Normalized defining polynomial
\( x^{18} - 9 x^{17} - 39 x^{16} + 468 x^{15} + 486 x^{14} - 9408 x^{13} - 2183 x^{12} + 94620 x^{11} + \cdots - 14641 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[18, 0]$ |
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| Discriminant: |
\(1220909192731128673051014533203125\)
\(\medspace = 3^{24}\cdot 5^{9}\cdot 19^{12}\)
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| Root discriminant: | \(68.89\) |
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| Galois root discriminant: | $3^{4/3}5^{1/2}19^{2/3}\approx 68.88887234517178$ | ||
| Ramified primes: |
\(3\), \(5\), \(19\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_3\times S_3$ |
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| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{11}a^{9}-\frac{4}{11}a^{8}+\frac{2}{11}a^{7}+\frac{4}{11}a^{6}-\frac{1}{11}a^{4}+\frac{4}{11}a^{3}-\frac{2}{11}a^{2}-\frac{4}{11}a$, $\frac{1}{11}a^{10}-\frac{3}{11}a^{8}+\frac{1}{11}a^{7}+\frac{5}{11}a^{6}-\frac{1}{11}a^{5}+\frac{3}{11}a^{3}-\frac{1}{11}a^{2}-\frac{5}{11}a$, $\frac{1}{11}a^{11}-\frac{1}{11}a$, $\frac{1}{55}a^{12}-\frac{1}{55}a^{11}-\frac{1}{55}a^{10}-\frac{1}{55}a^{9}-\frac{26}{55}a^{8}+\frac{8}{55}a^{7}-\frac{4}{11}a^{6}-\frac{2}{11}a^{5}+\frac{1}{55}a^{4}+\frac{26}{55}a^{3}-\frac{9}{55}a^{2}-\frac{23}{55}a-\frac{1}{5}$, $\frac{1}{55}a^{13}-\frac{2}{55}a^{11}-\frac{2}{55}a^{10}-\frac{2}{55}a^{9}-\frac{8}{55}a^{8}-\frac{17}{55}a^{7}+\frac{3}{11}a^{6}-\frac{9}{55}a^{5}+\frac{2}{55}a^{4}+\frac{7}{55}a^{3}-\frac{27}{55}a^{2}-\frac{24}{55}a-\frac{1}{5}$, $\frac{1}{55}a^{14}+\frac{1}{55}a^{11}+\frac{1}{55}a^{10}-\frac{14}{55}a^{8}+\frac{1}{55}a^{7}+\frac{16}{55}a^{6}-\frac{23}{55}a^{5}-\frac{1}{55}a^{4}+\frac{5}{11}a^{3}-\frac{12}{55}a^{2}-\frac{17}{55}a-\frac{2}{5}$, $\frac{1}{19075045}a^{15}-\frac{152276}{19075045}a^{14}+\frac{86659}{19075045}a^{13}+\frac{9916}{3815009}a^{12}-\frac{400927}{19075045}a^{11}-\frac{436578}{19075045}a^{10}+\frac{744319}{19075045}a^{9}-\frac{2316311}{19075045}a^{8}+\frac{8290709}{19075045}a^{7}+\frac{3474591}{19075045}a^{6}-\frac{3590479}{19075045}a^{5}-\frac{4004287}{19075045}a^{4}-\frac{871544}{3815009}a^{3}+\frac{1916001}{19075045}a^{2}-\frac{54798}{157645}a-\frac{62391}{157645}$, $\frac{1}{13753107445}a^{16}+\frac{237}{13753107445}a^{15}-\frac{9838106}{2750621489}a^{14}-\frac{87996831}{13753107445}a^{13}-\frac{3522142}{392945927}a^{12}-\frac{251842471}{13753107445}a^{11}-\frac{560183622}{13753107445}a^{10}+\frac{92752945}{2750621489}a^{9}+\frac{906764862}{13753107445}a^{8}+\frac{127503654}{13753107445}a^{7}-\frac{63673602}{1250282495}a^{6}+\frac{5466594906}{13753107445}a^{5}-\frac{5215189344}{13753107445}a^{4}-\frac{2079045478}{13753107445}a^{3}-\frac{677774387}{2750621489}a^{2}+\frac{5310916}{12138665}a+\frac{8874984}{22732409}$, $\frac{1}{10\cdots 65}a^{17}-\frac{91392134059}{10\cdots 65}a^{16}-\frac{3816483961996}{92\cdots 15}a^{15}-\frac{29\cdots 11}{10\cdots 65}a^{14}+\frac{29\cdots 91}{10\cdots 65}a^{13}-\frac{32\cdots 78}{10\cdots 65}a^{12}-\frac{34\cdots 37}{10\cdots 65}a^{11}+\frac{37\cdots 08}{20\cdots 53}a^{10}+\frac{20\cdots 16}{10\cdots 65}a^{9}+\frac{22\cdots 26}{10\cdots 65}a^{8}-\frac{23\cdots 99}{10\cdots 65}a^{7}-\frac{42\cdots 40}{20\cdots 53}a^{6}+\frac{44\cdots 59}{10\cdots 65}a^{5}-\frac{29\cdots 32}{10\cdots 65}a^{4}+\frac{68\cdots 28}{10\cdots 65}a^{3}-\frac{38\cdots 52}{10\cdots 65}a^{2}-\frac{17\cdots 11}{92\cdots 15}a-\frac{15\cdots 17}{84\cdots 65}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $11$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $17$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{384064711232724}{98\cdots 55}a^{17}-\frac{33\cdots 09}{98\cdots 55}a^{16}-\frac{17\cdots 88}{98\cdots 55}a^{15}+\frac{18\cdots 02}{98\cdots 55}a^{14}+\frac{53\cdots 14}{17\cdots 41}a^{13}-\frac{39\cdots 74}{98\cdots 55}a^{12}-\frac{37\cdots 42}{12\cdots 15}a^{11}+\frac{12\cdots 40}{28\cdots 93}a^{10}+\frac{20\cdots 54}{98\cdots 55}a^{9}-\frac{27\cdots 84}{98\cdots 55}a^{8}-\frac{11\cdots 38}{98\cdots 55}a^{7}+\frac{92\cdots 26}{98\cdots 55}a^{6}+\frac{42\cdots 58}{11\cdots 65}a^{5}-\frac{31\cdots 55}{19\cdots 51}a^{4}-\frac{47\cdots 06}{98\cdots 55}a^{3}+\frac{13\cdots 08}{12\cdots 15}a^{2}+\frac{14\cdots 61}{81\cdots 55}a-\frac{44\cdots 14}{81\cdots 55}$, $\frac{2220420980664}{30\cdots 83}a^{17}-\frac{23132751252834}{30\cdots 83}a^{16}-\frac{55893566988954}{30\cdots 83}a^{15}+\frac{11\cdots 96}{30\cdots 83}a^{14}-\frac{455321170115112}{30\cdots 83}a^{13}-\frac{21\cdots 76}{30\cdots 83}a^{12}+\frac{24\cdots 98}{30\cdots 83}a^{11}+\frac{19\cdots 34}{30\cdots 83}a^{10}-\frac{232523789605004}{26713708850851}a^{9}-\frac{94\cdots 46}{30\cdots 83}a^{8}+\frac{12\cdots 22}{30\cdots 83}a^{7}+\frac{52\cdots 39}{67110049064333}a^{6}-\frac{27\cdots 77}{30\cdots 83}a^{5}-\frac{28\cdots 74}{30\cdots 83}a^{4}+\frac{24\cdots 27}{27\cdots 53}a^{3}+\frac{13\cdots 34}{30\cdots 83}a^{2}-\frac{75\cdots 93}{27\cdots 53}a-\frac{597589029571084}{250137455603423}$, $\frac{1192396032108}{30\cdots 83}a^{17}-\frac{10526285051418}{30\cdots 83}a^{16}-\frac{48446263194258}{30\cdots 83}a^{15}+\frac{551768389017216}{30\cdots 83}a^{14}+\frac{669992600376144}{30\cdots 83}a^{13}-\frac{10\cdots 63}{27\cdots 53}a^{12}-\frac{41\cdots 84}{30\cdots 83}a^{11}+\frac{11\cdots 45}{30\cdots 83}a^{10}+\frac{173212635536170}{293850797359361}a^{9}-\frac{60\cdots 68}{30\cdots 83}a^{8}-\frac{99\cdots 04}{30\cdots 83}a^{7}+\frac{16\cdots 86}{30\cdots 83}a^{6}+\frac{38\cdots 43}{30\cdots 83}a^{5}-\frac{19\cdots 56}{30\cdots 83}a^{4}-\frac{54\cdots 23}{30\cdots 83}a^{3}+\frac{73\cdots 09}{30\cdots 83}a^{2}+\frac{19\cdots 19}{27\cdots 53}a-\frac{226862166107114}{250137455603423}$, $\frac{46\cdots 92}{10\cdots 65}a^{17}-\frac{48\cdots 82}{10\cdots 65}a^{16}-\frac{67\cdots 92}{10\cdots 65}a^{15}+\frac{18\cdots 51}{10\cdots 65}a^{14}-\frac{20\cdots 71}{10\cdots 65}a^{13}-\frac{17\cdots 12}{10\cdots 65}a^{12}+\frac{46\cdots 77}{14\cdots 95}a^{11}-\frac{98\cdots 43}{14\cdots 95}a^{10}+\frac{97\cdots 28}{10\cdots 65}a^{9}+\frac{39\cdots 80}{20\cdots 53}a^{8}-\frac{34\cdots 06}{10\cdots 65}a^{7}-\frac{10\cdots 33}{10\cdots 65}a^{6}+\frac{49\cdots 21}{29\cdots 79}a^{5}+\frac{22\cdots 76}{10\cdots 65}a^{4}-\frac{28\cdots 06}{10\cdots 65}a^{3}-\frac{20\cdots 38}{14\cdots 95}a^{2}+\frac{25\cdots 54}{22\cdots 15}a+\frac{21\cdots 21}{84\cdots 65}$, $\frac{79\cdots 54}{10\cdots 65}a^{17}-\frac{73\cdots 22}{10\cdots 65}a^{16}-\frac{27\cdots 84}{10\cdots 65}a^{15}+\frac{33\cdots 96}{92\cdots 15}a^{14}+\frac{21\cdots 53}{92\cdots 15}a^{13}-\frac{72\cdots 38}{10\cdots 65}a^{12}+\frac{12\cdots 31}{14\cdots 95}a^{11}+\frac{98\cdots 57}{14\cdots 95}a^{10}-\frac{17\cdots 97}{10\cdots 65}a^{9}-\frac{34\cdots 67}{10\cdots 65}a^{8}+\frac{53\cdots 92}{10\cdots 65}a^{7}+\frac{86\cdots 26}{10\cdots 65}a^{6}+\frac{19\cdots 13}{14\cdots 95}a^{5}-\frac{94\cdots 86}{10\cdots 65}a^{4}-\frac{34\cdots 51}{24\cdots 65}a^{3}+\frac{41\cdots 37}{14\cdots 95}a^{2}+\frac{16\cdots 56}{22\cdots 15}a+\frac{26\cdots 82}{84\cdots 65}$, $\frac{79\cdots 72}{14\cdots 95}a^{17}-\frac{53\cdots 81}{10\cdots 65}a^{16}-\frac{36\cdots 48}{20\cdots 53}a^{15}+\frac{24\cdots 07}{92\cdots 15}a^{14}+\frac{17\cdots 65}{20\cdots 53}a^{13}-\frac{74\cdots 54}{14\cdots 95}a^{12}+\frac{44\cdots 61}{20\cdots 53}a^{11}+\frac{49\cdots 99}{10\cdots 65}a^{10}-\frac{29\cdots 84}{10\cdots 65}a^{9}-\frac{25\cdots 49}{10\cdots 65}a^{8}+\frac{13\cdots 47}{10\cdots 65}a^{7}+\frac{64\cdots 04}{10\cdots 65}a^{6}-\frac{53\cdots 82}{20\cdots 53}a^{5}-\frac{75\cdots 47}{10\cdots 65}a^{4}+\frac{25\cdots 06}{10\cdots 65}a^{3}+\frac{57\cdots 60}{20\cdots 53}a^{2}-\frac{22\cdots 43}{22\cdots 15}a-\frac{70\cdots 76}{84\cdots 65}$, $\frac{11\cdots 77}{10\cdots 65}a^{17}-\frac{10\cdots 57}{10\cdots 65}a^{16}-\frac{43\cdots 24}{10\cdots 65}a^{15}+\frac{53\cdots 46}{10\cdots 65}a^{14}+\frac{52\cdots 08}{10\cdots 65}a^{13}-\frac{10\cdots 18}{10\cdots 65}a^{12}-\frac{74\cdots 91}{32\cdots 45}a^{11}+\frac{29\cdots 85}{29\cdots 79}a^{10}+\frac{10\cdots 82}{10\cdots 65}a^{9}-\frac{10\cdots 01}{20\cdots 53}a^{8}-\frac{11\cdots 58}{10\cdots 65}a^{7}+\frac{13\cdots 13}{10\cdots 65}a^{6}+\frac{81\cdots 23}{14\cdots 95}a^{5}-\frac{33\cdots 30}{20\cdots 53}a^{4}-\frac{89\cdots 83}{10\cdots 65}a^{3}+\frac{11\cdots 73}{14\cdots 95}a^{2}+\frac{32\cdots 14}{92\cdots 15}a-\frac{10\cdots 09}{84\cdots 65}$, $\frac{97\cdots 83}{10\cdots 65}a^{17}-\frac{11\cdots 97}{14\cdots 95}a^{16}-\frac{45\cdots 46}{10\cdots 65}a^{15}+\frac{45\cdots 18}{10\cdots 65}a^{14}+\frac{19\cdots 09}{24\cdots 65}a^{13}-\frac{10\cdots 97}{10\cdots 65}a^{12}-\frac{71\cdots 76}{10\cdots 65}a^{11}+\frac{11\cdots 96}{10\cdots 65}a^{10}+\frac{21\cdots 30}{70\cdots 19}a^{9}-\frac{64\cdots 77}{92\cdots 15}a^{8}-\frac{91\cdots 38}{10\cdots 65}a^{7}+\frac{33\cdots 36}{14\cdots 95}a^{6}-\frac{92\cdots 86}{20\cdots 53}a^{5}-\frac{99\cdots 55}{29\cdots 79}a^{4}+\frac{19\cdots 51}{14\cdots 95}a^{3}+\frac{16\cdots 63}{10\cdots 65}a^{2}-\frac{63\cdots 57}{92\cdots 15}a-\frac{81\cdots 03}{84\cdots 65}$, $\frac{47\cdots 33}{10\cdots 65}a^{17}-\frac{36\cdots 44}{10\cdots 65}a^{16}-\frac{23\cdots 86}{10\cdots 65}a^{15}+\frac{19\cdots 18}{10\cdots 65}a^{14}+\frac{72\cdots 62}{14\cdots 95}a^{13}-\frac{41\cdots 63}{10\cdots 65}a^{12}-\frac{11\cdots 66}{18\cdots 23}a^{11}+\frac{43\cdots 52}{10\cdots 65}a^{10}+\frac{55\cdots 31}{10\cdots 65}a^{9}-\frac{23\cdots 06}{10\cdots 65}a^{8}-\frac{29\cdots 96}{10\cdots 65}a^{7}+\frac{14\cdots 77}{24\cdots 65}a^{6}+\frac{16\cdots 11}{20\cdots 53}a^{5}-\frac{63\cdots 66}{10\cdots 65}a^{4}-\frac{94\cdots 09}{10\cdots 65}a^{3}+\frac{15\cdots 57}{10\cdots 65}a^{2}+\frac{29\cdots 86}{92\cdots 15}a+\frac{41\cdots 24}{12\cdots 95}$, $\frac{19\cdots 37}{10\cdots 65}a^{17}-\frac{18\cdots 53}{10\cdots 65}a^{16}-\frac{63\cdots 07}{10\cdots 65}a^{15}+\frac{95\cdots 71}{10\cdots 65}a^{14}+\frac{31\cdots 28}{10\cdots 65}a^{13}-\frac{18\cdots 96}{10\cdots 65}a^{12}+\frac{74\cdots 07}{92\cdots 15}a^{11}+\frac{17\cdots 31}{10\cdots 65}a^{10}-\frac{22\cdots 21}{20\cdots 53}a^{9}-\frac{90\cdots 73}{10\cdots 65}a^{8}+\frac{52\cdots 58}{10\cdots 65}a^{7}+\frac{23\cdots 28}{10\cdots 65}a^{6}-\frac{22\cdots 36}{20\cdots 53}a^{5}-\frac{56\cdots 39}{20\cdots 53}a^{4}+\frac{11\cdots 98}{10\cdots 65}a^{3}+\frac{11\cdots 83}{10\cdots 65}a^{2}-\frac{38\cdots 22}{92\cdots 15}a-\frac{13\cdots 91}{84\cdots 65}$, $\frac{46\cdots 48}{10\cdots 65}a^{17}-\frac{95\cdots 33}{18\cdots 23}a^{16}-\frac{71\cdots 53}{10\cdots 65}a^{15}+\frac{25\cdots 79}{10\cdots 65}a^{14}-\frac{63\cdots 36}{20\cdots 53}a^{13}-\frac{44\cdots 54}{10\cdots 65}a^{12}+\frac{12\cdots 44}{13\cdots 45}a^{11}+\frac{54\cdots 03}{14\cdots 95}a^{10}-\frac{93\cdots 08}{10\cdots 65}a^{9}-\frac{16\cdots 56}{10\cdots 65}a^{8}+\frac{44\cdots 88}{10\cdots 65}a^{7}+\frac{66\cdots 98}{18\cdots 23}a^{6}-\frac{14\cdots 12}{14\cdots 95}a^{5}-\frac{34\cdots 69}{10\cdots 65}a^{4}+\frac{10\cdots 68}{10\cdots 65}a^{3}+\frac{77\cdots 92}{14\cdots 95}a^{2}-\frac{28\cdots 16}{92\cdots 15}a+\frac{37\cdots 07}{84\cdots 65}$, $\frac{35\cdots 28}{10\cdots 65}a^{17}-\frac{39\cdots 63}{10\cdots 65}a^{16}-\frac{47\cdots 51}{10\cdots 65}a^{15}+\frac{17\cdots 14}{10\cdots 65}a^{14}-\frac{24\cdots 47}{10\cdots 65}a^{13}-\frac{29\cdots 18}{10\cdots 65}a^{12}+\frac{62\cdots 71}{10\cdots 65}a^{11}+\frac{21\cdots 39}{10\cdots 65}a^{10}-\frac{53\cdots 88}{10\cdots 65}a^{9}-\frac{69\cdots 52}{10\cdots 65}a^{8}+\frac{18\cdots 58}{10\cdots 65}a^{7}+\frac{84\cdots 87}{10\cdots 65}a^{6}-\frac{11\cdots 82}{49\cdots 33}a^{5}-\frac{95\cdots 39}{10\cdots 65}a^{4}+\frac{58\cdots 64}{92\cdots 15}a^{3}-\frac{10\cdots 64}{10\cdots 65}a^{2}-\frac{33\cdots 09}{18\cdots 23}a+\frac{76\cdots 74}{84\cdots 65}$, $\frac{15\cdots 76}{10\cdots 65}a^{17}-\frac{48\cdots 82}{29\cdots 79}a^{16}-\frac{33\cdots 61}{92\cdots 15}a^{15}+\frac{85\cdots 79}{10\cdots 65}a^{14}-\frac{53\cdots 82}{10\cdots 65}a^{13}-\frac{33\cdots 60}{20\cdots 53}a^{12}+\frac{22\cdots 43}{10\cdots 65}a^{11}+\frac{16\cdots 39}{10\cdots 65}a^{10}-\frac{70\cdots 85}{29\cdots 79}a^{9}-\frac{89\cdots 28}{10\cdots 65}a^{8}+\frac{24\cdots 88}{20\cdots 53}a^{7}+\frac{38\cdots 53}{14\cdots 95}a^{6}-\frac{26\cdots 23}{10\cdots 65}a^{5}-\frac{11\cdots 47}{28\cdots 93}a^{4}+\frac{68\cdots 37}{29\cdots 79}a^{3}+\frac{18\cdots 93}{92\cdots 15}a^{2}-\frac{15\cdots 48}{18\cdots 23}a-\frac{10\cdots 69}{84\cdots 65}$, $\frac{47\cdots 96}{10\cdots 65}a^{17}-\frac{49\cdots 11}{10\cdots 65}a^{16}-\frac{10\cdots 48}{10\cdots 65}a^{15}+\frac{45\cdots 62}{20\cdots 53}a^{14}-\frac{24\cdots 74}{20\cdots 53}a^{13}-\frac{38\cdots 29}{10\cdots 65}a^{12}+\frac{45\cdots 42}{10\cdots 65}a^{11}+\frac{30\cdots 94}{10\cdots 65}a^{10}-\frac{37\cdots 18}{10\cdots 65}a^{9}-\frac{11\cdots 23}{10\cdots 65}a^{8}+\frac{10\cdots 73}{10\cdots 65}a^{7}+\frac{16\cdots 39}{10\cdots 65}a^{6}-\frac{26\cdots 66}{24\cdots 65}a^{5}-\frac{36\cdots 44}{10\cdots 65}a^{4}-\frac{17\cdots 53}{98\cdots 55}a^{3}-\frac{12\cdots 15}{18\cdots 23}a^{2}+\frac{56\cdots 08}{92\cdots 15}a+\frac{79\cdots 82}{84\cdots 65}$, $\frac{46\cdots 69}{10\cdots 65}a^{17}-\frac{56\cdots 87}{10\cdots 65}a^{16}-\frac{76\cdots 85}{20\cdots 53}a^{15}+\frac{38\cdots 36}{14\cdots 95}a^{14}-\frac{99\cdots 43}{20\cdots 53}a^{13}-\frac{46\cdots 08}{10\cdots 65}a^{12}+\frac{12\cdots 27}{10\cdots 65}a^{11}+\frac{79\cdots 27}{20\cdots 53}a^{10}-\frac{13\cdots 38}{10\cdots 65}a^{9}-\frac{17\cdots 94}{10\cdots 65}a^{8}+\frac{92\cdots 03}{14\cdots 95}a^{7}+\frac{79\cdots 01}{20\cdots 53}a^{6}-\frac{15\cdots 89}{10\cdots 65}a^{5}-\frac{48\cdots 02}{10\cdots 65}a^{4}+\frac{17\cdots 96}{10\cdots 65}a^{3}+\frac{26\cdots 71}{10\cdots 65}a^{2}-\frac{16\cdots 62}{26\cdots 89}a-\frac{21\cdots 78}{84\cdots 65}$, $\frac{23\cdots 43}{14\cdots 95}a^{17}-\frac{21\cdots 42}{14\cdots 95}a^{16}-\frac{93\cdots 61}{14\cdots 95}a^{15}+\frac{10\cdots 11}{14\cdots 95}a^{14}+\frac{12\cdots 38}{14\cdots 95}a^{13}-\frac{20\cdots 37}{14\cdots 95}a^{12}-\frac{17\cdots 03}{29\cdots 79}a^{11}+\frac{20\cdots 09}{14\cdots 95}a^{10}+\frac{11\cdots 41}{29\cdots 79}a^{9}-\frac{24\cdots 51}{35\cdots 95}a^{8}-\frac{41\cdots 91}{14\cdots 95}a^{7}+\frac{24\cdots 33}{14\cdots 95}a^{6}+\frac{13\cdots 58}{13\cdots 45}a^{5}-\frac{42\cdots 12}{26\cdots 89}a^{4}-\frac{38\cdots 33}{29\cdots 79}a^{3}+\frac{39\cdots 59}{14\cdots 65}a^{2}+\frac{79\cdots 99}{24\cdots 99}a+\frac{24\cdots 17}{11\cdots 65}$, $\frac{27\cdots 98}{20\cdots 53}a^{17}-\frac{14\cdots 56}{10\cdots 65}a^{16}-\frac{27\cdots 18}{92\cdots 15}a^{15}+\frac{63\cdots 68}{98\cdots 55}a^{14}-\frac{37\cdots 69}{10\cdots 65}a^{13}-\frac{10\cdots 07}{10\cdots 65}a^{12}+\frac{28\cdots 91}{20\cdots 53}a^{11}+\frac{82\cdots 87}{10\cdots 65}a^{10}-\frac{24\cdots 14}{20\cdots 53}a^{9}-\frac{26\cdots 68}{10\cdots 65}a^{8}+\frac{43\cdots 36}{10\cdots 65}a^{7}+\frac{17\cdots 24}{10\cdots 65}a^{6}-\frac{64\cdots 42}{10\cdots 65}a^{5}+\frac{63\cdots 39}{10\cdots 65}a^{4}+\frac{71\cdots 49}{10\cdots 65}a^{3}-\frac{63\cdots 32}{10\cdots 65}a^{2}-\frac{45\cdots 01}{92\cdots 15}a-\frac{24\cdots 67}{84\cdots 65}$
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| Regulator: | \( 180167785608 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 180167785608 \cdot 1}{2\cdot\sqrt{1220909192731128673051014533203125}}\cr\approx \mathstrut & 0.675842323421940 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3$ (as 18T3):
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.3.146205.1 x3, 3.3.361.1, 6.6.106879510125.1, 6.6.296065125.3 x2, 6.6.16290125.1, 9.9.3125263755565125.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 sibling: | 6.6.296065125.3 |
| Degree 9 sibling: | 9.9.3125263755565125.1 |
| Minimal sibling: | 6.6.296065125.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | ${\href{/padicField/11.1.0.1}{1} }^{18}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.24a1.1 | $x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 29 x^{12} + 24 x^{11} + 51 x^{10} + 36 x^{9} + 72 x^{8} + 60 x^{7} + 85 x^{6} + 78 x^{5} + 69 x^{4} + 44 x^{3} + 60 x^{2} + 48 x + 23$ | $3$ | $6$ | $24$ | $S_3 \times C_3$ | not computed |
|
\(5\)
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(19\)
| 19.3.3.6a1.3 | $x^{9} + 12 x^{7} + 51 x^{6} + 48 x^{5} + 408 x^{4} + 931 x^{3} + 816 x^{2} + 3468 x + 4932$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |
| 19.3.3.6a1.3 | $x^{9} + 12 x^{7} + 51 x^{6} + 48 x^{5} + 408 x^{4} + 931 x^{3} + 816 x^{2} + 3468 x + 4932$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |