Normalized defining polynomial
\( x^{18} - 3 x^{17} - 63 x^{16} + 147 x^{15} + 1245 x^{14} - 1968 x^{13} - 9306 x^{12} + 9264 x^{11} + \cdots - 1 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[18, 0]$ |
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| Discriminant: |
\(1303780692070187898696301025390625\)
\(\medspace = 3^{30}\cdot 5^{12}\cdot 11^{10}\)
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| Root discriminant: | \(69.14\) |
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| Galois root discriminant: | $3^{97/54}5^{3/4}11^{2/3}\approx 118.99821203205568$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{15}a^{12}-\frac{2}{15}a^{11}+\frac{2}{15}a^{10}-\frac{2}{15}a^{9}-\frac{1}{15}a^{8}-\frac{2}{5}a^{7}-\frac{2}{15}a^{6}+\frac{1}{5}a^{5}+\frac{1}{3}a^{4}-\frac{7}{15}a^{3}+\frac{1}{3}a^{2}-\frac{1}{15}a+\frac{1}{15}$, $\frac{1}{15}a^{13}-\frac{2}{15}a^{11}+\frac{2}{15}a^{10}-\frac{1}{3}a^{9}+\frac{7}{15}a^{8}+\frac{1}{15}a^{7}-\frac{1}{15}a^{6}-\frac{4}{15}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{15}a+\frac{2}{15}$, $\frac{1}{30}a^{14}-\frac{1}{30}a^{13}+\frac{1}{15}a^{10}+\frac{1}{10}a^{9}+\frac{1}{15}a^{8}+\frac{11}{30}a^{7}+\frac{4}{15}a^{6}-\frac{7}{30}a^{5}+\frac{13}{30}a^{4}-\frac{1}{30}a^{3}+\frac{1}{6}a^{2}-\frac{2}{15}a-\frac{1}{6}$, $\frac{1}{30}a^{15}-\frac{1}{30}a^{13}+\frac{1}{15}a^{11}-\frac{1}{6}a^{10}-\frac{1}{2}a^{9}-\frac{7}{30}a^{8}-\frac{1}{30}a^{7}+\frac{1}{30}a^{6}-\frac{7}{15}a^{5}+\frac{2}{5}a^{4}+\frac{7}{15}a^{3}-\frac{3}{10}a^{2}+\frac{1}{30}a+\frac{1}{6}$, $\frac{1}{60}a^{16}-\frac{1}{60}a^{13}-\frac{1}{30}a^{12}+\frac{1}{20}a^{11}+\frac{3}{20}a^{10}+\frac{1}{15}a^{9}+\frac{1}{12}a^{8}+\frac{1}{10}a^{7}-\frac{7}{15}a^{6}+\frac{23}{60}a^{5}+\frac{7}{60}a^{4}-\frac{1}{5}a^{3}-\frac{7}{30}a^{2}-\frac{5}{12}a-\frac{3}{20}$, $\frac{1}{31\cdots 20}a^{17}-\frac{14\cdots 21}{14\cdots 20}a^{16}-\frac{10\cdots 89}{74\cdots 10}a^{15}-\frac{17\cdots 83}{44\cdots 60}a^{14}-\frac{29\cdots 41}{10\cdots 40}a^{13}+\frac{12\cdots 97}{31\cdots 20}a^{12}-\frac{26\cdots 37}{15\cdots 10}a^{11}-\frac{70\cdots 93}{44\cdots 60}a^{10}+\frac{72\cdots 87}{31\cdots 20}a^{9}+\frac{28\cdots 07}{62\cdots 44}a^{8}+\frac{19\cdots 83}{78\cdots 55}a^{7}-\frac{68\cdots 51}{31\cdots 20}a^{6}+\frac{38\cdots 53}{78\cdots 55}a^{5}-\frac{19\cdots 51}{62\cdots 44}a^{4}-\frac{81\cdots 83}{52\cdots 70}a^{3}+\frac{55\cdots 17}{10\cdots 40}a^{2}+\frac{23\cdots 21}{52\cdots 70}a+\frac{19\cdots 91}{31\cdots 20}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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: |
$\frac{13\cdots 14}{26\cdots 85}a^{17}-\frac{61\cdots 78}{37\cdots 55}a^{16}-\frac{11\cdots 47}{37\cdots 55}a^{15}+\frac{61\cdots 00}{74\cdots 91}a^{14}+\frac{31\cdots 81}{52\cdots 37}a^{13}-\frac{29\cdots 72}{26\cdots 85}a^{12}-\frac{11\cdots 42}{26\cdots 85}a^{11}+\frac{21\cdots 71}{37\cdots 55}a^{10}+\frac{61\cdots 44}{52\cdots 37}a^{9}-\frac{52\cdots 79}{52\cdots 37}a^{8}-\frac{28\cdots 87}{26\cdots 85}a^{7}+\frac{12\cdots 68}{26\cdots 85}a^{6}+\frac{93\cdots 72}{26\cdots 85}a^{5}-\frac{35\cdots 43}{52\cdots 37}a^{4}-\frac{15\cdots 67}{52\cdots 37}a^{3}+\frac{11\cdots 38}{26\cdots 85}a^{2}+\frac{30\cdots 38}{26\cdots 85}a-\frac{21\cdots 18}{26\cdots 85}$, $\frac{37\cdots 04}{26\cdots 85}a^{17}-\frac{20\cdots 79}{37\cdots 55}a^{16}-\frac{31\cdots 06}{37\cdots 55}a^{15}+\frac{21\cdots 15}{74\cdots 91}a^{14}+\frac{80\cdots 84}{52\cdots 37}a^{13}-\frac{11\cdots 42}{26\cdots 85}a^{12}-\frac{26\cdots 01}{26\cdots 85}a^{11}+\frac{88\cdots 28}{37\cdots 55}a^{10}+\frac{12\cdots 25}{52\cdots 37}a^{9}-\frac{26\cdots 27}{52\cdots 37}a^{8}-\frac{37\cdots 62}{26\cdots 85}a^{7}+\frac{89\cdots 24}{26\cdots 85}a^{6}+\frac{75\cdots 11}{26\cdots 85}a^{5}-\frac{46\cdots 27}{52\cdots 37}a^{4}-\frac{73\cdots 68}{52\cdots 37}a^{3}+\frac{18\cdots 78}{26\cdots 85}a^{2}-\frac{31\cdots 81}{26\cdots 85}a-\frac{13\cdots 14}{26\cdots 85}$, $\frac{22\cdots 07}{26\cdots 85}a^{17}-\frac{23\cdots 50}{74\cdots 91}a^{16}-\frac{39\cdots 28}{74\cdots 91}a^{15}+\frac{11\cdots 52}{74\cdots 91}a^{14}+\frac{51\cdots 41}{52\cdots 37}a^{13}-\frac{59\cdots 96}{26\cdots 85}a^{12}-\frac{35\cdots 01}{52\cdots 37}a^{11}+\frac{89\cdots 43}{74\cdots 91}a^{10}+\frac{94\cdots 51}{52\cdots 37}a^{9}-\frac{12\cdots 71}{52\cdots 37}a^{8}-\frac{39\cdots 56}{26\cdots 85}a^{7}+\frac{78\cdots 31}{52\cdots 37}a^{6}+\frac{23\cdots 23}{52\cdots 37}a^{5}-\frac{18\cdots 92}{52\cdots 37}a^{4}-\frac{17\cdots 49}{52\cdots 37}a^{3}+\frac{71\cdots 09}{26\cdots 85}a^{2}-\frac{34\cdots 48}{52\cdots 37}a-\frac{97\cdots 10}{52\cdots 37}$, $\frac{17\cdots 87}{44\cdots 46}a^{17}-\frac{51\cdots 93}{44\cdots 46}a^{16}-\frac{55\cdots 79}{22\cdots 30}a^{15}+\frac{41\cdots 97}{74\cdots 10}a^{14}+\frac{10\cdots 13}{22\cdots 73}a^{13}-\frac{16\cdots 47}{22\cdots 30}a^{12}-\frac{40\cdots 54}{11\cdots 65}a^{11}+\frac{35\cdots 11}{11\cdots 65}a^{10}+\frac{39\cdots 07}{37\cdots 55}a^{9}-\frac{15\cdots 99}{37\cdots 55}a^{8}-\frac{24\cdots 31}{22\cdots 30}a^{7}-\frac{68\cdots 62}{11\cdots 65}a^{6}+\frac{83\cdots 88}{22\cdots 73}a^{5}+\frac{19\cdots 01}{22\cdots 30}a^{4}-\frac{11\cdots 86}{37\cdots 55}a^{3}-\frac{13\cdots 08}{11\cdots 65}a^{2}+\frac{66\cdots 29}{74\cdots 10}a+\frac{42\cdots 31}{11\cdots 65}$, $\frac{99\cdots 71}{74\cdots 10}a^{17}-\frac{29\cdots 31}{22\cdots 30}a^{16}-\frac{10\cdots 53}{11\cdots 65}a^{15}+\frac{30\cdots 49}{11\cdots 65}a^{14}+\frac{46\cdots 62}{22\cdots 73}a^{13}+\frac{10\cdots 77}{14\cdots 82}a^{12}-\frac{19\cdots 23}{11\cdots 65}a^{11}-\frac{91\cdots 19}{74\cdots 10}a^{10}+\frac{69\cdots 53}{11\cdots 65}a^{9}+\frac{11\cdots 83}{22\cdots 30}a^{8}-\frac{17\cdots 87}{22\cdots 30}a^{7}-\frac{50\cdots 73}{74\cdots 10}a^{6}+\frac{67\cdots 37}{22\cdots 30}a^{5}+\frac{10\cdots 93}{37\cdots 55}a^{4}-\frac{56\cdots 03}{22\cdots 30}a^{3}-\frac{32\cdots 31}{11\cdots 65}a^{2}+\frac{52\cdots 17}{11\cdots 65}a+\frac{42\cdots 29}{11\cdots 65}$, $\frac{14\cdots 13}{62\cdots 44}a^{17}-\frac{20\cdots 83}{22\cdots 30}a^{16}-\frac{20\cdots 57}{14\cdots 82}a^{15}+\frac{20\cdots 97}{44\cdots 60}a^{14}+\frac{38\cdots 03}{15\cdots 10}a^{13}-\frac{42\cdots 57}{62\cdots 44}a^{12}-\frac{16\cdots 03}{10\cdots 40}a^{11}+\frac{13\cdots 92}{37\cdots 55}a^{10}+\frac{10\cdots 27}{31\cdots 20}a^{9}-\frac{11\cdots 11}{15\cdots 10}a^{8}-\frac{22\cdots 33}{15\cdots 10}a^{7}+\frac{13\cdots 93}{31\cdots 20}a^{6}-\frac{20\cdots 81}{10\cdots 40}a^{5}-\frac{42\cdots 83}{52\cdots 70}a^{4}+\frac{24\cdots 73}{15\cdots 10}a^{3}+\frac{23\cdots 49}{62\cdots 44}a^{2}-\frac{72\cdots 41}{62\cdots 44}a+\frac{92\cdots 17}{15\cdots 11}$, $\frac{75\cdots 39}{10\cdots 40}a^{17}-\frac{16\cdots 97}{44\cdots 60}a^{16}-\frac{45\cdots 16}{11\cdots 65}a^{15}+\frac{89\cdots 61}{44\cdots 60}a^{14}+\frac{67\cdots 03}{10\cdots 40}a^{13}-\frac{10\cdots 03}{31\cdots 20}a^{12}-\frac{25\cdots 24}{78\cdots 55}a^{11}+\frac{87\cdots 83}{44\cdots 60}a^{10}+\frac{21\cdots 93}{62\cdots 44}a^{9}-\frac{14\cdots 37}{31\cdots 20}a^{8}+\frac{33\cdots 09}{52\cdots 70}a^{7}+\frac{12\cdots 71}{31\cdots 20}a^{6}-\frac{86\cdots 99}{31\cdots 22}a^{5}-\frac{38\cdots 01}{31\cdots 20}a^{4}-\frac{53\cdots 21}{15\cdots 10}a^{3}+\frac{37\cdots 73}{31\cdots 20}a^{2}+\frac{21\cdots 39}{31\cdots 22}a-\frac{18\cdots 47}{10\cdots 40}$, $\frac{70\cdots 23}{78\cdots 55}a^{17}-\frac{79\cdots 46}{37\cdots 55}a^{16}-\frac{22\cdots 18}{37\cdots 55}a^{15}+\frac{11\cdots 89}{11\cdots 65}a^{14}+\frac{97\cdots 74}{78\cdots 55}a^{13}-\frac{31\cdots 22}{26\cdots 85}a^{12}-\frac{81\cdots 77}{78\cdots 55}a^{11}+\frac{11\cdots 51}{22\cdots 73}a^{10}+\frac{97\cdots 31}{26\cdots 85}a^{9}-\frac{60\cdots 88}{78\cdots 55}a^{8}-\frac{43\cdots 52}{78\cdots 55}a^{7}+\frac{17\cdots 43}{26\cdots 85}a^{6}+\frac{26\cdots 79}{78\cdots 55}a^{5}-\frac{53\cdots 98}{15\cdots 11}a^{4}-\frac{20\cdots 94}{26\cdots 85}a^{3}+\frac{20\cdots 43}{26\cdots 85}a^{2}+\frac{45\cdots 71}{78\cdots 55}a-\frac{18\cdots 98}{26\cdots 85}$, $\frac{49\cdots 93}{44\cdots 60}a^{17}-\frac{17\cdots 97}{44\cdots 60}a^{16}-\frac{76\cdots 74}{11\cdots 65}a^{15}+\frac{85\cdots 53}{44\cdots 60}a^{14}+\frac{57\cdots 81}{44\cdots 60}a^{13}-\frac{12\cdots 03}{44\cdots 60}a^{12}-\frac{20\cdots 67}{22\cdots 30}a^{11}+\frac{61\cdots 39}{44\cdots 60}a^{10}+\frac{10\cdots 87}{44\cdots 60}a^{9}-\frac{38\cdots 49}{14\cdots 20}a^{8}-\frac{22\cdots 97}{11\cdots 65}a^{7}+\frac{60\cdots 79}{44\cdots 60}a^{6}+\frac{14\cdots 47}{22\cdots 30}a^{5}-\frac{10\cdots 09}{44\cdots 60}a^{4}-\frac{21\cdots 53}{37\cdots 55}a^{3}+\frac{20\cdots 09}{14\cdots 20}a^{2}+\frac{44\cdots 46}{11\cdots 65}a-\frac{56\cdots 73}{44\cdots 60}$, $\frac{28\cdots 38}{15\cdots 11}a^{17}-\frac{76\cdots 67}{14\cdots 20}a^{16}-\frac{12\cdots 37}{11\cdots 65}a^{15}+\frac{55\cdots 91}{22\cdots 73}a^{14}+\frac{72\cdots 31}{31\cdots 20}a^{13}-\frac{49\cdots 11}{15\cdots 10}a^{12}-\frac{54\cdots 77}{31\cdots 20}a^{11}+\frac{62\cdots 83}{44\cdots 60}a^{10}+\frac{42\cdots 46}{78\cdots 55}a^{9}-\frac{58\cdots 91}{31\cdots 20}a^{8}-\frac{96\cdots 51}{15\cdots 10}a^{7}+\frac{14\cdots 51}{78\cdots 55}a^{6}+\frac{81\cdots 23}{31\cdots 20}a^{5}+\frac{30\cdots 81}{10\cdots 40}a^{4}-\frac{20\cdots 52}{52\cdots 37}a^{3}-\frac{85\cdots 93}{15\cdots 10}a^{2}+\frac{61\cdots 27}{31\cdots 20}a+\frac{66\cdots 83}{31\cdots 20}$, $\frac{76\cdots 07}{52\cdots 70}a^{17}-\frac{11\cdots 59}{22\cdots 73}a^{16}-\frac{99\cdots 08}{11\cdots 65}a^{15}+\frac{20\cdots 37}{74\cdots 10}a^{14}+\frac{13\cdots 71}{78\cdots 55}a^{13}-\frac{61\cdots 17}{15\cdots 10}a^{12}-\frac{11\cdots 91}{10\cdots 74}a^{11}+\frac{46\cdots 06}{22\cdots 73}a^{10}+\frac{44\cdots 79}{15\cdots 10}a^{9}-\frac{10\cdots 64}{26\cdots 85}a^{8}-\frac{16\cdots 24}{78\cdots 55}a^{7}+\frac{74\cdots 61}{31\cdots 22}a^{6}+\frac{26\cdots 13}{52\cdots 70}a^{5}-\frac{38\cdots 61}{78\cdots 55}a^{4}-\frac{14\cdots 98}{15\cdots 11}a^{3}+\frac{16\cdots 01}{52\cdots 70}a^{2}-\frac{18\cdots 31}{52\cdots 70}a+\frac{13\cdots 76}{52\cdots 37}$, $\frac{79\cdots 33}{31\cdots 20}a^{17}-\frac{33\cdots 93}{44\cdots 60}a^{16}-\frac{59\cdots 89}{37\cdots 55}a^{15}+\frac{32\cdots 39}{89\cdots 92}a^{14}+\frac{98\cdots 11}{31\cdots 20}a^{13}-\frac{14\cdots 47}{31\cdots 20}a^{12}-\frac{18\cdots 62}{78\cdots 55}a^{11}+\frac{95\cdots 67}{44\cdots 60}a^{10}+\frac{72\cdots 57}{10\cdots 40}a^{9}-\frac{66\cdots 75}{20\cdots 48}a^{8}-\frac{11\cdots 00}{15\cdots 11}a^{7}+\frac{25\cdots 43}{31\cdots 20}a^{6}+\frac{40\cdots 23}{15\cdots 11}a^{5}+\frac{35\cdots 13}{31\cdots 20}a^{4}-\frac{18\cdots 67}{78\cdots 55}a^{3}-\frac{57\cdots 25}{62\cdots 44}a^{2}+\frac{56\cdots 03}{31\cdots 22}a-\frac{31\cdots 01}{10\cdots 40}$, $\frac{71\cdots 19}{31\cdots 20}a^{17}-\frac{10\cdots 99}{14\cdots 20}a^{16}-\frac{15\cdots 19}{11\cdots 65}a^{15}+\frac{54\cdots 89}{14\cdots 20}a^{14}+\frac{17\cdots 13}{62\cdots 44}a^{13}-\frac{15\cdots 13}{31\cdots 20}a^{12}-\frac{30\cdots 67}{15\cdots 10}a^{11}+\frac{22\cdots 23}{89\cdots 92}a^{10}+\frac{17\cdots 59}{31\cdots 20}a^{9}-\frac{13\cdots 13}{31\cdots 20}a^{8}-\frac{28\cdots 53}{52\cdots 70}a^{7}+\frac{72\cdots 37}{31\cdots 20}a^{6}+\frac{15\cdots 46}{78\cdots 55}a^{5}-\frac{12\cdots 21}{31\cdots 20}a^{4}-\frac{11\cdots 13}{52\cdots 70}a^{3}+\frac{90\cdots 59}{31\cdots 20}a^{2}+\frac{24\cdots 27}{78\cdots 55}a+\frac{85\cdots 83}{31\cdots 20}$, $\frac{22\cdots 67}{74\cdots 10}a^{17}+\frac{22\cdots 87}{74\cdots 10}a^{16}-\frac{16\cdots 09}{74\cdots 10}a^{15}-\frac{21\cdots 51}{74\cdots 10}a^{14}+\frac{64\cdots 18}{11\cdots 65}a^{13}+\frac{17\cdots 51}{22\cdots 30}a^{12}-\frac{21\cdots 37}{37\cdots 55}a^{11}-\frac{77\cdots 38}{11\cdots 65}a^{10}+\frac{26\cdots 64}{11\cdots 65}a^{9}+\frac{24\cdots 38}{11\cdots 65}a^{8}-\frac{28\cdots 09}{74\cdots 10}a^{7}-\frac{24\cdots 89}{11\cdots 65}a^{6}+\frac{22\cdots 26}{11\cdots 65}a^{5}+\frac{34\cdots 47}{44\cdots 46}a^{4}-\frac{14\cdots 22}{37\cdots 55}a^{3}-\frac{96\cdots 53}{11\cdots 65}a^{2}+\frac{56\cdots 27}{22\cdots 30}a+\frac{69\cdots 76}{37\cdots 55}$, $\frac{77\cdots 01}{26\cdots 85}a^{17}-\frac{39\cdots 67}{44\cdots 60}a^{16}-\frac{20\cdots 62}{11\cdots 65}a^{15}+\frac{32\cdots 97}{74\cdots 10}a^{14}+\frac{11\cdots 63}{31\cdots 20}a^{13}-\frac{89\cdots 81}{15\cdots 10}a^{12}-\frac{17\cdots 51}{62\cdots 44}a^{11}+\frac{38\cdots 63}{14\cdots 20}a^{10}+\frac{12\cdots 57}{15\cdots 10}a^{9}-\frac{11\cdots 77}{31\cdots 20}a^{8}-\frac{21\cdots 12}{26\cdots 85}a^{7}+\frac{60\cdots 01}{78\cdots 55}a^{6}+\frac{92\cdots 79}{31\cdots 20}a^{5}+\frac{97\cdots 83}{31\cdots 20}a^{4}-\frac{30\cdots 29}{10\cdots 74}a^{3}-\frac{12\cdots 69}{26\cdots 85}a^{2}+\frac{23\cdots 17}{62\cdots 44}a+\frac{16\cdots 91}{31\cdots 20}$, $\frac{87\cdots 61}{31\cdots 20}a^{17}-\frac{47\cdots 79}{44\cdots 60}a^{16}-\frac{75\cdots 39}{44\cdots 46}a^{15}+\frac{81\cdots 53}{14\cdots 20}a^{14}+\frac{19\cdots 63}{62\cdots 44}a^{13}-\frac{25\cdots 07}{31\cdots 20}a^{12}-\frac{10\cdots 79}{52\cdots 70}a^{11}+\frac{19\cdots 27}{44\cdots 60}a^{10}+\frac{16\cdots 37}{31\cdots 20}a^{9}-\frac{28\cdots 49}{31\cdots 20}a^{8}-\frac{60\cdots 09}{15\cdots 10}a^{7}+\frac{62\cdots 51}{10\cdots 40}a^{6}+\frac{37\cdots 73}{31\cdots 22}a^{5}-\frac{47\cdots 17}{31\cdots 20}a^{4}-\frac{12\cdots 58}{78\cdots 55}a^{3}+\frac{34\cdots 69}{31\cdots 20}a^{2}+\frac{11\cdots 29}{15\cdots 10}a+\frac{57\cdots 97}{31\cdots 20}$, $\frac{48\cdots 01}{31\cdots 20}a^{17}-\frac{68\cdots 31}{11\cdots 65}a^{16}-\frac{33\cdots 09}{37\cdots 55}a^{15}+\frac{95\cdots 09}{29\cdots 64}a^{14}+\frac{12\cdots 77}{78\cdots 55}a^{13}-\frac{50\cdots 31}{10\cdots 40}a^{12}-\frac{10\cdots 29}{10\cdots 40}a^{11}+\frac{29\cdots 43}{11\cdots 65}a^{10}+\frac{13\cdots 59}{62\cdots 44}a^{9}-\frac{89\cdots 97}{15\cdots 10}a^{8}-\frac{18\cdots 61}{31\cdots 22}a^{7}+\frac{12\cdots 71}{31\cdots 20}a^{6}-\frac{21\cdots 59}{31\cdots 20}a^{5}-\frac{15\cdots 03}{15\cdots 10}a^{4}+\frac{27\cdots 22}{78\cdots 55}a^{3}+\frac{13\cdots 09}{31\cdots 20}a^{2}-\frac{12\cdots 49}{31\cdots 20}a+\frac{27\cdots 59}{52\cdots 70}$
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| Regulator: | \( 39669265431.8 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 39669265431.8 \cdot 1}{2\cdot\sqrt{1303780692070187898696301025390625}}\cr\approx \mathstrut & 0.143999774588 \end{aligned}\] (assuming GRH)
Galois group
$\He_3:C_4$ (as 18T49):
| A solvable group of order 108 |
| The 14 conjugacy class representatives for $\He_3:C_4$ |
| Character table for $\He_3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.6.55130625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 sibling: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | 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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.9.30b15.7 | $x^{18} + 24 x^{17} + 258 x^{16} + 1734 x^{15} + 8295 x^{14} + 30198 x^{13} + 86982 x^{12} + 203016 x^{11} + 389556 x^{10} + 619403 x^{9} + 818211 x^{8} + 895902 x^{7} + 807108 x^{6} + 590100 x^{5} + 342360 x^{4} + 151992 x^{3} + 48528 x^{2} + 9888 x + 947$ | $9$ | $2$ | $30$ | 18T49 | not computed |
|
\(5\)
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 5.3.4.9a1.1 | $x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$ | $4$ | $3$ | $9$ | $C_{12}$ | $$[\ ]_{4}^{3}$$ | |
|
\(11\)
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 11.2.3.4a1.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 95 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
| 11.3.3.6a1.1 | $x^{9} + 6 x^{7} + 27 x^{6} + 12 x^{5} + 108 x^{4} + 251 x^{3} + 108 x^{2} + 486 x + 740$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |