Normalized defining polynomial
\( x^{27} - 78 x^{25} - 69 x^{24} + 2391 x^{23} + 3729 x^{22} - 36776 x^{21} - 78168 x^{20} + 303360 x^{19} + \cdots - 7561 \)
Invariants
| Degree: | $27$ |
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| Signature: | $(27, 0)$ |
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| Discriminant: |
\(1670605670104664083337543234069150946215895123101528358912\)
\(\medspace = 2^{18}\cdot 3^{27}\cdot 7^{18}\cdot 431^{9}\)
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| Root discriminant: | \(131.63\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(431\)
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| Discriminant root field: | \(\Q(\sqrt{1293}) \) | ||
| $\Aut(K/\Q)$: | $C_3^2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{6}a^{21}+\frac{1}{6}a^{18}-\frac{1}{2}a^{17}-\frac{1}{3}a^{15}-\frac{1}{2}a^{13}-\frac{1}{3}a^{12}-\frac{1}{2}a^{11}+\frac{1}{6}a^{9}-\frac{1}{3}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{6}a^{22}+\frac{1}{6}a^{19}-\frac{1}{2}a^{17}-\frac{1}{3}a^{16}+\frac{1}{6}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{6}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{7}-\frac{1}{2}a^{6}-\frac{1}{6}a^{4}-\frac{1}{2}a^{2}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{6}a^{23}+\frac{1}{6}a^{20}+\frac{1}{6}a^{17}-\frac{1}{3}a^{14}-\frac{1}{2}a^{12}-\frac{1}{3}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{6}a^{8}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{36}a^{24}-\frac{1}{12}a^{23}-\frac{1}{12}a^{22}-\frac{1}{36}a^{21}+\frac{1}{12}a^{20}+\frac{1}{12}a^{19}+\frac{1}{18}a^{18}-\frac{1}{12}a^{17}+\frac{2}{9}a^{15}-\frac{1}{4}a^{14}-\frac{1}{12}a^{13}-\frac{4}{9}a^{12}+\frac{1}{12}a^{11}-\frac{5}{12}a^{10}+\frac{2}{9}a^{9}-\frac{1}{4}a^{8}+\frac{1}{6}a^{7}+\frac{5}{12}a^{6}-\frac{1}{6}a^{5}+\frac{1}{3}a^{4}-\frac{1}{18}a^{3}+\frac{1}{6}a^{2}+\frac{5}{12}a-\frac{7}{36}$, $\frac{1}{36}a^{25}+\frac{1}{18}a^{22}+\frac{1}{6}a^{20}+\frac{5}{36}a^{19}+\frac{1}{12}a^{18}+\frac{1}{12}a^{17}+\frac{1}{18}a^{16}-\frac{1}{12}a^{15}-\frac{1}{2}a^{14}+\frac{5}{36}a^{13}-\frac{1}{4}a^{12}-\frac{1}{3}a^{11}+\frac{11}{36}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{5}{12}a^{6}-\frac{1}{2}a^{5}-\frac{7}{18}a^{4}-\frac{1}{2}a^{3}+\frac{1}{12}a^{2}-\frac{5}{18}a+\frac{5}{12}$, $\frac{1}{13\cdots 12}a^{26}-\frac{25\cdots 45}{66\cdots 06}a^{25}+\frac{57\cdots 73}{66\cdots 06}a^{24}-\frac{22\cdots 37}{33\cdots 03}a^{23}-\frac{32\cdots 49}{66\cdots 06}a^{22}+\frac{24\cdots 85}{66\cdots 06}a^{21}-\frac{17\cdots 95}{13\cdots 12}a^{20}-\frac{25\cdots 31}{13\cdots 12}a^{19}+\frac{18\cdots 53}{13\cdots 12}a^{18}+\frac{27\cdots 87}{66\cdots 06}a^{17}-\frac{57\cdots 49}{13\cdots 12}a^{16}-\frac{56\cdots 49}{66\cdots 06}a^{15}-\frac{53\cdots 03}{13\cdots 12}a^{14}+\frac{37\cdots 09}{13\cdots 12}a^{13}+\frac{16\cdots 77}{66\cdots 06}a^{12}-\frac{18\cdots 89}{13\cdots 12}a^{11}+\frac{21\cdots 81}{13\cdots 12}a^{10}-\frac{15\cdots 05}{13\cdots 12}a^{9}-\frac{17\cdots 05}{14\cdots 68}a^{8}+\frac{91\cdots 93}{44\cdots 04}a^{7}-\frac{11\cdots 10}{11\cdots 01}a^{6}-\frac{33\cdots 93}{66\cdots 06}a^{5}-\frac{31\cdots 25}{66\cdots 06}a^{4}-\frac{11\cdots 51}{13\cdots 12}a^{3}-\frac{23\cdots 61}{66\cdots 06}a^{2}-\frac{52\cdots 27}{13\cdots 12}a-\frac{87\cdots 23}{66\cdots 06}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $32$ (assuming GRH) |
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Unit group
| Rank: | $26$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{65\cdots 49}{33\cdots 03}a^{26}-\frac{51\cdots 06}{33\cdots 03}a^{25}-\frac{50\cdots 83}{33\cdots 03}a^{24}-\frac{50\cdots 46}{33\cdots 03}a^{23}+\frac{15\cdots 45}{33\cdots 03}a^{22}+\frac{11\cdots 90}{33\cdots 03}a^{21}-\frac{24\cdots 65}{33\cdots 03}a^{20}-\frac{31\cdots 40}{33\cdots 03}a^{19}+\frac{22\cdots 11}{33\cdots 03}a^{18}+\frac{36\cdots 10}{33\cdots 03}a^{17}-\frac{11\cdots 64}{33\cdots 03}a^{16}-\frac{23\cdots 34}{33\cdots 03}a^{15}+\frac{31\cdots 11}{33\cdots 03}a^{14}+\frac{82\cdots 22}{33\cdots 03}a^{13}-\frac{39\cdots 93}{33\cdots 03}a^{12}-\frac{16\cdots 14}{33\cdots 03}a^{11}+\frac{42\cdots 67}{33\cdots 03}a^{10}+\frac{17\cdots 58}{33\cdots 03}a^{9}+\frac{10\cdots 85}{11\cdots 01}a^{8}-\frac{34\cdots 64}{11\cdots 01}a^{7}-\frac{80\cdots 87}{11\cdots 01}a^{6}+\frac{34\cdots 92}{33\cdots 03}a^{5}+\frac{65\cdots 82}{33\cdots 03}a^{4}-\frac{66\cdots 04}{33\cdots 03}a^{3}-\frac{41\cdots 61}{33\cdots 03}a^{2}+\frac{56\cdots 88}{33\cdots 03}a-\frac{62\cdots 77}{33\cdots 03}$, $\frac{65\cdots 49}{33\cdots 03}a^{26}+\frac{51\cdots 06}{33\cdots 03}a^{25}+\frac{50\cdots 83}{33\cdots 03}a^{24}+\frac{50\cdots 46}{33\cdots 03}a^{23}-\frac{15\cdots 45}{33\cdots 03}a^{22}-\frac{11\cdots 90}{33\cdots 03}a^{21}+\frac{24\cdots 65}{33\cdots 03}a^{20}+\frac{31\cdots 40}{33\cdots 03}a^{19}-\frac{22\cdots 11}{33\cdots 03}a^{18}-\frac{36\cdots 10}{33\cdots 03}a^{17}+\frac{11\cdots 64}{33\cdots 03}a^{16}+\frac{23\cdots 34}{33\cdots 03}a^{15}-\frac{31\cdots 11}{33\cdots 03}a^{14}-\frac{82\cdots 22}{33\cdots 03}a^{13}+\frac{39\cdots 93}{33\cdots 03}a^{12}+\frac{16\cdots 14}{33\cdots 03}a^{11}-\frac{42\cdots 67}{33\cdots 03}a^{10}-\frac{17\cdots 58}{33\cdots 03}a^{9}-\frac{10\cdots 85}{11\cdots 01}a^{8}+\frac{34\cdots 64}{11\cdots 01}a^{7}+\frac{80\cdots 87}{11\cdots 01}a^{6}-\frac{34\cdots 92}{33\cdots 03}a^{5}-\frac{65\cdots 82}{33\cdots 03}a^{4}+\frac{66\cdots 04}{33\cdots 03}a^{3}+\frac{41\cdots 61}{33\cdots 03}a^{2}-\frac{56\cdots 88}{33\cdots 03}a+\frac{62\cdots 74}{33\cdots 03}$, $\frac{16\cdots 95}{66\cdots 06}a^{26}-\frac{23\cdots 27}{11\cdots 01}a^{25}-\frac{12\cdots 93}{66\cdots 06}a^{24}-\frac{38\cdots 55}{33\cdots 03}a^{23}+\frac{44\cdots 39}{74\cdots 34}a^{22}+\frac{14\cdots 39}{33\cdots 03}a^{21}-\frac{64\cdots 89}{66\cdots 06}a^{20}-\frac{43\cdots 10}{37\cdots 67}a^{19}+\frac{57\cdots 93}{66\cdots 06}a^{18}+\frac{45\cdots 83}{33\cdots 03}a^{17}-\frac{32\cdots 19}{74\cdots 34}a^{16}-\frac{28\cdots 51}{33\cdots 03}a^{15}+\frac{81\cdots 13}{66\cdots 06}a^{14}+\frac{11\cdots 57}{37\cdots 67}a^{13}-\frac{10\cdots 37}{66\cdots 06}a^{12}-\frac{20\cdots 93}{33\cdots 03}a^{11}+\frac{63\cdots 39}{22\cdots 02}a^{10}+\frac{21\cdots 52}{33\cdots 03}a^{9}+\frac{22\cdots 07}{22\cdots 02}a^{8}-\frac{13\cdots 12}{37\cdots 67}a^{7}-\frac{18\cdots 55}{22\cdots 02}a^{6}+\frac{42\cdots 61}{33\cdots 03}a^{5}+\frac{16\cdots 35}{74\cdots 34}a^{4}-\frac{79\cdots 19}{33\cdots 03}a^{3}-\frac{87\cdots 71}{66\cdots 06}a^{2}+\frac{22\cdots 85}{11\cdots 01}a-\frac{14\cdots 49}{66\cdots 06}$, $\frac{47\cdots 91}{22\cdots 02}a^{26}-\frac{56\cdots 15}{33\cdots 03}a^{25}-\frac{22\cdots 33}{13\cdots 12}a^{24}-\frac{72\cdots 01}{44\cdots 04}a^{23}+\frac{68\cdots 63}{13\cdots 12}a^{22}+\frac{52\cdots 57}{13\cdots 12}a^{21}-\frac{36\cdots 55}{44\cdots 04}a^{20}-\frac{13\cdots 89}{13\cdots 12}a^{19}+\frac{48\cdots 15}{66\cdots 06}a^{18}+\frac{53\cdots 03}{44\cdots 04}a^{17}-\frac{12\cdots 83}{33\cdots 03}a^{16}-\frac{25\cdots 39}{33\cdots 03}a^{15}+\frac{15\cdots 17}{14\cdots 68}a^{14}+\frac{35\cdots 35}{13\cdots 12}a^{13}-\frac{43\cdots 41}{33\cdots 03}a^{12}-\frac{23\cdots 97}{44\cdots 04}a^{11}+\frac{19\cdots 33}{13\cdots 12}a^{10}+\frac{37\cdots 37}{66\cdots 06}a^{9}+\frac{44\cdots 73}{44\cdots 04}a^{8}-\frac{24\cdots 33}{74\cdots 34}a^{7}-\frac{11\cdots 95}{14\cdots 68}a^{6}+\frac{12\cdots 54}{11\cdots 01}a^{5}+\frac{14\cdots 61}{66\cdots 06}a^{4}-\frac{72\cdots 32}{33\cdots 03}a^{3}-\frac{14\cdots 72}{11\cdots 01}a^{2}+\frac{24\cdots 37}{13\cdots 12}a-\frac{27\cdots 43}{13\cdots 12}$, $\frac{24\cdots 19}{66\cdots 06}a^{26}+\frac{10\cdots 81}{37\cdots 67}a^{25}+\frac{19\cdots 89}{66\cdots 06}a^{24}+\frac{93\cdots 83}{33\cdots 03}a^{23}-\frac{19\cdots 49}{22\cdots 02}a^{22}-\frac{22\cdots 63}{33\cdots 03}a^{21}+\frac{94\cdots 93}{66\cdots 06}a^{20}+\frac{19\cdots 38}{11\cdots 01}a^{19}-\frac{84\cdots 97}{66\cdots 06}a^{18}-\frac{69\cdots 73}{33\cdots 03}a^{17}+\frac{14\cdots 07}{22\cdots 02}a^{16}+\frac{43\cdots 63}{33\cdots 03}a^{15}-\frac{11\cdots 57}{66\cdots 06}a^{14}-\frac{51\cdots 51}{11\cdots 01}a^{13}+\frac{15\cdots 33}{66\cdots 06}a^{12}+\frac{30\cdots 61}{33\cdots 03}a^{11}-\frac{18\cdots 97}{74\cdots 34}a^{10}-\frac{32\cdots 08}{33\cdots 03}a^{9}-\frac{38\cdots 65}{22\cdots 02}a^{8}+\frac{21\cdots 68}{37\cdots 67}a^{7}+\frac{30\cdots 15}{22\cdots 02}a^{6}-\frac{65\cdots 67}{33\cdots 03}a^{5}-\frac{82\cdots 29}{22\cdots 02}a^{4}+\frac{12\cdots 71}{33\cdots 03}a^{3}+\frac{15\cdots 93}{66\cdots 06}a^{2}-\frac{35\cdots 94}{11\cdots 01}a+\frac{23\cdots 17}{66\cdots 06}$, $\frac{15\cdots 13}{66\cdots 06}a^{26}+\frac{72\cdots 69}{33\cdots 03}a^{25}+\frac{24\cdots 27}{13\cdots 12}a^{24}-\frac{62\cdots 91}{13\cdots 12}a^{23}-\frac{74\cdots 69}{13\cdots 12}a^{22}-\frac{47\cdots 91}{13\cdots 12}a^{21}+\frac{11\cdots 15}{13\cdots 12}a^{20}+\frac{13\cdots 43}{13\cdots 12}a^{19}-\frac{52\cdots 67}{66\cdots 06}a^{18}-\frac{15\cdots 71}{13\cdots 12}a^{17}+\frac{13\cdots 92}{33\cdots 03}a^{16}+\frac{24\cdots 91}{33\cdots 03}a^{15}-\frac{14\cdots 51}{13\cdots 12}a^{14}-\frac{34\cdots 85}{13\cdots 12}a^{13}+\frac{46\cdots 24}{33\cdots 03}a^{12}+\frac{65\cdots 57}{13\cdots 12}a^{11}-\frac{36\cdots 75}{13\cdots 12}a^{10}-\frac{31\cdots 59}{66\cdots 06}a^{9}-\frac{31\cdots 95}{44\cdots 04}a^{8}+\frac{55\cdots 89}{22\cdots 02}a^{7}+\frac{21\cdots 43}{44\cdots 04}a^{6}-\frac{24\cdots 68}{33\cdots 03}a^{5}-\frac{63\cdots 17}{66\cdots 06}a^{4}+\frac{40\cdots 79}{33\cdots 03}a^{3}+\frac{24\cdots 21}{33\cdots 03}a^{2}-\frac{12\cdots 87}{13\cdots 12}a+\frac{14\cdots 81}{13\cdots 12}$, $\frac{66\cdots 53}{33\cdots 03}a^{26}-\frac{59\cdots 52}{37\cdots 67}a^{25}-\frac{20\cdots 11}{13\cdots 12}a^{24}-\frac{18\cdots 79}{13\cdots 12}a^{23}+\frac{71\cdots 77}{14\cdots 68}a^{22}+\frac{48\cdots 05}{13\cdots 12}a^{21}-\frac{10\cdots 23}{13\cdots 12}a^{20}-\frac{42\cdots 39}{44\cdots 04}a^{19}+\frac{22\cdots 36}{33\cdots 03}a^{18}+\frac{14\cdots 09}{13\cdots 12}a^{17}-\frac{77\cdots 73}{22\cdots 02}a^{16}-\frac{23\cdots 76}{33\cdots 03}a^{15}+\frac{12\cdots 63}{13\cdots 12}a^{14}+\frac{37\cdots 75}{14\cdots 68}a^{13}-\frac{82\cdots 87}{66\cdots 06}a^{12}-\frac{65\cdots 63}{13\cdots 12}a^{11}+\frac{23\cdots 55}{14\cdots 68}a^{10}+\frac{34\cdots 55}{66\cdots 06}a^{9}+\frac{40\cdots 43}{44\cdots 04}a^{8}-\frac{68\cdots 71}{22\cdots 02}a^{7}-\frac{31\cdots 55}{44\cdots 04}a^{6}+\frac{34\cdots 06}{33\cdots 03}a^{5}+\frac{71\cdots 10}{37\cdots 67}a^{4}-\frac{66\cdots 14}{33\cdots 03}a^{3}-\frac{78\cdots 51}{66\cdots 06}a^{2}+\frac{74\cdots 67}{44\cdots 04}a-\frac{24\cdots 09}{13\cdots 12}$, $\frac{24\cdots 40}{11\cdots 01}a^{26}+\frac{56\cdots 28}{33\cdots 03}a^{25}+\frac{37\cdots 03}{22\cdots 02}a^{24}+\frac{14\cdots 99}{74\cdots 34}a^{23}-\frac{35\cdots 33}{66\cdots 06}a^{22}-\frac{91\cdots 59}{22\cdots 02}a^{21}+\frac{18\cdots 07}{22\cdots 02}a^{20}+\frac{71\cdots 29}{66\cdots 06}a^{19}-\frac{82\cdots 67}{11\cdots 01}a^{18}-\frac{27\cdots 71}{22\cdots 02}a^{17}+\frac{12\cdots 79}{33\cdots 03}a^{16}+\frac{29\cdots 44}{37\cdots 67}a^{15}-\frac{77\cdots 43}{74\cdots 34}a^{14}-\frac{18\cdots 67}{66\cdots 06}a^{13}+\frac{14\cdots 29}{11\cdots 01}a^{12}+\frac{12\cdots 73}{22\cdots 02}a^{11}-\frac{45\cdots 77}{66\cdots 06}a^{10}-\frac{21\cdots 75}{37\cdots 67}a^{9}-\frac{24\cdots 49}{22\cdots 02}a^{8}+\frac{37\cdots 69}{11\cdots 01}a^{7}+\frac{61\cdots 51}{74\cdots 34}a^{6}-\frac{42\cdots 74}{37\cdots 67}a^{5}-\frac{76\cdots 33}{33\cdots 03}a^{4}+\frac{24\cdots 78}{11\cdots 01}a^{3}+\frac{16\cdots 43}{11\cdots 01}a^{2}-\frac{12\cdots 35}{66\cdots 06}a+\frac{44\cdots 21}{22\cdots 02}$, $\frac{73\cdots 15}{22\cdots 02}a^{26}-\frac{86\cdots 27}{33\cdots 03}a^{25}-\frac{11\cdots 21}{44\cdots 04}a^{24}-\frac{11\cdots 43}{44\cdots 04}a^{23}+\frac{10\cdots 17}{13\cdots 12}a^{22}+\frac{89\cdots 35}{14\cdots 68}a^{21}-\frac{18\cdots 75}{14\cdots 68}a^{20}-\frac{21\cdots 35}{13\cdots 12}a^{19}+\frac{24\cdots 73}{22\cdots 02}a^{18}+\frac{82\cdots 29}{44\cdots 04}a^{17}-\frac{19\cdots 03}{33\cdots 03}a^{16}-\frac{12\cdots 81}{11\cdots 01}a^{15}+\frac{70\cdots 73}{44\cdots 04}a^{14}+\frac{55\cdots 45}{13\cdots 12}a^{13}-\frac{74\cdots 83}{37\cdots 67}a^{12}-\frac{12\cdots 33}{14\cdots 68}a^{11}+\frac{28\cdots 79}{13\cdots 12}a^{10}+\frac{19\cdots 63}{22\cdots 02}a^{9}+\frac{68\cdots 11}{44\cdots 04}a^{8}-\frac{38\cdots 11}{74\cdots 34}a^{7}-\frac{53\cdots 55}{44\cdots 04}a^{6}+\frac{19\cdots 79}{11\cdots 01}a^{5}+\frac{22\cdots 97}{66\cdots 06}a^{4}-\frac{12\cdots 82}{37\cdots 67}a^{3}-\frac{23\cdots 45}{11\cdots 01}a^{2}+\frac{38\cdots 27}{13\cdots 12}a-\frac{46\cdots 37}{14\cdots 68}$, $\frac{17\cdots 13}{66\cdots 06}a^{26}-\frac{36\cdots 95}{14\cdots 68}a^{25}-\frac{27\cdots 75}{13\cdots 12}a^{24}+\frac{12\cdots 95}{13\cdots 12}a^{23}+\frac{28\cdots 51}{44\cdots 04}a^{22}+\frac{52\cdots 81}{13\cdots 12}a^{21}-\frac{13\cdots 69}{13\cdots 12}a^{20}-\frac{24\cdots 77}{22\cdots 02}a^{19}+\frac{12\cdots 67}{13\cdots 12}a^{18}+\frac{45\cdots 16}{33\cdots 03}a^{17}-\frac{10\cdots 77}{22\cdots 02}a^{16}-\frac{11\cdots 13}{13\cdots 12}a^{15}+\frac{18\cdots 29}{13\cdots 12}a^{14}+\frac{34\cdots 99}{11\cdots 01}a^{13}-\frac{25\cdots 01}{13\cdots 12}a^{12}-\frac{82\cdots 71}{13\cdots 12}a^{11}+\frac{61\cdots 11}{74\cdots 34}a^{10}+\frac{87\cdots 93}{13\cdots 12}a^{9}+\frac{12\cdots 85}{22\cdots 02}a^{8}-\frac{58\cdots 79}{14\cdots 68}a^{7}-\frac{67\cdots 63}{11\cdots 01}a^{6}+\frac{44\cdots 38}{33\cdots 03}a^{5}+\frac{37\cdots 67}{22\cdots 02}a^{4}-\frac{16\cdots 75}{66\cdots 06}a^{3}-\frac{94\cdots 17}{13\cdots 12}a^{2}+\frac{86\cdots 27}{44\cdots 04}a-\frac{15\cdots 99}{66\cdots 06}$, $\frac{16\cdots 65}{44\cdots 04}a^{26}-\frac{19\cdots 29}{66\cdots 06}a^{25}-\frac{37\cdots 77}{13\cdots 12}a^{24}-\frac{39\cdots 19}{14\cdots 68}a^{23}+\frac{11\cdots 67}{13\cdots 12}a^{22}+\frac{88\cdots 95}{13\cdots 12}a^{21}-\frac{51\cdots 99}{37\cdots 67}a^{20}-\frac{11\cdots 79}{66\cdots 06}a^{19}+\frac{16\cdots 75}{13\cdots 12}a^{18}+\frac{90\cdots 49}{44\cdots 04}a^{17}-\frac{84\cdots 85}{13\cdots 12}a^{16}-\frac{85\cdots 47}{66\cdots 06}a^{15}+\frac{38\cdots 53}{22\cdots 02}a^{14}+\frac{30\cdots 13}{66\cdots 06}a^{13}-\frac{73\cdots 85}{33\cdots 03}a^{12}-\frac{66\cdots 33}{74\cdots 34}a^{11}+\frac{16\cdots 09}{66\cdots 06}a^{10}+\frac{12\cdots 51}{13\cdots 12}a^{9}+\frac{37\cdots 81}{22\cdots 02}a^{8}-\frac{24\cdots 47}{44\cdots 04}a^{7}-\frac{19\cdots 09}{14\cdots 68}a^{6}+\frac{14\cdots 59}{74\cdots 34}a^{5}+\frac{23\cdots 21}{66\cdots 06}a^{4}-\frac{47\cdots 45}{13\cdots 12}a^{3}-\frac{24\cdots 24}{11\cdots 01}a^{2}+\frac{10\cdots 48}{33\cdots 03}a-\frac{44\cdots 23}{13\cdots 12}$, $\frac{61\cdots 13}{74\cdots 34}a^{26}-\frac{72\cdots 05}{11\cdots 01}a^{25}-\frac{21\cdots 92}{33\cdots 03}a^{24}-\frac{48\cdots 27}{74\cdots 34}a^{23}+\frac{73\cdots 15}{37\cdots 67}a^{22}+\frac{10\cdots 95}{66\cdots 06}a^{21}-\frac{70\cdots 99}{22\cdots 02}a^{20}-\frac{44\cdots 13}{11\cdots 01}a^{19}+\frac{18\cdots 49}{66\cdots 06}a^{18}+\frac{34\cdots 11}{74\cdots 34}a^{17}-\frac{15\cdots 20}{11\cdots 01}a^{16}-\frac{19\cdots 23}{66\cdots 06}a^{15}+\frac{44\cdots 35}{11\cdots 01}a^{14}+\frac{38\cdots 85}{37\cdots 67}a^{13}-\frac{33\cdots 59}{66\cdots 06}a^{12}-\frac{22\cdots 04}{11\cdots 01}a^{11}+\frac{57\cdots 28}{11\cdots 01}a^{10}+\frac{14\cdots 85}{66\cdots 06}a^{9}+\frac{43\cdots 82}{11\cdots 01}a^{8}-\frac{28\cdots 57}{22\cdots 02}a^{7}-\frac{67\cdots 17}{22\cdots 02}a^{6}+\frac{16\cdots 16}{37\cdots 67}a^{5}+\frac{61\cdots 21}{74\cdots 34}a^{4}-\frac{56\cdots 17}{66\cdots 06}a^{3}-\frac{57\cdots 51}{11\cdots 01}a^{2}+\frac{53\cdots 49}{74\cdots 34}a-\frac{26\cdots 66}{33\cdots 03}$, $\frac{48\cdots 65}{33\cdots 03}a^{26}+\frac{83\cdots 61}{74\cdots 34}a^{25}+\frac{75\cdots 83}{66\cdots 06}a^{24}+\frac{91\cdots 09}{66\cdots 06}a^{23}-\frac{77\cdots 55}{22\cdots 02}a^{22}-\frac{91\cdots 09}{33\cdots 03}a^{21}+\frac{18\cdots 54}{33\cdots 03}a^{20}+\frac{52\cdots 15}{74\cdots 34}a^{19}-\frac{33\cdots 07}{66\cdots 06}a^{18}-\frac{27\cdots 24}{33\cdots 03}a^{17}+\frac{56\cdots 41}{22\cdots 02}a^{16}+\frac{17\cdots 68}{33\cdots 03}a^{15}-\frac{23\cdots 68}{33\cdots 03}a^{14}-\frac{41\cdots 23}{22\cdots 02}a^{13}+\frac{57\cdots 91}{66\cdots 06}a^{12}+\frac{12\cdots 04}{33\cdots 03}a^{11}-\frac{41\cdots 00}{11\cdots 01}a^{10}-\frac{12\cdots 85}{33\cdots 03}a^{9}-\frac{27\cdots 52}{37\cdots 67}a^{8}+\frac{50\cdots 17}{22\cdots 02}a^{7}+\frac{12\cdots 45}{22\cdots 02}a^{6}-\frac{25\cdots 10}{33\cdots 03}a^{5}-\frac{16\cdots 18}{11\cdots 01}a^{4}+\frac{98\cdots 01}{66\cdots 06}a^{3}+\frac{31\cdots 67}{33\cdots 03}a^{2}-\frac{94\cdots 71}{74\cdots 34}a+\frac{94\cdots 55}{66\cdots 06}$, $\frac{58\cdots 41}{11\cdots 01}a^{26}-\frac{27\cdots 55}{66\cdots 06}a^{25}-\frac{13\cdots 77}{33\cdots 03}a^{24}-\frac{92\cdots 73}{22\cdots 02}a^{23}+\frac{42\cdots 08}{33\cdots 03}a^{22}+\frac{32\cdots 65}{33\cdots 03}a^{21}-\frac{44\cdots 15}{22\cdots 02}a^{20}-\frac{84\cdots 81}{33\cdots 03}a^{19}+\frac{12\cdots 53}{66\cdots 06}a^{18}+\frac{66\cdots 69}{22\cdots 02}a^{17}-\frac{30\cdots 44}{33\cdots 03}a^{16}-\frac{62\cdots 33}{33\cdots 03}a^{15}+\frac{28\cdots 81}{11\cdots 01}a^{14}+\frac{22\cdots 39}{33\cdots 03}a^{13}-\frac{21\cdots 27}{66\cdots 06}a^{12}-\frac{29\cdots 87}{22\cdots 02}a^{11}+\frac{21\cdots 09}{66\cdots 06}a^{10}+\frac{92\cdots 81}{66\cdots 06}a^{9}+\frac{55\cdots 35}{22\cdots 02}a^{8}-\frac{91\cdots 71}{11\cdots 01}a^{7}-\frac{72\cdots 27}{37\cdots 67}a^{6}+\frac{31\cdots 42}{11\cdots 01}a^{5}+\frac{17\cdots 67}{33\cdots 03}a^{4}-\frac{35\cdots 19}{66\cdots 06}a^{3}-\frac{24\cdots 17}{74\cdots 34}a^{2}+\frac{15\cdots 24}{33\cdots 03}a-\frac{33\cdots 35}{66\cdots 06}$, $\frac{82\cdots 53}{66\cdots 06}a^{26}+\frac{13\cdots 13}{13\cdots 12}a^{25}+\frac{14\cdots 67}{14\cdots 68}a^{24}+\frac{12\cdots 03}{13\cdots 12}a^{23}-\frac{39\cdots 01}{13\cdots 12}a^{22}-\frac{99\cdots 57}{44\cdots 04}a^{21}+\frac{62\cdots 53}{13\cdots 12}a^{20}+\frac{19\cdots 18}{33\cdots 03}a^{19}-\frac{18\cdots 27}{44\cdots 04}a^{18}-\frac{46\cdots 29}{66\cdots 06}a^{17}+\frac{71\cdots 42}{33\cdots 03}a^{16}+\frac{19\cdots 69}{44\cdots 04}a^{15}-\frac{78\cdots 47}{13\cdots 12}a^{14}-\frac{10\cdots 23}{66\cdots 06}a^{13}+\frac{33\cdots 25}{44\cdots 04}a^{12}+\frac{40\cdots 51}{13\cdots 12}a^{11}-\frac{27\cdots 34}{33\cdots 03}a^{10}-\frac{14\cdots 79}{44\cdots 04}a^{9}-\frac{21\cdots 79}{37\cdots 67}a^{8}+\frac{85\cdots 37}{44\cdots 04}a^{7}+\frac{33\cdots 65}{74\cdots 34}a^{6}-\frac{21\cdots 54}{33\cdots 03}a^{5}-\frac{82\cdots 09}{66\cdots 06}a^{4}+\frac{27\cdots 41}{22\cdots 02}a^{3}+\frac{10\cdots 25}{13\cdots 12}a^{2}-\frac{14\cdots 77}{13\cdots 12}a+\frac{86\cdots 37}{74\cdots 34}$, $\frac{29\cdots 03}{44\cdots 04}a^{26}-\frac{57\cdots 02}{11\cdots 01}a^{25}-\frac{11\cdots 85}{22\cdots 02}a^{24}-\frac{39\cdots 25}{74\cdots 34}a^{23}+\frac{35\cdots 99}{22\cdots 02}a^{22}+\frac{27\cdots 13}{22\cdots 02}a^{21}-\frac{37\cdots 93}{14\cdots 68}a^{20}-\frac{14\cdots 01}{44\cdots 04}a^{19}+\frac{10\cdots 79}{44\cdots 04}a^{18}+\frac{27\cdots 35}{74\cdots 34}a^{17}-\frac{51\cdots 59}{44\cdots 04}a^{16}-\frac{52\cdots 83}{22\cdots 02}a^{15}+\frac{47\cdots 59}{14\cdots 68}a^{14}+\frac{37\cdots 81}{44\cdots 04}a^{13}-\frac{44\cdots 22}{11\cdots 01}a^{12}-\frac{24\cdots 25}{14\cdots 68}a^{11}+\frac{17\cdots 09}{44\cdots 04}a^{10}+\frac{77\cdots 45}{44\cdots 04}a^{9}+\frac{13\cdots 11}{44\cdots 04}a^{8}-\frac{45\cdots 21}{44\cdots 04}a^{7}-\frac{27\cdots 62}{11\cdots 01}a^{6}+\frac{13\cdots 26}{37\cdots 67}a^{5}+\frac{74\cdots 31}{11\cdots 01}a^{4}-\frac{29\cdots 23}{44\cdots 04}a^{3}-\frac{46\cdots 99}{11\cdots 01}a^{2}+\frac{85\cdots 55}{14\cdots 68}a-\frac{47\cdots 95}{74\cdots 34}$, $\frac{32\cdots 67}{13\cdots 12}a^{26}-\frac{22\cdots 59}{11\cdots 01}a^{25}-\frac{84\cdots 37}{44\cdots 04}a^{24}-\frac{18\cdots 77}{13\cdots 12}a^{23}+\frac{26\cdots 89}{44\cdots 04}a^{22}+\frac{64\cdots 73}{14\cdots 68}a^{21}-\frac{62\cdots 29}{66\cdots 06}a^{20}-\frac{12\cdots 62}{11\cdots 01}a^{19}+\frac{37\cdots 43}{44\cdots 04}a^{18}+\frac{18\cdots 13}{13\cdots 12}a^{17}-\frac{63\cdots 97}{14\cdots 68}a^{16}-\frac{95\cdots 12}{11\cdots 01}a^{15}+\frac{40\cdots 46}{33\cdots 03}a^{14}+\frac{33\cdots 87}{11\cdots 01}a^{13}-\frac{57\cdots 76}{37\cdots 67}a^{12}-\frac{40\cdots 25}{66\cdots 06}a^{11}+\frac{10\cdots 19}{37\cdots 67}a^{10}+\frac{28\cdots 81}{44\cdots 04}a^{9}+\frac{23\cdots 53}{22\cdots 02}a^{8}-\frac{17\cdots 63}{44\cdots 04}a^{7}-\frac{37\cdots 49}{44\cdots 04}a^{6}+\frac{87\cdots 37}{66\cdots 06}a^{5}+\frac{26\cdots 56}{11\cdots 01}a^{4}-\frac{36\cdots 49}{14\cdots 68}a^{3}-\frac{47\cdots 49}{33\cdots 03}a^{2}+\frac{78\cdots 35}{37\cdots 67}a-\frac{34\cdots 33}{14\cdots 68}$, $\frac{67\cdots 99}{13\cdots 12}a^{26}+\frac{26\cdots 79}{66\cdots 06}a^{25}+\frac{26\cdots 01}{66\cdots 06}a^{24}+\frac{27\cdots 29}{66\cdots 06}a^{23}-\frac{81\cdots 99}{66\cdots 06}a^{22}-\frac{31\cdots 16}{33\cdots 03}a^{21}+\frac{25\cdots 19}{13\cdots 12}a^{20}+\frac{32\cdots 21}{13\cdots 12}a^{19}-\frac{23\cdots 61}{13\cdots 12}a^{18}-\frac{19\cdots 19}{66\cdots 06}a^{17}+\frac{11\cdots 27}{13\cdots 12}a^{16}+\frac{12\cdots 97}{66\cdots 06}a^{15}-\frac{32\cdots 83}{13\cdots 12}a^{14}-\frac{85\cdots 17}{13\cdots 12}a^{13}+\frac{10\cdots 78}{33\cdots 03}a^{12}+\frac{16\cdots 49}{13\cdots 12}a^{11}-\frac{39\cdots 37}{13\cdots 12}a^{10}-\frac{17\cdots 61}{13\cdots 12}a^{9}-\frac{10\cdots 53}{44\cdots 04}a^{8}+\frac{35\cdots 17}{44\cdots 04}a^{7}+\frac{21\cdots 07}{11\cdots 01}a^{6}-\frac{90\cdots 89}{33\cdots 03}a^{5}-\frac{34\cdots 15}{66\cdots 06}a^{4}+\frac{69\cdots 07}{13\cdots 12}a^{3}+\frac{10\cdots 33}{33\cdots 03}a^{2}-\frac{59\cdots 93}{13\cdots 12}a+\frac{16\cdots 09}{33\cdots 03}$, $\frac{48\cdots 95}{37\cdots 67}a^{26}-\frac{68\cdots 33}{66\cdots 06}a^{25}-\frac{34\cdots 57}{33\cdots 03}a^{24}-\frac{12\cdots 87}{11\cdots 01}a^{23}+\frac{10\cdots 19}{33\cdots 03}a^{22}+\frac{81\cdots 25}{33\cdots 03}a^{21}-\frac{56\cdots 97}{11\cdots 01}a^{20}-\frac{21\cdots 39}{33\cdots 03}a^{19}+\frac{14\cdots 51}{33\cdots 03}a^{18}+\frac{55\cdots 17}{74\cdots 34}a^{17}-\frac{76\cdots 67}{33\cdots 03}a^{16}-\frac{15\cdots 79}{33\cdots 03}a^{15}+\frac{14\cdots 71}{22\cdots 02}a^{14}+\frac{11\cdots 41}{66\cdots 06}a^{13}-\frac{26\cdots 73}{33\cdots 03}a^{12}-\frac{36\cdots 86}{11\cdots 01}a^{11}+\frac{20\cdots 17}{33\cdots 03}a^{10}+\frac{23\cdots 13}{66\cdots 06}a^{9}+\frac{71\cdots 00}{11\cdots 01}a^{8}-\frac{46\cdots 31}{22\cdots 02}a^{7}-\frac{55\cdots 88}{11\cdots 01}a^{6}+\frac{15\cdots 39}{22\cdots 02}a^{5}+\frac{45\cdots 89}{33\cdots 03}a^{4}-\frac{45\cdots 24}{33\cdots 03}a^{3}-\frac{95\cdots 68}{11\cdots 01}a^{2}+\frac{77\cdots 65}{66\cdots 06}a-\frac{85\cdots 71}{66\cdots 06}$, $\frac{15\cdots 63}{44\cdots 04}a^{26}+\frac{18\cdots 05}{66\cdots 06}a^{25}+\frac{11\cdots 91}{44\cdots 04}a^{24}+\frac{12\cdots 53}{44\cdots 04}a^{23}-\frac{11\cdots 27}{13\cdots 12}a^{22}-\frac{94\cdots 71}{14\cdots 68}a^{21}+\frac{97\cdots 29}{74\cdots 34}a^{20}+\frac{11\cdots 93}{66\cdots 06}a^{19}-\frac{52\cdots 39}{44\cdots 04}a^{18}-\frac{86\cdots 51}{44\cdots 04}a^{17}+\frac{79\cdots 27}{13\cdots 12}a^{16}+\frac{13\cdots 06}{11\cdots 01}a^{15}-\frac{18\cdots 10}{11\cdots 01}a^{14}-\frac{28\cdots 51}{66\cdots 06}a^{13}+\frac{15\cdots 29}{74\cdots 34}a^{12}+\frac{63\cdots 79}{74\cdots 34}a^{11}-\frac{13\cdots 39}{66\cdots 06}a^{10}-\frac{40\cdots 69}{44\cdots 04}a^{9}-\frac{12\cdots 87}{74\cdots 34}a^{8}+\frac{79\cdots 01}{14\cdots 68}a^{7}+\frac{56\cdots 29}{44\cdots 04}a^{6}-\frac{40\cdots 73}{22\cdots 02}a^{5}-\frac{11\cdots 00}{33\cdots 03}a^{4}+\frac{52\cdots 83}{14\cdots 68}a^{3}+\frac{48\cdots 83}{22\cdots 02}a^{2}-\frac{20\cdots 19}{66\cdots 06}a+\frac{49\cdots 37}{14\cdots 68}$, $\frac{68\cdots 09}{44\cdots 04}a^{26}-\frac{53\cdots 43}{44\cdots 04}a^{25}-\frac{79\cdots 61}{66\cdots 06}a^{24}-\frac{28\cdots 87}{22\cdots 02}a^{23}+\frac{81\cdots 19}{22\cdots 02}a^{22}+\frac{18\cdots 05}{66\cdots 06}a^{21}-\frac{87\cdots 87}{14\cdots 68}a^{20}-\frac{16\cdots 01}{22\cdots 02}a^{19}+\frac{17\cdots 29}{33\cdots 03}a^{18}+\frac{38\cdots 95}{44\cdots 04}a^{17}-\frac{39\cdots 43}{14\cdots 68}a^{16}-\frac{72\cdots 85}{13\cdots 12}a^{15}+\frac{32\cdots 13}{44\cdots 04}a^{14}+\frac{21\cdots 63}{11\cdots 01}a^{13}-\frac{12\cdots 61}{13\cdots 12}a^{12}-\frac{56\cdots 95}{14\cdots 68}a^{11}+\frac{92\cdots 23}{11\cdots 01}a^{10}+\frac{13\cdots 81}{33\cdots 03}a^{9}+\frac{27\cdots 71}{37\cdots 67}a^{8}-\frac{26\cdots 82}{11\cdots 01}a^{7}-\frac{25\cdots 75}{44\cdots 04}a^{6}+\frac{18\cdots 73}{22\cdots 02}a^{5}+\frac{17\cdots 64}{11\cdots 01}a^{4}-\frac{20\cdots 67}{13\cdots 12}a^{3}-\frac{43\cdots 01}{44\cdots 04}a^{2}+\frac{59\cdots 19}{44\cdots 04}a-\frac{19\cdots 07}{13\cdots 12}$, $\frac{22\cdots 49}{13\cdots 12}a^{26}-\frac{89\cdots 87}{66\cdots 06}a^{25}-\frac{19\cdots 13}{14\cdots 68}a^{24}-\frac{18\cdots 49}{13\cdots 12}a^{23}+\frac{54\cdots 63}{13\cdots 12}a^{22}+\frac{46\cdots 07}{14\cdots 68}a^{21}-\frac{21\cdots 72}{33\cdots 03}a^{20}-\frac{27\cdots 89}{33\cdots 03}a^{19}+\frac{25\cdots 93}{44\cdots 04}a^{18}+\frac{12\cdots 91}{13\cdots 12}a^{17}-\frac{39\cdots 59}{13\cdots 12}a^{16}-\frac{13\cdots 45}{22\cdots 02}a^{15}+\frac{54\cdots 57}{66\cdots 06}a^{14}+\frac{14\cdots 07}{66\cdots 06}a^{13}-\frac{76\cdots 37}{74\cdots 34}a^{12}-\frac{28\cdots 11}{66\cdots 06}a^{11}+\frac{62\cdots 57}{66\cdots 06}a^{10}+\frac{20\cdots 03}{44\cdots 04}a^{9}+\frac{18\cdots 55}{22\cdots 02}a^{8}-\frac{39\cdots 21}{14\cdots 68}a^{7}-\frac{94\cdots 05}{14\cdots 68}a^{6}+\frac{30\cdots 79}{33\cdots 03}a^{5}+\frac{11\cdots 59}{66\cdots 06}a^{4}-\frac{25\cdots 85}{14\cdots 68}a^{3}-\frac{72\cdots 35}{66\cdots 06}a^{2}+\frac{49\cdots 09}{33\cdots 03}a-\frac{24\cdots 15}{14\cdots 68}$, $\frac{57\cdots 47}{11\cdots 01}a^{26}-\frac{54\cdots 99}{13\cdots 12}a^{25}-\frac{52\cdots 01}{13\cdots 12}a^{24}-\frac{54\cdots 07}{14\cdots 68}a^{23}+\frac{16\cdots 85}{13\cdots 12}a^{22}+\frac{12\cdots 67}{13\cdots 12}a^{21}-\frac{87\cdots 25}{44\cdots 04}a^{20}-\frac{81\cdots 48}{33\cdots 03}a^{19}+\frac{23\cdots 67}{13\cdots 12}a^{18}+\frac{31\cdots 02}{11\cdots 01}a^{17}-\frac{59\cdots 39}{66\cdots 06}a^{16}-\frac{24\cdots 37}{13\cdots 12}a^{15}+\frac{11\cdots 57}{44\cdots 04}a^{14}+\frac{21\cdots 18}{33\cdots 03}a^{13}-\frac{42\cdots 17}{13\cdots 12}a^{12}-\frac{56\cdots 29}{44\cdots 04}a^{11}+\frac{28\cdots 11}{66\cdots 06}a^{10}+\frac{18\cdots 71}{13\cdots 12}a^{9}+\frac{25\cdots 08}{11\cdots 01}a^{8}-\frac{11\cdots 13}{14\cdots 68}a^{7}-\frac{20\cdots 30}{11\cdots 01}a^{6}+\frac{20\cdots 01}{74\cdots 34}a^{5}+\frac{17\cdots 68}{33\cdots 03}a^{4}-\frac{34\cdots 05}{66\cdots 06}a^{3}-\frac{46\cdots 35}{14\cdots 68}a^{2}+\frac{59\cdots 45}{13\cdots 12}a-\frac{32\cdots 05}{66\cdots 06}$, $\frac{79\cdots 95}{33\cdots 03}a^{26}-\frac{24\cdots 77}{13\cdots 12}a^{25}-\frac{24\cdots 41}{13\cdots 12}a^{24}-\frac{25\cdots 49}{13\cdots 12}a^{23}+\frac{76\cdots 53}{13\cdots 12}a^{22}+\frac{58\cdots 33}{13\cdots 12}a^{21}-\frac{12\cdots 01}{13\cdots 12}a^{20}-\frac{38\cdots 76}{33\cdots 03}a^{19}+\frac{10\cdots 25}{13\cdots 12}a^{18}+\frac{89\cdots 17}{66\cdots 06}a^{17}-\frac{27\cdots 55}{66\cdots 06}a^{16}-\frac{11\cdots 89}{13\cdots 12}a^{15}+\frac{15\cdots 33}{13\cdots 12}a^{14}+\frac{99\cdots 88}{33\cdots 03}a^{13}-\frac{19\cdots 87}{13\cdots 12}a^{12}-\frac{78\cdots 65}{13\cdots 12}a^{11}+\frac{47\cdots 61}{33\cdots 03}a^{10}+\frac{83\cdots 49}{13\cdots 12}a^{9}+\frac{25\cdots 41}{22\cdots 02}a^{8}-\frac{16\cdots 13}{44\cdots 04}a^{7}-\frac{32\cdots 55}{37\cdots 67}a^{6}+\frac{84\cdots 05}{66\cdots 06}a^{5}+\frac{80\cdots 10}{33\cdots 03}a^{4}-\frac{80\cdots 89}{33\cdots 03}a^{3}-\frac{20\cdots 01}{13\cdots 12}a^{2}+\frac{27\cdots 89}{13\cdots 12}a-\frac{15\cdots 75}{66\cdots 06}$, $\frac{19\cdots 69}{44\cdots 04}a^{26}+\frac{46\cdots 07}{13\cdots 12}a^{25}+\frac{50\cdots 93}{14\cdots 68}a^{24}+\frac{15\cdots 81}{44\cdots 04}a^{23}-\frac{14\cdots 05}{13\cdots 12}a^{22}-\frac{36\cdots 61}{44\cdots 04}a^{21}+\frac{18\cdots 85}{11\cdots 01}a^{20}+\frac{28\cdots 71}{13\cdots 12}a^{19}-\frac{16\cdots 50}{11\cdots 01}a^{18}-\frac{18\cdots 39}{74\cdots 34}a^{17}+\frac{10\cdots 13}{13\cdots 12}a^{16}+\frac{23\cdots 99}{14\cdots 68}a^{15}-\frac{23\cdots 01}{11\cdots 01}a^{14}-\frac{74\cdots 69}{13\cdots 12}a^{13}+\frac{11\cdots 79}{44\cdots 04}a^{12}+\frac{24\cdots 77}{22\cdots 02}a^{11}-\frac{35\cdots 69}{13\cdots 12}a^{10}-\frac{86\cdots 17}{74\cdots 34}a^{9}-\frac{31\cdots 69}{14\cdots 68}a^{8}+\frac{25\cdots 71}{37\cdots 67}a^{7}+\frac{36\cdots 77}{22\cdots 02}a^{6}-\frac{26\cdots 90}{11\cdots 01}a^{5}-\frac{30\cdots 63}{66\cdots 06}a^{4}+\frac{20\cdots 87}{44\cdots 04}a^{3}+\frac{41\cdots 15}{14\cdots 68}a^{2}-\frac{25\cdots 25}{66\cdots 06}a+\frac{47\cdots 25}{11\cdots 01}$, $\frac{80\cdots 11}{33\cdots 03}a^{26}-\frac{78\cdots 03}{37\cdots 67}a^{25}-\frac{27\cdots 65}{14\cdots 68}a^{24}-\frac{46\cdots 17}{13\cdots 12}a^{23}+\frac{25\cdots 49}{44\cdots 04}a^{22}+\frac{58\cdots 33}{14\cdots 68}a^{21}-\frac{12\cdots 85}{13\cdots 12}a^{20}-\frac{48\cdots 05}{44\cdots 04}a^{19}+\frac{93\cdots 29}{11\cdots 01}a^{18}+\frac{17\cdots 59}{13\cdots 12}a^{17}-\frac{32\cdots 69}{74\cdots 34}a^{16}-\frac{92\cdots 19}{11\cdots 01}a^{15}+\frac{16\cdots 23}{13\cdots 12}a^{14}+\frac{13\cdots 31}{44\cdots 04}a^{13}-\frac{12\cdots 33}{74\cdots 34}a^{12}-\frac{80\cdots 41}{13\cdots 12}a^{11}+\frac{87\cdots 49}{14\cdots 68}a^{10}+\frac{14\cdots 63}{22\cdots 02}a^{9}+\frac{11\cdots 23}{14\cdots 68}a^{8}-\frac{44\cdots 76}{11\cdots 01}a^{7}-\frac{34\cdots 57}{44\cdots 04}a^{6}+\frac{46\cdots 78}{33\cdots 03}a^{5}+\frac{25\cdots 75}{11\cdots 01}a^{4}-\frac{99\cdots 98}{37\cdots 67}a^{3}-\frac{92\cdots 07}{66\cdots 06}a^{2}+\frac{99\cdots 77}{44\cdots 04}a-\frac{11\cdots 27}{44\cdots 04}$
|
| |
| Regulator: | \( 41400174381407900000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 24 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{27}\cdot(2\pi)^{0}\cdot 41400174381407900000 \cdot 4}{2\cdot\sqrt{1670605670104664083337543234069150946215895123101528358912}}\cr\approx \mathstrut & 0.271897411748645 \end{aligned}\] (assuming GRH)
Galois group
$C_3^3:S_3$ (as 27T46):
| A solvable group of order 162 |
| The 30 conjugacy class representatives for $C_3^3:S_3$ |
| Character table for $C_3^3:S_3$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), deg 3, deg 9, deg 9, deg 9, deg 9 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 27 siblings: | data not computed |
| Minimal sibling: | 27.27.2291640151035204503892377550163444370666522802608406528.1 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}{,}\,{\href{/padicField/13.3.0.1}{3} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{9}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.3.0.1}{3} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{9}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.3.0.1}{3} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.9.6.1 | $x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |
| 2.9.6.1 | $x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
| 2.9.6.1 | $x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
|
\(3\)
| 3.9.9.6 | $x^{9} + 6 x^{7} + 3 x^{6} + 12 x^{5} + 12 x^{4} + 14 x^{3} + 12 x^{2} + 12 x + 7$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $$[\frac{3}{2}]_{2}^{3}$$ |
| 3.9.9.6 | $x^{9} + 6 x^{7} + 3 x^{6} + 12 x^{5} + 12 x^{4} + 14 x^{3} + 12 x^{2} + 12 x + 7$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $$[\frac{3}{2}]_{2}^{3}$$ | |
| 3.9.9.6 | $x^{9} + 6 x^{7} + 3 x^{6} + 12 x^{5} + 12 x^{4} + 14 x^{3} + 12 x^{2} + 12 x + 7$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $$[\frac{3}{2}]_{2}^{3}$$ | |
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |
| 7.18.12.1 | $x^{18} + 3 x^{16} + 15 x^{15} + 15 x^{14} + 48 x^{13} + 109 x^{12} + 171 x^{11} + 333 x^{10} + 497 x^{9} + 717 x^{8} + 1032 x^{7} + 1216 x^{6} + 1296 x^{5} + 1143 x^{4} + 783 x^{3} + 432 x^{2} + 162 x + 34$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $$[\ ]_{3}^{6}$$ | |
|
\(431\)
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | ||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ | ||||
| Deg $6$ | $2$ | $3$ | $3$ |