a^{13}-\frac{13\cdots 37}{23\cdots 72}a^{12}+\frac{14\cdots 33}{17\cdots 79}a^{11}+\frac{50\cdots 69}{23\cdots 72}a^{10}+\frac{16\cdots 41}{23\cdots 72}a^{9}-\frac{19\cdots 32}{17\cdots 79}a^{8}-\frac{83\cdots 99}{52\cdots 37}a^{7}-\frac{11\cdots 76}{19\cdots 31}a^{6}-\frac{75\cdots 51}{69\cdots 16}a^{5}-\frac{24\cdots 97}{20\cdots 48}a^{4}-\frac{49\cdots 61}{77\cdots 24}a^{3}-\frac{16\cdots 89}{36\cdots 44}a^{2}+\frac{29\cdots 97}{28\cdots 12}a+\frac{22\cdots 11}{79\cdots 17}$, $\frac{27\cdots 36}{52\cdots 37}a^{19}-\frac{24\cdots 79}{16\cdots 98}a^{18}-\frac{23\cdots 86}{17\cdots 79}a^{17}-\frac{68\cdots 36}{52\cdots 37}a^{16}+\frac{33\cdots 19}{23\cdots 72}a^{15}+\frac{22\cdots 37}{34\cdots 58}a^{14}-\frac{15\cdots 21}{23\cdots 72}a^{13}-\frac{70\cdots 47}{11\cdots 86}a^{12}+\frac{61\cdots 63}{69\cdots 16}a^{11}+\frac{46\cdots 59}{23\cdots 72}a^{10}+\frac{19\cdots 93}{23\cdots 72}a^{9}-\frac{58\cdots 23}{17\cdots 79}a^{8}-\frac{15\cdots 73}{10\cdots 74}a^{7}-\frac{14\cdots 51}{23\cdots 72}a^{6}-\frac{13\cdots 11}{99\cdots 88}a^{5}-\frac{12\cdots 24}{74\cdots 91}a^{4}-\frac{26\cdots 93}{23\cdots 72}a^{3}-\frac{18\cdots 63}{10\cdots 79}a^{2}+\frac{11\cdots 33}{71\cdots 53}a+\frac{18\cdots 55}{31\cdots 68}$
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| Regulator: | \( 8321502208490000 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{4}\cdot 8321502208490000 \cdot 2}{2\cdot\sqrt{41041796771414445392922107616798059242325016576}}\cr\approx \mathstrut & 0.262221303186952 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.C_3^4:S_5$ (as 20T1035):
| A non-solvable group of order 9953280 |
| The 108 conjugacy class representatives for $C_2^{10}.C_3^4:S_5$ |
| Character table for $C_2^{10}.C_3^4:S_5$ |
Intermediate fields
| 5.3.4511.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
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 | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.10.0.1}{10} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.30a10.2 | $x^{20} + 6 x^{17} + 6 x^{15} + 12 x^{14} + 24 x^{12} + 10 x^{11} + 14 x^{10} + 30 x^{9} + 3 x^{8} + 34 x^{7} + 12 x^{6} + 14 x^{5} + 20 x^{4} + 16 x^{2} + 4 x + 7$ | $4$ | $5$ | $30$ | 20T74 | not computed |
|
\(3\)
| 3.5.1.0a1.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ |
| 3.5.3.20a46.3 | $x^{15} + 6 x^{14} + 3 x^{13} + 6 x^{12} + 9 x^{11} + 33 x^{10} + 24 x^{9} + 30 x^{8} + 36 x^{7} + 66 x^{6} + 51 x^{5} + 42 x^{4} + 47 x^{3} + 54 x^{2} + 51 x + 10$ | $3$ | $5$ | $20$ | 15T26 | $$[2, 2, 2, 2]^{5}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.1.2.1a1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.3.2.3a1.1 | $x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 57 x + 121$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 13.6.1.0a1.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(347\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
|
\(5145233\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |