Normalized defining polynomial
\( x^{36} + 68 x^{34} + 2018 x^{32} + 34520 x^{30} + 379376 x^{28} + 2832128 x^{26} + 14836664 x^{24} + \cdots + 512 \)
Invariants
| Degree: | $36$ |
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| Signature: | $[0, 18]$ |
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| Discriminant: |
\(52733281945045886724167383478270850720626086921526306402773390818541568\)
\(\medspace = 2^{99}\cdot 19^{32}\)
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| Root discriminant: | \(92.15\) |
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| Galois root discriminant: | $2^{11/4}19^{8/9}\approx 92.151488696787$ | ||
| Ramified primes: |
\(2\), \(19\)
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| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{36}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(304=2^{4}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{304}(1,·)$, $\chi_{304}(131,·)$, $\chi_{304}(9,·)$, $\chi_{304}(11,·)$, $\chi_{304}(17,·)$, $\chi_{304}(275,·)$, $\chi_{304}(73,·)$, $\chi_{304}(25,·)$, $\chi_{304}(153,·)$, $\chi_{304}(283,·)$, $\chi_{304}(289,·)$, $\chi_{304}(35,·)$, $\chi_{304}(49,·)$, $\chi_{304}(169,·)$, $\chi_{304}(43,·)$, $\chi_{304}(177,·)$, $\chi_{304}(115,·)$, $\chi_{304}(137,·)$, $\chi_{304}(187,·)$, $\chi_{304}(123,·)$, $\chi_{304}(267,·)$, $\chi_{304}(161,·)$, $\chi_{304}(201,·)$, $\chi_{304}(291,·)$, $\chi_{304}(81,·)$, $\chi_{304}(163,·)$, $\chi_{304}(139,·)$, $\chi_{304}(225,·)$, $\chi_{304}(99,·)$, $\chi_{304}(195,·)$, $\chi_{304}(273,·)$, $\chi_{304}(233,·)$, $\chi_{304}(235,·)$, $\chi_{304}(83,·)$, $\chi_{304}(121,·)$, $\chi_{304}(251,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{32}a^{22}$, $\frac{1}{32}a^{23}$, $\frac{1}{64}a^{24}$, $\frac{1}{64}a^{25}$, $\frac{1}{64}a^{26}$, $\frac{1}{64}a^{27}$, $\frac{1}{128}a^{28}$, $\frac{1}{128}a^{29}$, $\frac{1}{128}a^{30}$, $\frac{1}{128}a^{31}$, $\frac{1}{256}a^{32}$, $\frac{1}{256}a^{33}$, $\frac{1}{98\cdots 96}a^{34}-\frac{30\cdots 55}{98\cdots 96}a^{32}-\frac{78\cdots 55}{49\cdots 48}a^{30}-\frac{55\cdots 83}{30\cdots 28}a^{28}-\frac{30\cdots 93}{12\cdots 12}a^{26}-\frac{16\cdots 75}{24\cdots 24}a^{24}+\frac{13\cdots 61}{12\cdots 12}a^{22}-\frac{48\cdots 07}{12\cdots 12}a^{20}-\frac{24\cdots 37}{15\cdots 64}a^{18}+\frac{89\cdots 68}{38\cdots 41}a^{16}+\frac{10\cdots 25}{30\cdots 28}a^{14}+\frac{43\cdots 96}{38\cdots 41}a^{12}-\frac{10\cdots 83}{15\cdots 64}a^{10}-\frac{16\cdots 41}{15\cdots 64}a^{8}+\frac{11\cdots 97}{76\cdots 82}a^{6}-\frac{74\cdots 08}{38\cdots 41}a^{4}+\frac{12\cdots 05}{38\cdots 41}a^{2}-\frac{10\cdots 26}{38\cdots 41}$, $\frac{1}{98\cdots 96}a^{35}-\frac{30\cdots 55}{98\cdots 96}a^{33}-\frac{78\cdots 55}{49\cdots 48}a^{31}-\frac{55\cdots 83}{30\cdots 28}a^{29}-\frac{30\cdots 93}{12\cdots 12}a^{27}-\frac{16\cdots 75}{24\cdots 24}a^{25}+\frac{13\cdots 61}{12\cdots 12}a^{23}-\frac{48\cdots 07}{12\cdots 12}a^{21}-\frac{24\cdots 37}{15\cdots 64}a^{19}+\frac{89\cdots 68}{38\cdots 41}a^{17}+\frac{10\cdots 25}{30\cdots 28}a^{15}+\frac{43\cdots 96}{38\cdots 41}a^{13}-\frac{10\cdots 83}{15\cdots 64}a^{11}-\frac{16\cdots 41}{15\cdots 64}a^{9}+\frac{11\cdots 97}{76\cdots 82}a^{7}-\frac{74\cdots 08}{38\cdots 41}a^{5}+\frac{12\cdots 05}{38\cdots 41}a^{3}-\frac{10\cdots 26}{38\cdots 41}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{37}\times C_{294409}$, which has order $10893133$ (assuming GRH) |
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| Narrow class group: | $C_{294409}\times C_{37}$, which has order $10893133$ (assuming GRH) |
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| Relative class number: | $10893133$ (assuming GRH) |
Unit group
| Rank: | $17$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{58\cdots 59}{30\cdots 28}a^{34}+\frac{12\cdots 17}{98\cdots 96}a^{32}+\frac{13\cdots 19}{38\cdots 41}a^{30}+\frac{27\cdots 11}{49\cdots 48}a^{28}+\frac{70\cdots 09}{12\cdots 12}a^{26}+\frac{47\cdots 37}{12\cdots 12}a^{24}+\frac{10\cdots 05}{61\cdots 56}a^{22}+\frac{69\cdots 67}{12\cdots 12}a^{20}+\frac{38\cdots 85}{30\cdots 28}a^{18}+\frac{11\cdots 91}{61\cdots 56}a^{16}+\frac{70\cdots 07}{38\cdots 41}a^{14}+\frac{33\cdots 57}{30\cdots 28}a^{12}+\frac{23\cdots 95}{76\cdots 82}a^{10}-\frac{25\cdots 19}{76\cdots 82}a^{8}-\frac{16\cdots 00}{38\cdots 41}a^{6}-\frac{33\cdots 49}{38\cdots 41}a^{4}-\frac{21\cdots 96}{38\cdots 41}a^{2}-\frac{46\cdots 27}{38\cdots 41}$, $\frac{59\cdots 55}{38\cdots 41}a^{34}+\frac{32\cdots 39}{30\cdots 28}a^{32}+\frac{37\cdots 35}{12\cdots 12}a^{30}+\frac{31\cdots 15}{61\cdots 56}a^{28}+\frac{33\cdots 43}{61\cdots 56}a^{26}+\frac{11\cdots 35}{30\cdots 28}a^{24}+\frac{28\cdots 27}{15\cdots 64}a^{22}+\frac{45\cdots 49}{76\cdots 82}a^{20}+\frac{19\cdots 91}{15\cdots 64}a^{18}+\frac{11\cdots 95}{76\cdots 82}a^{16}+\frac{15\cdots 83}{38\cdots 41}a^{14}-\frac{68\cdots 38}{38\cdots 41}a^{12}-\frac{11\cdots 70}{38\cdots 41}a^{10}-\frac{37\cdots 95}{15\cdots 64}a^{8}-\frac{38\cdots 16}{38\cdots 41}a^{6}-\frac{77\cdots 02}{38\cdots 41}a^{4}-\frac{52\cdots 96}{38\cdots 41}a^{2}-\frac{43\cdots 58}{38\cdots 41}$, $\frac{40\cdots 25}{61\cdots 56}a^{34}+\frac{26\cdots 39}{61\cdots 56}a^{32}+\frac{37\cdots 43}{30\cdots 28}a^{30}+\frac{59\cdots 91}{30\cdots 28}a^{28}+\frac{15\cdots 61}{76\cdots 82}a^{26}+\frac{50\cdots 95}{38\cdots 41}a^{24}+\frac{44\cdots 71}{76\cdots 82}a^{22}+\frac{21\cdots 65}{12\cdots 12}a^{20}+\frac{11\cdots 85}{30\cdots 28}a^{18}+\frac{14\cdots 85}{30\cdots 28}a^{16}+\frac{55\cdots 61}{15\cdots 64}a^{14}+\frac{18\cdots 19}{30\cdots 28}a^{12}-\frac{11\cdots 93}{76\cdots 82}a^{10}-\frac{10\cdots 59}{76\cdots 82}a^{8}-\frac{20\cdots 06}{38\cdots 41}a^{6}-\frac{69\cdots 87}{76\cdots 82}a^{4}-\frac{21\cdots 86}{38\cdots 41}a^{2}-\frac{58\cdots 39}{38\cdots 41}$, $\frac{50\cdots 33}{61\cdots 56}a^{34}+\frac{33\cdots 19}{61\cdots 56}a^{32}+\frac{49\cdots 79}{30\cdots 28}a^{30}+\frac{83\cdots 31}{30\cdots 28}a^{28}+\frac{22\cdots 89}{76\cdots 82}a^{26}+\frac{81\cdots 63}{38\cdots 41}a^{24}+\frac{16\cdots 79}{15\cdots 64}a^{22}+\frac{47\cdots 73}{12\cdots 12}a^{20}+\frac{30\cdots 61}{30\cdots 28}a^{18}+\frac{11\cdots 17}{61\cdots 56}a^{16}+\frac{37\cdots 81}{15\cdots 64}a^{14}+\frac{71\cdots 99}{30\cdots 28}a^{12}+\frac{11\cdots 31}{76\cdots 82}a^{10}+\frac{50\cdots 63}{76\cdots 82}a^{8}+\frac{66\cdots 02}{38\cdots 41}a^{6}+\frac{18\cdots 81}{76\cdots 82}a^{4}+\frac{52\cdots 90}{38\cdots 41}a^{2}+\frac{50\cdots 96}{38\cdots 41}$, $\frac{77\cdots 03}{49\cdots 48}a^{34}+\frac{20\cdots 87}{98\cdots 96}a^{32}+\frac{23\cdots 91}{24\cdots 24}a^{30}+\frac{11\cdots 53}{49\cdots 48}a^{28}+\frac{21\cdots 71}{61\cdots 56}a^{26}+\frac{82\cdots 57}{24\cdots 24}a^{24}+\frac{13\cdots 81}{61\cdots 56}a^{22}+\frac{58\cdots 63}{61\cdots 56}a^{20}+\frac{89\cdots 03}{30\cdots 28}a^{18}+\frac{38\cdots 37}{61\cdots 56}a^{16}+\frac{13\cdots 97}{15\cdots 64}a^{14}+\frac{13\cdots 23}{15\cdots 64}a^{12}+\frac{22\cdots 16}{38\cdots 41}a^{10}+\frac{18\cdots 19}{76\cdots 82}a^{8}+\frac{22\cdots 86}{38\cdots 41}a^{6}+\frac{28\cdots 18}{38\cdots 41}a^{4}+\frac{14\cdots 80}{38\cdots 41}a^{2}-\frac{58\cdots 85}{38\cdots 41}$, $\frac{12\cdots 01}{76\cdots 82}a^{34}+\frac{18\cdots 45}{15\cdots 64}a^{32}+\frac{15\cdots 92}{38\cdots 41}a^{30}+\frac{58\cdots 85}{76\cdots 82}a^{28}+\frac{36\cdots 14}{38\cdots 41}a^{26}+\frac{31\cdots 68}{38\cdots 41}a^{24}+\frac{74\cdots 37}{15\cdots 64}a^{22}+\frac{15\cdots 13}{76\cdots 82}a^{20}+\frac{47\cdots 69}{76\cdots 82}a^{18}+\frac{83\cdots 47}{61\cdots 56}a^{16}+\frac{80\cdots 30}{38\cdots 41}a^{14}+\frac{87\cdots 85}{38\cdots 41}a^{12}+\frac{64\cdots 62}{38\cdots 41}a^{10}+\frac{30\cdots 11}{38\cdots 41}a^{8}+\frac{86\cdots 08}{38\cdots 41}a^{6}+\frac{12\cdots 84}{38\cdots 41}a^{4}+\frac{74\cdots 76}{38\cdots 41}a^{2}+\frac{70\cdots 94}{38\cdots 41}$, $\frac{59\cdots 55}{38\cdots 41}a^{34}+\frac{32\cdots 39}{30\cdots 28}a^{32}+\frac{37\cdots 35}{12\cdots 12}a^{30}+\frac{31\cdots 15}{61\cdots 56}a^{28}+\frac{33\cdots 43}{61\cdots 56}a^{26}+\frac{11\cdots 35}{30\cdots 28}a^{24}+\frac{28\cdots 27}{15\cdots 64}a^{22}+\frac{45\cdots 49}{76\cdots 82}a^{20}+\frac{19\cdots 91}{15\cdots 64}a^{18}+\frac{11\cdots 95}{76\cdots 82}a^{16}+\frac{15\cdots 83}{38\cdots 41}a^{14}-\frac{68\cdots 38}{38\cdots 41}a^{12}-\frac{11\cdots 70}{38\cdots 41}a^{10}-\frac{37\cdots 95}{15\cdots 64}a^{8}-\frac{38\cdots 16}{38\cdots 41}a^{6}-\frac{77\cdots 02}{38\cdots 41}a^{4}-\frac{52\cdots 96}{38\cdots 41}a^{2}+\frac{34\cdots 83}{38\cdots 41}$, $\frac{55\cdots 77}{61\cdots 56}a^{34}+\frac{37\cdots 45}{61\cdots 56}a^{32}+\frac{54\cdots 23}{30\cdots 28}a^{30}+\frac{14\cdots 57}{49\cdots 48}a^{28}+\frac{19\cdots 07}{61\cdots 56}a^{26}+\frac{29\cdots 37}{12\cdots 12}a^{24}+\frac{91\cdots 49}{76\cdots 82}a^{22}+\frac{51\cdots 57}{12\cdots 12}a^{20}+\frac{15\cdots 73}{15\cdots 64}a^{18}+\frac{55\cdots 71}{30\cdots 28}a^{16}+\frac{81\cdots 52}{38\cdots 41}a^{14}+\frac{24\cdots 67}{15\cdots 64}a^{12}+\frac{52\cdots 21}{76\cdots 82}a^{10}+\frac{33\cdots 64}{38\cdots 41}a^{8}-\frac{16\cdots 77}{38\cdots 41}a^{6}-\frac{12\cdots 71}{76\cdots 82}a^{4}-\frac{49\cdots 38}{38\cdots 41}a^{2}-\frac{49\cdots 83}{38\cdots 41}$, $\frac{18\cdots 97}{98\cdots 96}a^{34}+\frac{15\cdots 73}{12\cdots 12}a^{32}+\frac{92\cdots 43}{24\cdots 24}a^{30}+\frac{98\cdots 83}{15\cdots 64}a^{28}+\frac{17\cdots 15}{24\cdots 24}a^{26}+\frac{15\cdots 17}{30\cdots 28}a^{24}+\frac{16\cdots 47}{61\cdots 56}a^{22}+\frac{38\cdots 06}{38\cdots 41}a^{20}+\frac{16\cdots 51}{61\cdots 56}a^{18}+\frac{38\cdots 63}{76\cdots 82}a^{16}+\frac{10\cdots 85}{15\cdots 64}a^{14}+\frac{24\cdots 12}{38\cdots 41}a^{12}+\frac{32\cdots 93}{76\cdots 82}a^{10}+\frac{68\cdots 96}{38\cdots 41}a^{8}+\frac{18\cdots 74}{38\cdots 41}a^{6}+\frac{26\cdots 08}{38\cdots 41}a^{4}+\frac{16\cdots 49}{38\cdots 41}a^{2}+\frac{22\cdots 77}{38\cdots 41}$, $\frac{24\cdots 97}{98\cdots 96}a^{34}+\frac{20\cdots 51}{12\cdots 12}a^{32}+\frac{12\cdots 87}{24\cdots 24}a^{30}+\frac{25\cdots 57}{30\cdots 28}a^{28}+\frac{22\cdots 67}{24\cdots 24}a^{26}+\frac{19\cdots 77}{30\cdots 28}a^{24}+\frac{20\cdots 15}{61\cdots 56}a^{22}+\frac{14\cdots 57}{12\cdots 12}a^{20}+\frac{18\cdots 21}{61\cdots 56}a^{18}+\frac{16\cdots 37}{30\cdots 28}a^{16}+\frac{54\cdots 23}{76\cdots 82}a^{14}+\frac{19\cdots 15}{30\cdots 28}a^{12}+\frac{15\cdots 00}{38\cdots 41}a^{10}+\frac{12\cdots 33}{76\cdots 82}a^{8}+\frac{16\cdots 68}{38\cdots 41}a^{6}+\frac{45\cdots 29}{76\cdots 82}a^{4}+\frac{14\cdots 63}{38\cdots 41}a^{2}+\frac{24\cdots 20}{38\cdots 41}$, $\frac{18\cdots 01}{98\cdots 96}a^{34}+\frac{14\cdots 55}{98\cdots 96}a^{32}+\frac{12\cdots 91}{24\cdots 24}a^{30}+\frac{29\cdots 11}{30\cdots 28}a^{28}+\frac{29\cdots 79}{24\cdots 24}a^{26}+\frac{38\cdots 45}{38\cdots 41}a^{24}+\frac{36\cdots 31}{61\cdots 56}a^{22}+\frac{95\cdots 69}{38\cdots 41}a^{20}+\frac{45\cdots 55}{61\cdots 56}a^{18}+\frac{23\cdots 81}{15\cdots 64}a^{16}+\frac{35\cdots 09}{15\cdots 64}a^{14}+\frac{93\cdots 68}{38\cdots 41}a^{12}+\frac{13\cdots 01}{76\cdots 82}a^{10}+\frac{30\cdots 52}{38\cdots 41}a^{8}+\frac{87\cdots 58}{38\cdots 41}a^{6}+\frac{13\cdots 96}{38\cdots 41}a^{4}+\frac{98\cdots 37}{38\cdots 41}a^{2}+\frac{24\cdots 24}{38\cdots 41}$, $\frac{11\cdots 89}{98\cdots 96}a^{34}+\frac{98\cdots 69}{12\cdots 12}a^{32}+\frac{58\cdots 19}{24\cdots 24}a^{30}+\frac{31\cdots 03}{76\cdots 82}a^{28}+\frac{10\cdots 71}{24\cdots 24}a^{26}+\frac{80\cdots 83}{24\cdots 24}a^{24}+\frac{10\cdots 91}{61\cdots 56}a^{22}+\frac{18\cdots 15}{30\cdots 28}a^{20}+\frac{98\cdots 59}{61\cdots 56}a^{18}+\frac{90\cdots 21}{30\cdots 28}a^{16}+\frac{59\cdots 79}{15\cdots 64}a^{14}+\frac{13\cdots 85}{38\cdots 41}a^{12}+\frac{16\cdots 83}{76\cdots 82}a^{10}+\frac{12\cdots 79}{15\cdots 64}a^{8}+\frac{74\cdots 50}{38\cdots 41}a^{6}+\frac{10\cdots 26}{38\cdots 41}a^{4}+\frac{71\cdots 69}{38\cdots 41}a^{2}+\frac{17\cdots 64}{38\cdots 41}$, $\frac{16\cdots 17}{98\cdots 96}a^{34}+\frac{14\cdots 17}{12\cdots 12}a^{32}+\frac{84\cdots 73}{24\cdots 24}a^{30}+\frac{36\cdots 17}{61\cdots 56}a^{28}+\frac{15\cdots 43}{24\cdots 24}a^{26}+\frac{73\cdots 41}{15\cdots 64}a^{24}+\frac{15\cdots 39}{61\cdots 56}a^{22}+\frac{71\cdots 63}{76\cdots 82}a^{20}+\frac{15\cdots 87}{61\cdots 56}a^{18}+\frac{18\cdots 34}{38\cdots 41}a^{16}+\frac{10\cdots 53}{15\cdots 64}a^{14}+\frac{25\cdots 50}{38\cdots 41}a^{12}+\frac{34\cdots 33}{76\cdots 82}a^{10}+\frac{31\cdots 79}{15\cdots 64}a^{8}+\frac{21\cdots 90}{38\cdots 41}a^{6}+\frac{34\cdots 10}{38\cdots 41}a^{4}+\frac{21\cdots 45}{38\cdots 41}a^{2}+\frac{26\cdots 76}{38\cdots 41}$, $\frac{21\cdots 85}{98\cdots 96}a^{34}+\frac{32\cdots 69}{24\cdots 24}a^{32}+\frac{39\cdots 07}{12\cdots 12}a^{30}+\frac{48\cdots 09}{12\cdots 12}a^{28}+\frac{60\cdots 51}{24\cdots 24}a^{26}+\frac{10\cdots 87}{61\cdots 56}a^{24}-\frac{54\cdots 65}{61\cdots 56}a^{22}-\frac{11\cdots 81}{15\cdots 64}a^{20}-\frac{18\cdots 89}{61\cdots 56}a^{18}-\frac{12\cdots 81}{15\cdots 64}a^{16}-\frac{22\cdots 83}{15\cdots 64}a^{14}-\frac{66\cdots 25}{38\cdots 41}a^{12}-\frac{10\cdots 03}{76\cdots 82}a^{10}-\frac{27\cdots 14}{38\cdots 41}a^{8}-\frac{87\cdots 74}{38\cdots 41}a^{6}-\frac{29\cdots 31}{76\cdots 82}a^{4}-\frac{10\cdots 95}{38\cdots 41}a^{2}-\frac{26\cdots 36}{38\cdots 41}$, $\frac{19\cdots 25}{98\cdots 96}a^{34}+\frac{16\cdots 33}{12\cdots 12}a^{32}+\frac{10\cdots 31}{24\cdots 24}a^{30}+\frac{10\cdots 53}{15\cdots 64}a^{28}+\frac{19\cdots 11}{24\cdots 24}a^{26}+\frac{18\cdots 61}{30\cdots 28}a^{24}+\frac{19\cdots 95}{61\cdots 56}a^{22}+\frac{91\cdots 25}{76\cdots 82}a^{20}+\frac{19\cdots 03}{61\cdots 56}a^{18}+\frac{38\cdots 51}{61\cdots 56}a^{16}+\frac{13\cdots 05}{15\cdots 64}a^{14}+\frac{33\cdots 97}{38\cdots 41}a^{12}+\frac{44\cdots 17}{76\cdots 82}a^{10}+\frac{99\cdots 07}{38\cdots 41}a^{8}+\frac{26\cdots 82}{38\cdots 41}a^{6}+\frac{39\cdots 92}{38\cdots 41}a^{4}+\frac{24\cdots 25}{38\cdots 41}a^{2}+\frac{33\cdots 12}{38\cdots 41}$, $\frac{60\cdots 21}{98\cdots 96}a^{34}+\frac{20\cdots 47}{49\cdots 48}a^{32}+\frac{15\cdots 59}{12\cdots 12}a^{30}+\frac{50\cdots 49}{24\cdots 24}a^{28}+\frac{54\cdots 73}{24\cdots 24}a^{26}+\frac{39\cdots 81}{24\cdots 24}a^{24}+\frac{31\cdots 21}{38\cdots 41}a^{22}+\frac{36\cdots 69}{12\cdots 12}a^{20}+\frac{47\cdots 99}{61\cdots 56}a^{18}+\frac{44\cdots 33}{30\cdots 28}a^{16}+\frac{14\cdots 03}{76\cdots 82}a^{14}+\frac{14\cdots 45}{76\cdots 82}a^{12}+\frac{46\cdots 47}{38\cdots 41}a^{10}+\frac{40\cdots 95}{76\cdots 82}a^{8}+\frac{54\cdots 21}{38\cdots 41}a^{6}+\frac{16\cdots 39}{76\cdots 82}a^{4}+\frac{53\cdots 00}{38\cdots 41}a^{2}+\frac{64\cdots 17}{38\cdots 41}$, $\frac{56\cdots 45}{98\cdots 96}a^{34}+\frac{50\cdots 87}{12\cdots 12}a^{32}+\frac{32\cdots 07}{24\cdots 24}a^{30}+\frac{73\cdots 83}{30\cdots 28}a^{28}+\frac{70\cdots 09}{24\cdots 24}a^{26}+\frac{14\cdots 23}{61\cdots 56}a^{24}+\frac{40\cdots 09}{30\cdots 28}a^{22}+\frac{16\cdots 11}{30\cdots 28}a^{20}+\frac{97\cdots 33}{61\cdots 56}a^{18}+\frac{51\cdots 51}{15\cdots 64}a^{16}+\frac{77\cdots 05}{15\cdots 64}a^{14}+\frac{16\cdots 77}{30\cdots 28}a^{12}+\frac{30\cdots 45}{76\cdots 82}a^{10}+\frac{14\cdots 49}{76\cdots 82}a^{8}+\frac{22\cdots 59}{38\cdots 41}a^{6}+\frac{71\cdots 41}{76\cdots 82}a^{4}+\frac{23\cdots 19}{38\cdots 41}a^{2}+\frac{30\cdots 28}{38\cdots 41}$
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| Regulator: | \( 28122649019657.055 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 28122649019657.055 \cdot 10893133}{2\cdot\sqrt{52733281945045886724167383478270850720626086921526306402773390818541568}}\cr\approx \mathstrut & 0.155372540556310 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 36 |
| The 36 conjugacy class representatives for $C_{36}$ |
| Character table for $C_{36}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.3.361.1, 4.0.2048.2, 6.6.66724352.1, \(\Q(\zeta_{19})^+\), 12.0.145887695661298614272.69, 18.18.38713951190154487490850848768.1 |
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 | $36$ | $36$ | ${\href{/padicField/7.3.0.1}{3} }^{12}$ | ${\href{/padicField/11.12.0.1}{12} }^{3}$ | $36$ | ${\href{/padicField/17.9.0.1}{9} }^{4}$ | R | ${\href{/padicField/23.9.0.1}{9} }^{4}$ | $36$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{9}$ | $18^{2}$ | $36$ | $18^{2}$ | $36$ | $36$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $36$ | $4$ | $9$ | $99$ | |||
|
\(19\)
| Deg $36$ | $9$ | $4$ | $32$ |