Normalized defining polynomial
\( x^{27} - 117 x^{25} - 72 x^{24} + 5508 x^{23} + 7281 x^{22} - 134253 x^{21} - 286200 x^{20} + \cdots - 675379 \)
Invariants
| Degree: | $27$ |
| |
| Signature: | $(27, 0)$ |
| |
| Discriminant: |
\(124252631053426325344275434705435089635453266149328961\)
\(\medspace = 3^{73}\cdot 107^{9}\)
|
| |
| Root discriminant: | \(92.57\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(3\), \(107\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{321}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{19}+\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{20}+\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{21}+\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{22}+\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{23}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{24}-\frac{1}{3}a^{12}-\frac{1}{3}a^{9}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{25}-\frac{1}{3}a^{13}-\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{28\cdots 03}a^{26}+\frac{38\cdots 38}{28\cdots 03}a^{25}+\frac{50\cdots 59}{28\cdots 03}a^{24}+\frac{36\cdots 92}{28\cdots 03}a^{23}+\frac{61\cdots 85}{94\cdots 01}a^{22}-\frac{37\cdots 22}{28\cdots 03}a^{21}-\frac{63\cdots 24}{28\cdots 03}a^{20}-\frac{10\cdots 43}{94\cdots 01}a^{19}+\frac{39\cdots 47}{28\cdots 03}a^{18}+\frac{32\cdots 86}{94\cdots 01}a^{17}-\frac{59\cdots 08}{28\cdots 03}a^{16}+\frac{33\cdots 22}{28\cdots 03}a^{15}-\frac{12\cdots 53}{28\cdots 03}a^{14}-\frac{13\cdots 47}{28\cdots 03}a^{13}+\frac{22\cdots 12}{94\cdots 01}a^{12}+\frac{62\cdots 14}{94\cdots 01}a^{11}-\frac{14\cdots 87}{28\cdots 03}a^{10}-\frac{37\cdots 59}{28\cdots 03}a^{9}+\frac{13\cdots 50}{28\cdots 03}a^{8}-\frac{27\cdots 68}{94\cdots 01}a^{7}+\frac{21\cdots 06}{94\cdots 01}a^{6}-\frac{13\cdots 93}{94\cdots 01}a^{5}-\frac{12\cdots 47}{28\cdots 03}a^{4}+\frac{61\cdots 96}{28\cdots 03}a^{3}-\frac{13\cdots 79}{28\cdots 03}a^{2}+\frac{13\cdots 48}{28\cdots 03}a+\frac{44\cdots 40}{94\cdots 01}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
| |
| Narrow class group: | $C_{3}$, which has order $3$ (assuming GRH) |
|
Unit group
| Rank: | $26$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{10\cdots 30}{15\cdots 03}a^{26}-\frac{12\cdots 07}{15\cdots 03}a^{25}-\frac{12\cdots 25}{15\cdots 03}a^{24}+\frac{63\cdots 03}{15\cdots 03}a^{23}+\frac{58\cdots 47}{15\cdots 03}a^{22}+\frac{12\cdots 37}{15\cdots 03}a^{21}-\frac{14\cdots 65}{15\cdots 03}a^{20}-\frac{14\cdots 07}{15\cdots 03}a^{19}+\frac{62\cdots 34}{46\cdots 09}a^{18}+\frac{37\cdots 80}{15\cdots 03}a^{17}-\frac{16\cdots 20}{15\cdots 03}a^{16}-\frac{46\cdots 66}{15\cdots 03}a^{15}+\frac{50\cdots 57}{15\cdots 03}a^{14}+\frac{29\cdots 87}{15\cdots 03}a^{13}+\frac{12\cdots 22}{15\cdots 03}a^{12}-\frac{84\cdots 87}{15\cdots 03}a^{11}-\frac{12\cdots 98}{15\cdots 03}a^{10}+\frac{12\cdots 41}{46\cdots 09}a^{9}+\frac{25\cdots 22}{15\cdots 03}a^{8}+\frac{21\cdots 05}{15\cdots 03}a^{7}-\frac{54\cdots 34}{15\cdots 03}a^{6}-\frac{21\cdots 90}{15\cdots 03}a^{5}-\frac{17\cdots 82}{15\cdots 03}a^{4}-\frac{71\cdots 81}{15\cdots 03}a^{3}-\frac{15\cdots 79}{15\cdots 03}a^{2}-\frac{16\cdots 97}{15\cdots 03}a-\frac{19\cdots 57}{46\cdots 09}$, $\frac{77\cdots 06}{15\cdots 03}a^{26}+\frac{88\cdots 40}{15\cdots 03}a^{25}+\frac{89\cdots 18}{15\cdots 03}a^{24}-\frac{46\cdots 73}{15\cdots 03}a^{23}-\frac{42\cdots 66}{15\cdots 03}a^{22}-\frac{85\cdots 94}{15\cdots 03}a^{21}+\frac{10\cdots 27}{15\cdots 03}a^{20}+\frac{10\cdots 00}{15\cdots 03}a^{19}-\frac{44\cdots 58}{46\cdots 09}a^{18}-\frac{26\cdots 32}{15\cdots 03}a^{17}+\frac{11\cdots 78}{15\cdots 03}a^{16}+\frac{33\cdots 49}{15\cdots 03}a^{15}-\frac{36\cdots 72}{15\cdots 03}a^{14}-\frac{21\cdots 96}{15\cdots 03}a^{13}-\frac{88\cdots 40}{15\cdots 03}a^{12}+\frac{60\cdots 96}{15\cdots 03}a^{11}+\frac{91\cdots 02}{15\cdots 03}a^{10}-\frac{88\cdots 90}{46\cdots 09}a^{9}-\frac{18\cdots 54}{15\cdots 03}a^{8}-\frac{15\cdots 53}{15\cdots 03}a^{7}+\frac{39\cdots 87}{15\cdots 03}a^{6}+\frac{15\cdots 23}{15\cdots 03}a^{5}+\frac{12\cdots 90}{15\cdots 03}a^{4}+\frac{51\cdots 20}{15\cdots 03}a^{3}+\frac{11\cdots 99}{15\cdots 03}a^{2}+\frac{11\cdots 05}{15\cdots 03}a+\frac{13\cdots 95}{46\cdots 09}$, $\frac{12\cdots 96}{28\cdots 03}a^{26}+\frac{46\cdots 44}{94\cdots 01}a^{25}+\frac{47\cdots 65}{94\cdots 01}a^{24}-\frac{73\cdots 04}{28\cdots 03}a^{23}-\frac{67\cdots 02}{28\cdots 03}a^{22}-\frac{46\cdots 30}{94\cdots 01}a^{21}+\frac{16\cdots 33}{28\cdots 03}a^{20}+\frac{16\cdots 66}{28\cdots 03}a^{19}-\frac{23\cdots 43}{28\cdots 03}a^{18}-\frac{43\cdots 12}{28\cdots 03}a^{17}+\frac{61\cdots 53}{94\cdots 01}a^{16}+\frac{53\cdots 28}{28\cdots 03}a^{15}-\frac{57\cdots 16}{28\cdots 03}a^{14}-\frac{33\cdots 98}{28\cdots 03}a^{13}-\frac{14\cdots 98}{28\cdots 03}a^{12}+\frac{96\cdots 10}{28\cdots 03}a^{11}+\frac{14\cdots 61}{28\cdots 03}a^{10}-\frac{46\cdots 28}{28\cdots 03}a^{9}-\frac{29\cdots 40}{28\cdots 03}a^{8}-\frac{81\cdots 22}{94\cdots 01}a^{7}+\frac{62\cdots 49}{28\cdots 03}a^{6}+\frac{25\cdots 84}{28\cdots 03}a^{5}+\frac{20\cdots 41}{28\cdots 03}a^{4}+\frac{82\cdots 92}{28\cdots 03}a^{3}+\frac{17\cdots 61}{28\cdots 03}a^{2}+\frac{18\cdots 58}{28\cdots 03}a+\frac{24\cdots 07}{94\cdots 01}$, $\frac{13\cdots 10}{28\cdots 03}a^{26}+\frac{51\cdots 44}{94\cdots 01}a^{25}+\frac{15\cdots 63}{28\cdots 03}a^{24}-\frac{81\cdots 50}{28\cdots 03}a^{23}-\frac{75\cdots 64}{28\cdots 03}a^{22}-\frac{15\cdots 22}{28\cdots 03}a^{21}+\frac{18\cdots 19}{28\cdots 03}a^{20}+\frac{18\cdots 76}{28\cdots 03}a^{19}-\frac{26\cdots 17}{28\cdots 03}a^{18}-\frac{48\cdots 92}{28\cdots 03}a^{17}+\frac{68\cdots 70}{94\cdots 01}a^{16}+\frac{59\cdots 36}{28\cdots 03}a^{15}-\frac{63\cdots 42}{28\cdots 03}a^{14}-\frac{37\cdots 64}{28\cdots 03}a^{13}-\frac{52\cdots 07}{94\cdots 01}a^{12}+\frac{10\cdots 36}{28\cdots 03}a^{11}+\frac{16\cdots 04}{28\cdots 03}a^{10}-\frac{51\cdots 49}{28\cdots 03}a^{9}-\frac{33\cdots 78}{28\cdots 03}a^{8}-\frac{90\cdots 12}{94\cdots 01}a^{7}+\frac{23\cdots 23}{94\cdots 01}a^{6}+\frac{28\cdots 33}{28\cdots 03}a^{5}+\frac{22\cdots 44}{28\cdots 03}a^{4}+\frac{91\cdots 97}{28\cdots 03}a^{3}+\frac{19\cdots 60}{28\cdots 03}a^{2}+\frac{20\cdots 11}{28\cdots 03}a+\frac{27\cdots 44}{94\cdots 01}$, $\frac{20\cdots 46}{28\cdots 03}a^{26}-\frac{73\cdots 98}{94\cdots 01}a^{25}-\frac{78\cdots 36}{94\cdots 01}a^{24}+\frac{10\cdots 94}{28\cdots 03}a^{23}+\frac{36\cdots 44}{94\cdots 01}a^{22}+\frac{27\cdots 30}{28\cdots 03}a^{21}-\frac{27\cdots 06}{28\cdots 03}a^{20}-\frac{93\cdots 32}{94\cdots 01}a^{19}+\frac{12\cdots 06}{94\cdots 01}a^{18}+\frac{71\cdots 50}{28\cdots 03}a^{17}-\frac{10\cdots 62}{94\cdots 01}a^{16}-\frac{88\cdots 62}{28\cdots 03}a^{15}+\frac{91\cdots 08}{28\cdots 03}a^{14}+\frac{18\cdots 72}{94\cdots 01}a^{13}+\frac{24\cdots 65}{28\cdots 03}a^{12}-\frac{15\cdots 92}{28\cdots 03}a^{11}-\frac{81\cdots 45}{94\cdots 01}a^{10}+\frac{72\cdots 87}{28\cdots 03}a^{9}+\frac{49\cdots 91}{28\cdots 03}a^{8}+\frac{13\cdots 02}{94\cdots 01}a^{7}-\frac{99\cdots 88}{28\cdots 03}a^{6}-\frac{41\cdots 91}{28\cdots 03}a^{5}-\frac{11\cdots 56}{94\cdots 01}a^{4}-\frac{45\cdots 75}{94\cdots 01}a^{3}-\frac{30\cdots 45}{28\cdots 03}a^{2}-\frac{10\cdots 13}{94\cdots 01}a-\frac{12\cdots 79}{28\cdots 03}$, $\frac{17\cdots 66}{28\cdots 03}a^{26}+\frac{19\cdots 38}{28\cdots 03}a^{25}+\frac{67\cdots 04}{94\cdots 01}a^{24}-\frac{10\cdots 04}{28\cdots 03}a^{23}-\frac{94\cdots 04}{28\cdots 03}a^{22}-\frac{65\cdots 08}{94\cdots 01}a^{21}+\frac{23\cdots 44}{28\cdots 03}a^{20}+\frac{23\cdots 50}{28\cdots 03}a^{19}-\frac{11\cdots 48}{94\cdots 01}a^{18}-\frac{60\cdots 64}{28\cdots 03}a^{17}+\frac{25\cdots 73}{28\cdots 03}a^{16}+\frac{74\cdots 80}{28\cdots 03}a^{15}-\frac{80\cdots 96}{28\cdots 03}a^{14}-\frac{47\cdots 56}{28\cdots 03}a^{13}-\frac{19\cdots 69}{28\cdots 03}a^{12}+\frac{13\cdots 76}{28\cdots 03}a^{11}+\frac{20\cdots 17}{28\cdots 03}a^{10}-\frac{65\cdots 41}{28\cdots 03}a^{9}-\frac{41\cdots 18}{28\cdots 03}a^{8}-\frac{34\cdots 80}{28\cdots 03}a^{7}+\frac{88\cdots 98}{28\cdots 03}a^{6}+\frac{35\cdots 55}{28\cdots 03}a^{5}+\frac{28\cdots 02}{28\cdots 03}a^{4}+\frac{11\cdots 23}{28\cdots 03}a^{3}+\frac{25\cdots 78}{28\cdots 03}a^{2}+\frac{26\cdots 22}{28\cdots 03}a+\frac{10\cdots 53}{28\cdots 03}$, $\frac{43\cdots 98}{28\cdots 03}a^{26}+\frac{49\cdots 08}{28\cdots 03}a^{25}+\frac{50\cdots 27}{28\cdots 03}a^{24}-\frac{25\cdots 69}{28\cdots 03}a^{23}-\frac{23\cdots 82}{28\cdots 03}a^{22}-\frac{16\cdots 10}{94\cdots 01}a^{21}+\frac{59\cdots 33}{28\cdots 03}a^{20}+\frac{58\cdots 00}{28\cdots 03}a^{19}-\frac{84\cdots 66}{28\cdots 03}a^{18}-\frac{15\cdots 55}{28\cdots 03}a^{17}+\frac{65\cdots 88}{28\cdots 03}a^{16}+\frac{62\cdots 45}{94\cdots 01}a^{15}-\frac{20\cdots 12}{28\cdots 03}a^{14}-\frac{11\cdots 26}{28\cdots 03}a^{13}-\frac{50\cdots 65}{28\cdots 03}a^{12}+\frac{34\cdots 68}{28\cdots 03}a^{11}+\frac{51\cdots 18}{28\cdots 03}a^{10}-\frac{16\cdots 01}{28\cdots 03}a^{9}-\frac{10\cdots 17}{28\cdots 03}a^{8}-\frac{86\cdots 41}{28\cdots 03}a^{7}+\frac{22\cdots 19}{28\cdots 03}a^{6}+\frac{89\cdots 46}{28\cdots 03}a^{5}+\frac{71\cdots 82}{28\cdots 03}a^{4}+\frac{29\cdots 94}{28\cdots 03}a^{3}+\frac{63\cdots 92}{28\cdots 03}a^{2}+\frac{66\cdots 90}{28\cdots 03}a+\frac{87\cdots 80}{94\cdots 01}$, $\frac{57\cdots 28}{28\cdots 03}a^{26}+\frac{21\cdots 38}{94\cdots 01}a^{25}+\frac{22\cdots 15}{94\cdots 01}a^{24}-\frac{34\cdots 73}{28\cdots 03}a^{23}-\frac{10\cdots 34}{94\cdots 01}a^{22}-\frac{59\cdots 84}{28\cdots 03}a^{21}+\frac{77\cdots 99}{28\cdots 03}a^{20}+\frac{24\cdots 20}{94\cdots 01}a^{19}-\frac{10\cdots 70}{28\cdots 03}a^{18}-\frac{19\cdots 13}{28\cdots 03}a^{17}+\frac{28\cdots 72}{94\cdots 01}a^{16}+\frac{24\cdots 79}{28\cdots 03}a^{15}-\frac{26\cdots 84}{28\cdots 03}a^{14}-\frac{51\cdots 74}{94\cdots 01}a^{13}-\frac{21\cdots 80}{94\cdots 01}a^{12}+\frac{44\cdots 68}{28\cdots 03}a^{11}+\frac{22\cdots 07}{94\cdots 01}a^{10}-\frac{73\cdots 62}{94\cdots 01}a^{9}-\frac{13\cdots 79}{28\cdots 03}a^{8}-\frac{36\cdots 71}{94\cdots 01}a^{7}+\frac{29\cdots 77}{28\cdots 03}a^{6}+\frac{11\cdots 19}{28\cdots 03}a^{5}+\frac{30\cdots 29}{94\cdots 01}a^{4}+\frac{37\cdots 23}{28\cdots 03}a^{3}+\frac{81\cdots 74}{28\cdots 03}a^{2}+\frac{28\cdots 91}{94\cdots 01}a+\frac{33\cdots 97}{28\cdots 03}$, $\frac{89\cdots 06}{28\cdots 03}a^{26}-\frac{32\cdots 71}{94\cdots 01}a^{25}-\frac{34\cdots 48}{94\cdots 01}a^{24}+\frac{48\cdots 35}{28\cdots 03}a^{23}+\frac{48\cdots 50}{28\cdots 03}a^{22}+\frac{11\cdots 89}{28\cdots 03}a^{21}-\frac{12\cdots 34}{28\cdots 03}a^{20}-\frac{41\cdots 45}{94\cdots 01}a^{19}+\frac{17\cdots 55}{28\cdots 03}a^{18}+\frac{10\cdots 44}{94\cdots 01}a^{17}-\frac{44\cdots 13}{94\cdots 01}a^{16}-\frac{13\cdots 18}{94\cdots 01}a^{15}+\frac{41\cdots 44}{28\cdots 03}a^{14}+\frac{24\cdots 90}{28\cdots 03}a^{13}+\frac{10\cdots 18}{28\cdots 03}a^{12}-\frac{70\cdots 59}{28\cdots 03}a^{11}-\frac{10\cdots 67}{28\cdots 03}a^{10}+\frac{33\cdots 51}{28\cdots 03}a^{9}+\frac{21\cdots 63}{28\cdots 03}a^{8}+\frac{59\cdots 95}{94\cdots 01}a^{7}-\frac{15\cdots 39}{94\cdots 01}a^{6}-\frac{18\cdots 37}{28\cdots 03}a^{5}-\frac{49\cdots 99}{94\cdots 01}a^{4}-\frac{20\cdots 23}{94\cdots 01}a^{3}-\frac{43\cdots 63}{94\cdots 01}a^{2}-\frac{13\cdots 10}{28\cdots 03}a-\frac{18\cdots 56}{94\cdots 01}$, $\frac{42\cdots 20}{28\cdots 03}a^{26}+\frac{47\cdots 05}{28\cdots 03}a^{25}+\frac{49\cdots 51}{28\cdots 03}a^{24}-\frac{83\cdots 59}{94\cdots 01}a^{23}-\frac{23\cdots 13}{28\cdots 03}a^{22}-\frac{48\cdots 66}{28\cdots 03}a^{21}+\frac{57\cdots 38}{28\cdots 03}a^{20}+\frac{56\cdots 21}{28\cdots 03}a^{19}-\frac{27\cdots 28}{94\cdots 01}a^{18}-\frac{49\cdots 52}{94\cdots 01}a^{17}+\frac{21\cdots 52}{94\cdots 01}a^{16}+\frac{18\cdots 01}{28\cdots 03}a^{15}-\frac{19\cdots 44}{28\cdots 03}a^{14}-\frac{11\cdots 38}{28\cdots 03}a^{13}-\frac{16\cdots 82}{94\cdots 01}a^{12}+\frac{33\cdots 62}{28\cdots 03}a^{11}+\frac{50\cdots 85}{28\cdots 03}a^{10}-\frac{15\cdots 92}{28\cdots 03}a^{9}-\frac{10\cdots 35}{28\cdots 03}a^{8}-\frac{83\cdots 20}{28\cdots 03}a^{7}+\frac{21\cdots 36}{28\cdots 03}a^{6}+\frac{86\cdots 70}{28\cdots 03}a^{5}+\frac{23\cdots 56}{94\cdots 01}a^{4}+\frac{28\cdots 50}{28\cdots 03}a^{3}+\frac{60\cdots 85}{28\cdots 03}a^{2}+\frac{64\cdots 98}{28\cdots 03}a+\frac{25\cdots 01}{28\cdots 03}$, $\frac{20\cdots 22}{28\cdots 03}a^{26}+\frac{23\cdots 73}{28\cdots 03}a^{25}+\frac{79\cdots 57}{94\cdots 01}a^{24}-\frac{41\cdots 02}{94\cdots 01}a^{23}-\frac{11\cdots 13}{28\cdots 03}a^{22}-\frac{72\cdots 16}{94\cdots 01}a^{21}+\frac{92\cdots 73}{94\cdots 01}a^{20}+\frac{27\cdots 00}{28\cdots 03}a^{19}-\frac{13\cdots 70}{94\cdots 01}a^{18}-\frac{23\cdots 35}{94\cdots 01}a^{17}+\frac{30\cdots 02}{28\cdots 03}a^{16}+\frac{29\cdots 02}{94\cdots 01}a^{15}-\frac{32\cdots 67}{94\cdots 01}a^{14}-\frac{18\cdots 17}{94\cdots 01}a^{13}-\frac{23\cdots 04}{28\cdots 03}a^{12}+\frac{16\cdots 84}{28\cdots 03}a^{11}+\frac{80\cdots 57}{94\cdots 01}a^{10}-\frac{78\cdots 35}{28\cdots 03}a^{9}-\frac{49\cdots 58}{28\cdots 03}a^{8}-\frac{39\cdots 11}{28\cdots 03}a^{7}+\frac{35\cdots 13}{94\cdots 01}a^{6}+\frac{13\cdots 31}{94\cdots 01}a^{5}+\frac{33\cdots 73}{28\cdots 03}a^{4}+\frac{13\cdots 74}{28\cdots 03}a^{3}+\frac{29\cdots 40}{28\cdots 03}a^{2}+\frac{30\cdots 28}{28\cdots 03}a+\frac{12\cdots 57}{28\cdots 03}$, $\frac{13\cdots 96}{28\cdots 03}a^{26}-\frac{50\cdots 55}{94\cdots 01}a^{25}-\frac{15\cdots 92}{28\cdots 03}a^{24}+\frac{26\cdots 87}{94\cdots 01}a^{23}+\frac{72\cdots 36}{28\cdots 03}a^{22}+\frac{15\cdots 74}{28\cdots 03}a^{21}-\frac{18\cdots 56}{28\cdots 03}a^{20}-\frac{17\cdots 82}{28\cdots 03}a^{19}+\frac{85\cdots 69}{94\cdots 01}a^{18}+\frac{46\cdots 03}{28\cdots 03}a^{17}-\frac{66\cdots 38}{94\cdots 01}a^{16}-\frac{57\cdots 67}{28\cdots 03}a^{15}+\frac{20\cdots 28}{94\cdots 01}a^{14}+\frac{12\cdots 03}{94\cdots 01}a^{13}+\frac{15\cdots 14}{28\cdots 03}a^{12}-\frac{34\cdots 18}{94\cdots 01}a^{11}-\frac{15\cdots 76}{28\cdots 03}a^{10}+\frac{50\cdots 88}{28\cdots 03}a^{9}+\frac{10\cdots 50}{94\cdots 01}a^{8}+\frac{87\cdots 58}{94\cdots 01}a^{7}-\frac{68\cdots 18}{28\cdots 03}a^{6}-\frac{90\cdots 75}{94\cdots 01}a^{5}-\frac{21\cdots 88}{28\cdots 03}a^{4}-\frac{29\cdots 10}{94\cdots 01}a^{3}-\frac{19\cdots 27}{28\cdots 03}a^{2}-\frac{67\cdots 20}{94\cdots 01}a-\frac{79\cdots 72}{28\cdots 03}$, $\frac{65\cdots 16}{94\cdots 01}a^{26}-\frac{22\cdots 55}{28\cdots 03}a^{25}-\frac{76\cdots 42}{94\cdots 01}a^{24}+\frac{11\cdots 60}{28\cdots 03}a^{23}+\frac{35\cdots 07}{94\cdots 01}a^{22}+\frac{20\cdots 88}{28\cdots 03}a^{21}-\frac{26\cdots 52}{28\cdots 03}a^{20}-\frac{25\cdots 23}{28\cdots 03}a^{19}+\frac{37\cdots 91}{28\cdots 03}a^{18}+\frac{22\cdots 47}{94\cdots 01}a^{17}-\frac{29\cdots 53}{28\cdots 03}a^{16}-\frac{84\cdots 11}{28\cdots 03}a^{15}+\frac{92\cdots 49}{28\cdots 03}a^{14}+\frac{53\cdots 55}{28\cdots 03}a^{13}+\frac{73\cdots 86}{94\cdots 01}a^{12}-\frac{15\cdots 96}{28\cdots 03}a^{11}-\frac{23\cdots 23}{28\cdots 03}a^{10}+\frac{25\cdots 37}{94\cdots 01}a^{9}+\frac{15\cdots 64}{94\cdots 01}a^{8}+\frac{12\cdots 28}{94\cdots 01}a^{7}-\frac{10\cdots 28}{28\cdots 03}a^{6}-\frac{39\cdots 68}{28\cdots 03}a^{5}-\frac{31\cdots 04}{28\cdots 03}a^{4}-\frac{12\cdots 11}{28\cdots 03}a^{3}-\frac{28\cdots 11}{28\cdots 03}a^{2}-\frac{98\cdots 43}{94\cdots 01}a-\frac{11\cdots 76}{28\cdots 03}$, $\frac{10\cdots 74}{94\cdots 01}a^{26}+\frac{34\cdots 60}{28\cdots 03}a^{25}+\frac{35\cdots 21}{28\cdots 03}a^{24}-\frac{18\cdots 33}{28\cdots 03}a^{23}-\frac{16\cdots 20}{28\cdots 03}a^{22}-\frac{34\cdots 46}{28\cdots 03}a^{21}+\frac{41\cdots 66}{28\cdots 03}a^{20}+\frac{40\cdots 19}{28\cdots 03}a^{19}-\frac{59\cdots 60}{28\cdots 03}a^{18}-\frac{35\cdots 37}{94\cdots 01}a^{17}+\frac{15\cdots 03}{94\cdots 01}a^{16}+\frac{13\cdots 11}{28\cdots 03}a^{15}-\frac{14\cdots 88}{28\cdots 03}a^{14}-\frac{27\cdots 95}{94\cdots 01}a^{13}-\frac{35\cdots 97}{28\cdots 03}a^{12}+\frac{23\cdots 79}{28\cdots 03}a^{11}+\frac{12\cdots 02}{94\cdots 01}a^{10}-\frac{11\cdots 40}{28\cdots 03}a^{9}-\frac{24\cdots 82}{94\cdots 01}a^{8}-\frac{60\cdots 85}{28\cdots 03}a^{7}+\frac{52\cdots 34}{94\cdots 01}a^{6}+\frac{62\cdots 84}{28\cdots 03}a^{5}+\frac{49\cdots 94}{28\cdots 03}a^{4}+\frac{67\cdots 67}{94\cdots 01}a^{3}+\frac{44\cdots 12}{28\cdots 03}a^{2}+\frac{46\cdots 66}{28\cdots 03}a+\frac{60\cdots 03}{94\cdots 01}$, $\frac{56\cdots 44}{28\cdots 03}a^{26}+\frac{64\cdots 31}{28\cdots 03}a^{25}+\frac{65\cdots 27}{28\cdots 03}a^{24}-\frac{33\cdots 29}{28\cdots 03}a^{23}-\frac{30\cdots 68}{28\cdots 03}a^{22}-\frac{63\cdots 35}{28\cdots 03}a^{21}+\frac{76\cdots 29}{28\cdots 03}a^{20}+\frac{25\cdots 80}{94\cdots 01}a^{19}-\frac{36\cdots 11}{94\cdots 01}a^{18}-\frac{19\cdots 37}{28\cdots 03}a^{17}+\frac{84\cdots 96}{28\cdots 03}a^{16}+\frac{24\cdots 31}{28\cdots 03}a^{15}-\frac{26\cdots 26}{28\cdots 03}a^{14}-\frac{15\cdots 02}{28\cdots 03}a^{13}-\frac{21\cdots 02}{94\cdots 01}a^{12}+\frac{44\cdots 59}{28\cdots 03}a^{11}+\frac{66\cdots 18}{28\cdots 03}a^{10}-\frac{21\cdots 73}{28\cdots 03}a^{9}-\frac{13\cdots 64}{28\cdots 03}a^{8}-\frac{36\cdots 98}{94\cdots 01}a^{7}+\frac{95\cdots 24}{94\cdots 01}a^{6}+\frac{11\cdots 89}{28\cdots 03}a^{5}+\frac{30\cdots 57}{94\cdots 01}a^{4}+\frac{37\cdots 98}{28\cdots 03}a^{3}+\frac{81\cdots 22}{28\cdots 03}a^{2}+\frac{85\cdots 07}{28\cdots 03}a+\frac{33\cdots 76}{28\cdots 03}$, $\frac{87\cdots 19}{94\cdots 01}a^{26}+\frac{96\cdots 75}{94\cdots 01}a^{25}+\frac{10\cdots 02}{94\cdots 01}a^{24}-\frac{14\cdots 44}{28\cdots 03}a^{23}-\frac{47\cdots 36}{94\cdots 01}a^{22}-\frac{11\cdots 04}{94\cdots 01}a^{21}+\frac{35\cdots 19}{28\cdots 03}a^{20}+\frac{35\cdots 49}{28\cdots 03}a^{19}-\frac{16\cdots 28}{94\cdots 01}a^{18}-\frac{92\cdots 34}{28\cdots 03}a^{17}+\frac{12\cdots 86}{94\cdots 01}a^{16}+\frac{11\cdots 73}{28\cdots 03}a^{15}-\frac{12\cdots 66}{28\cdots 03}a^{14}-\frac{71\cdots 36}{28\cdots 03}a^{13}-\frac{10\cdots 00}{94\cdots 01}a^{12}+\frac{20\cdots 82}{28\cdots 03}a^{11}+\frac{31\cdots 36}{28\cdots 03}a^{10}-\frac{96\cdots 58}{28\cdots 03}a^{9}-\frac{63\cdots 52}{28\cdots 03}a^{8}-\frac{17\cdots 61}{94\cdots 01}a^{7}+\frac{13\cdots 10}{28\cdots 03}a^{6}+\frac{17\cdots 04}{94\cdots 01}a^{5}+\frac{43\cdots 12}{28\cdots 03}a^{4}+\frac{58\cdots 51}{94\cdots 01}a^{3}+\frac{38\cdots 49}{28\cdots 03}a^{2}+\frac{13\cdots 70}{94\cdots 01}a+\frac{15\cdots 69}{28\cdots 03}$, $\frac{20\cdots 02}{28\cdots 03}a^{26}-\frac{78\cdots 12}{94\cdots 01}a^{25}-\frac{80\cdots 82}{94\cdots 01}a^{24}+\frac{12\cdots 23}{28\cdots 03}a^{23}+\frac{11\cdots 34}{28\cdots 03}a^{22}+\frac{22\cdots 41}{28\cdots 03}a^{21}-\frac{93\cdots 14}{94\cdots 01}a^{20}-\frac{27\cdots 55}{28\cdots 03}a^{19}+\frac{39\cdots 73}{28\cdots 03}a^{18}+\frac{23\cdots 31}{94\cdots 01}a^{17}-\frac{30\cdots 87}{28\cdots 03}a^{16}-\frac{88\cdots 78}{28\cdots 03}a^{15}+\frac{32\cdots 40}{94\cdots 01}a^{14}+\frac{56\cdots 83}{28\cdots 03}a^{13}+\frac{23\cdots 06}{28\cdots 03}a^{12}-\frac{53\cdots 59}{94\cdots 01}a^{11}-\frac{24\cdots 16}{28\cdots 03}a^{10}+\frac{26\cdots 38}{94\cdots 01}a^{9}+\frac{49\cdots 52}{28\cdots 03}a^{8}+\frac{40\cdots 00}{28\cdots 03}a^{7}-\frac{10\cdots 07}{28\cdots 03}a^{6}-\frac{42\cdots 67}{28\cdots 03}a^{5}-\frac{11\cdots 27}{94\cdots 01}a^{4}-\frac{13\cdots 84}{28\cdots 03}a^{3}-\frac{98\cdots 01}{94\cdots 01}a^{2}-\frac{31\cdots 80}{28\cdots 03}a-\frac{12\cdots 17}{28\cdots 03}$, $\frac{17\cdots 54}{28\cdots 03}a^{26}-\frac{20\cdots 15}{28\cdots 03}a^{25}-\frac{68\cdots 13}{94\cdots 01}a^{24}+\frac{34\cdots 28}{94\cdots 01}a^{23}+\frac{96\cdots 95}{28\cdots 03}a^{22}+\frac{20\cdots 80}{28\cdots 03}a^{21}-\frac{24\cdots 23}{28\cdots 03}a^{20}-\frac{23\cdots 36}{28\cdots 03}a^{19}+\frac{11\cdots 16}{94\cdots 01}a^{18}+\frac{61\cdots 79}{28\cdots 03}a^{17}-\frac{26\cdots 38}{28\cdots 03}a^{16}-\frac{25\cdots 31}{94\cdots 01}a^{15}+\frac{82\cdots 68}{28\cdots 03}a^{14}+\frac{48\cdots 83}{28\cdots 03}a^{13}+\frac{68\cdots 90}{94\cdots 01}a^{12}-\frac{13\cdots 86}{28\cdots 03}a^{11}-\frac{70\cdots 03}{94\cdots 01}a^{10}+\frac{66\cdots 44}{28\cdots 03}a^{9}+\frac{42\cdots 55}{28\cdots 03}a^{8}+\frac{11\cdots 45}{94\cdots 01}a^{7}-\frac{30\cdots 99}{94\cdots 01}a^{6}-\frac{36\cdots 82}{28\cdots 03}a^{5}-\frac{29\cdots 32}{28\cdots 03}a^{4}-\frac{11\cdots 35}{28\cdots 03}a^{3}-\frac{85\cdots 91}{94\cdots 01}a^{2}-\frac{26\cdots 77}{28\cdots 03}a-\frac{10\cdots 32}{28\cdots 03}$, $\frac{44\cdots 35}{94\cdots 01}a^{26}-\frac{15\cdots 11}{28\cdots 03}a^{25}-\frac{15\cdots 08}{28\cdots 03}a^{24}+\frac{78\cdots 44}{28\cdots 03}a^{23}+\frac{71\cdots 80}{28\cdots 03}a^{22}+\frac{48\cdots 75}{94\cdots 01}a^{21}-\frac{17\cdots 06}{28\cdots 03}a^{20}-\frac{58\cdots 14}{94\cdots 01}a^{19}+\frac{84\cdots 58}{94\cdots 01}a^{18}+\frac{45\cdots 08}{28\cdots 03}a^{17}-\frac{65\cdots 03}{94\cdots 01}a^{16}-\frac{18\cdots 51}{94\cdots 01}a^{15}+\frac{61\cdots 94}{28\cdots 03}a^{14}+\frac{11\cdots 14}{94\cdots 01}a^{13}+\frac{15\cdots 64}{28\cdots 03}a^{12}-\frac{10\cdots 04}{28\cdots 03}a^{11}-\frac{51\cdots 31}{94\cdots 01}a^{10}+\frac{16\cdots 25}{94\cdots 01}a^{9}+\frac{31\cdots 83}{28\cdots 03}a^{8}+\frac{25\cdots 52}{28\cdots 03}a^{7}-\frac{67\cdots 88}{28\cdots 03}a^{6}-\frac{26\cdots 55}{28\cdots 03}a^{5}-\frac{21\cdots 32}{28\cdots 03}a^{4}-\frac{87\cdots 81}{28\cdots 03}a^{3}-\frac{63\cdots 59}{94\cdots 01}a^{2}-\frac{66\cdots 97}{94\cdots 01}a-\frac{26\cdots 25}{94\cdots 01}$, $\frac{21\cdots 07}{94\cdots 01}a^{26}+\frac{24\cdots 52}{94\cdots 01}a^{25}+\frac{25\cdots 38}{94\cdots 01}a^{24}-\frac{38\cdots 85}{28\cdots 03}a^{23}-\frac{11\cdots 82}{94\cdots 01}a^{22}-\frac{72\cdots 15}{28\cdots 03}a^{21}+\frac{87\cdots 85}{28\cdots 03}a^{20}+\frac{86\cdots 48}{28\cdots 03}a^{19}-\frac{12\cdots 01}{28\cdots 03}a^{18}-\frac{75\cdots 15}{94\cdots 01}a^{17}+\frac{96\cdots 41}{28\cdots 03}a^{16}+\frac{92\cdots 10}{94\cdots 01}a^{15}-\frac{30\cdots 14}{28\cdots 03}a^{14}-\frac{17\cdots 89}{28\cdots 03}a^{13}-\frac{74\cdots 31}{28\cdots 03}a^{12}+\frac{50\cdots 49}{28\cdots 03}a^{11}+\frac{76\cdots 49}{28\cdots 03}a^{10}-\frac{24\cdots 36}{28\cdots 03}a^{9}-\frac{51\cdots 96}{94\cdots 01}a^{8}-\frac{12\cdots 77}{28\cdots 03}a^{7}+\frac{10\cdots 97}{94\cdots 01}a^{6}+\frac{13\cdots 86}{28\cdots 03}a^{5}+\frac{10\cdots 65}{28\cdots 03}a^{4}+\frac{42\cdots 59}{28\cdots 03}a^{3}+\frac{93\cdots 47}{28\cdots 03}a^{2}+\frac{32\cdots 17}{94\cdots 01}a+\frac{38\cdots 57}{28\cdots 03}$, $\frac{70\cdots 24}{28\cdots 03}a^{26}+\frac{79\cdots 23}{28\cdots 03}a^{25}+\frac{81\cdots 13}{28\cdots 03}a^{24}-\frac{41\cdots 94}{28\cdots 03}a^{23}-\frac{12\cdots 42}{94\cdots 01}a^{22}-\frac{80\cdots 61}{28\cdots 03}a^{21}+\frac{95\cdots 61}{28\cdots 03}a^{20}+\frac{93\cdots 40}{28\cdots 03}a^{19}-\frac{45\cdots 49}{94\cdots 01}a^{18}-\frac{81\cdots 63}{94\cdots 01}a^{17}+\frac{10\cdots 02}{28\cdots 03}a^{16}+\frac{30\cdots 03}{28\cdots 03}a^{15}-\frac{32\cdots 90}{28\cdots 03}a^{14}-\frac{19\cdots 42}{28\cdots 03}a^{13}-\frac{27\cdots 07}{94\cdots 01}a^{12}+\frac{18\cdots 51}{94\cdots 01}a^{11}+\frac{83\cdots 15}{28\cdots 03}a^{10}-\frac{26\cdots 51}{28\cdots 03}a^{9}-\frac{16\cdots 51}{28\cdots 03}a^{8}-\frac{13\cdots 08}{28\cdots 03}a^{7}+\frac{35\cdots 78}{28\cdots 03}a^{6}+\frac{14\cdots 43}{28\cdots 03}a^{5}+\frac{11\cdots 57}{28\cdots 03}a^{4}+\frac{46\cdots 43}{28\cdots 03}a^{3}+\frac{10\cdots 11}{28\cdots 03}a^{2}+\frac{35\cdots 89}{94\cdots 01}a+\frac{42\cdots 35}{28\cdots 03}$, $\frac{11\cdots 25}{94\cdots 01}a^{26}-\frac{57\cdots 23}{94\cdots 01}a^{25}-\frac{38\cdots 10}{28\cdots 03}a^{24}+\frac{16\cdots 56}{28\cdots 03}a^{23}+\frac{16\cdots 14}{28\cdots 03}a^{22}-\frac{21\cdots 15}{94\cdots 01}a^{21}-\frac{42\cdots 79}{28\cdots 03}a^{20}+\frac{12\cdots 39}{28\cdots 03}a^{19}+\frac{67\cdots 10}{28\cdots 03}a^{18}-\frac{42\cdots 56}{94\cdots 01}a^{17}-\frac{71\cdots 37}{28\cdots 03}a^{16}+\frac{62\cdots 13}{28\cdots 03}a^{15}+\frac{16\cdots 95}{94\cdots 01}a^{14}+\frac{53\cdots 08}{28\cdots 03}a^{13}-\frac{21\cdots 49}{28\cdots 03}a^{12}-\frac{53\cdots 51}{94\cdots 01}a^{11}+\frac{15\cdots 95}{94\cdots 01}a^{10}+\frac{24\cdots 19}{94\cdots 01}a^{9}-\frac{88\cdots 34}{94\cdots 01}a^{8}-\frac{12\cdots 46}{28\cdots 03}a^{7}-\frac{71\cdots 37}{28\cdots 03}a^{6}+\frac{47\cdots 21}{28\cdots 03}a^{5}+\frac{81\cdots 54}{28\cdots 03}a^{4}+\frac{44\cdots 37}{28\cdots 03}a^{3}+\frac{11\cdots 13}{28\cdots 03}a^{2}+\frac{13\cdots 93}{28\cdots 03}a+\frac{56\cdots 85}{28\cdots 03}$, $\frac{18\cdots 02}{94\cdots 01}a^{26}-\frac{60\cdots 93}{28\cdots 03}a^{25}-\frac{65\cdots 57}{28\cdots 03}a^{24}+\frac{99\cdots 50}{94\cdots 01}a^{23}+\frac{30\cdots 68}{28\cdots 03}a^{22}+\frac{79\cdots 32}{28\cdots 03}a^{21}-\frac{76\cdots 31}{28\cdots 03}a^{20}-\frac{78\cdots 17}{28\cdots 03}a^{19}+\frac{10\cdots 12}{28\cdots 03}a^{18}+\frac{67\cdots 42}{94\cdots 01}a^{17}-\frac{83\cdots 62}{28\cdots 03}a^{16}-\frac{82\cdots 91}{94\cdots 01}a^{15}+\frac{85\cdots 48}{94\cdots 01}a^{14}+\frac{15\cdots 42}{28\cdots 03}a^{13}+\frac{23\cdots 78}{94\cdots 01}a^{12}-\frac{44\cdots 47}{28\cdots 03}a^{11}-\frac{68\cdots 08}{28\cdots 03}a^{10}+\frac{67\cdots 55}{94\cdots 01}a^{9}+\frac{45\cdots 57}{94\cdots 01}a^{8}+\frac{11\cdots 72}{28\cdots 03}a^{7}-\frac{28\cdots 91}{28\cdots 03}a^{6}-\frac{39\cdots 94}{94\cdots 01}a^{5}-\frac{94\cdots 17}{28\cdots 03}a^{4}-\frac{38\cdots 07}{28\cdots 03}a^{3}-\frac{84\cdots 63}{28\cdots 03}a^{2}-\frac{88\cdots 35}{28\cdots 03}a-\frac{34\cdots 44}{28\cdots 03}$, $\frac{14\cdots 50}{28\cdots 03}a^{26}-\frac{54\cdots 50}{94\cdots 01}a^{25}-\frac{16\cdots 44}{28\cdots 03}a^{24}+\frac{28\cdots 18}{94\cdots 01}a^{23}+\frac{78\cdots 13}{28\cdots 03}a^{22}+\frac{16\cdots 24}{28\cdots 03}a^{21}-\frac{19\cdots 95}{28\cdots 03}a^{20}-\frac{63\cdots 09}{94\cdots 01}a^{19}+\frac{27\cdots 37}{28\cdots 03}a^{18}+\frac{50\cdots 62}{28\cdots 03}a^{17}-\frac{21\cdots 80}{28\cdots 03}a^{16}-\frac{61\cdots 33}{28\cdots 03}a^{15}+\frac{66\cdots 30}{28\cdots 03}a^{14}+\frac{39\cdots 03}{28\cdots 03}a^{13}+\frac{16\cdots 35}{28\cdots 03}a^{12}-\frac{11\cdots 36}{28\cdots 03}a^{11}-\frac{56\cdots 93}{94\cdots 01}a^{10}+\frac{18\cdots 54}{94\cdots 01}a^{9}+\frac{34\cdots 13}{28\cdots 03}a^{8}+\frac{28\cdots 69}{28\cdots 03}a^{7}-\frac{24\cdots 70}{94\cdots 01}a^{6}-\frac{29\cdots 67}{28\cdots 03}a^{5}-\frac{78\cdots 32}{94\cdots 01}a^{4}-\frac{95\cdots 74}{28\cdots 03}a^{3}-\frac{69\cdots 11}{94\cdots 01}a^{2}-\frac{72\cdots 42}{94\cdots 01}a-\frac{86\cdots 50}{28\cdots 03}$, $\frac{22\cdots 90}{28\cdots 03}a^{26}-\frac{85\cdots 19}{94\cdots 01}a^{25}-\frac{26\cdots 04}{28\cdots 03}a^{24}+\frac{13\cdots 75}{28\cdots 03}a^{23}+\frac{12\cdots 46}{28\cdots 03}a^{22}+\frac{84\cdots 35}{94\cdots 01}a^{21}-\frac{30\cdots 47}{28\cdots 03}a^{20}-\frac{30\cdots 66}{28\cdots 03}a^{19}+\frac{43\cdots 89}{28\cdots 03}a^{18}+\frac{78\cdots 98}{28\cdots 03}a^{17}-\frac{33\cdots 91}{28\cdots 03}a^{16}-\frac{97\cdots 42}{28\cdots 03}a^{15}+\frac{10\cdots 82}{28\cdots 03}a^{14}+\frac{61\cdots 30}{28\cdots 03}a^{13}+\frac{25\cdots 55}{28\cdots 03}a^{12}-\frac{17\cdots 14}{28\cdots 03}a^{11}-\frac{26\cdots 61}{28\cdots 03}a^{10}+\frac{85\cdots 16}{28\cdots 03}a^{9}+\frac{18\cdots 05}{94\cdots 01}a^{8}+\frac{44\cdots 68}{28\cdots 03}a^{7}-\frac{11\cdots 98}{28\cdots 03}a^{6}-\frac{15\cdots 81}{94\cdots 01}a^{5}-\frac{36\cdots 93}{28\cdots 03}a^{4}-\frac{14\cdots 27}{28\cdots 03}a^{3}-\frac{32\cdots 03}{28\cdots 03}a^{2}-\frac{11\cdots 31}{94\cdots 01}a-\frac{44\cdots 91}{94\cdots 01}$, $\frac{29\cdots 71}{94\cdots 01}a^{26}+\frac{99\cdots 95}{28\cdots 03}a^{25}+\frac{10\cdots 35}{28\cdots 03}a^{24}-\frac{51\cdots 60}{28\cdots 03}a^{23}-\frac{15\cdots 44}{94\cdots 01}a^{22}-\frac{32\cdots 81}{94\cdots 01}a^{21}+\frac{11\cdots 80}{28\cdots 03}a^{20}+\frac{38\cdots 02}{94\cdots 01}a^{19}-\frac{56\cdots 85}{94\cdots 01}a^{18}-\frac{10\cdots 28}{94\cdots 01}a^{17}+\frac{13\cdots 06}{28\cdots 03}a^{16}+\frac{37\cdots 67}{28\cdots 03}a^{15}-\frac{40\cdots 47}{28\cdots 03}a^{14}-\frac{79\cdots 18}{94\cdots 01}a^{13}-\frac{10\cdots 32}{28\cdots 03}a^{12}+\frac{68\cdots 02}{28\cdots 03}a^{11}+\frac{34\cdots 08}{94\cdots 01}a^{10}-\frac{32\cdots 32}{28\cdots 03}a^{9}-\frac{69\cdots 78}{94\cdots 01}a^{8}-\frac{17\cdots 83}{28\cdots 03}a^{7}+\frac{14\cdots 86}{94\cdots 01}a^{6}+\frac{17\cdots 14}{28\cdots 03}a^{5}+\frac{47\cdots 88}{94\cdots 01}a^{4}+\frac{57\cdots 77}{28\cdots 03}a^{3}+\frac{12\cdots 58}{28\cdots 03}a^{2}+\frac{44\cdots 74}{94\cdots 01}a+\frac{52\cdots 95}{28\cdots 03}$
|
| |
| Regulator: | \( 407456336900721150 \) (assuming GRH) |
| |
| Unit signature rank: | \( 27 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{27}\cdot(2\pi)^{0}\cdot 407456336900721150 \cdot 3}{2\cdot\sqrt{124252631053426325344275434705435089635453266149328961}}\cr\approx \mathstrut & 0.232717702994696 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:C_{18}$ (as 27T47):
| A solvable group of order 162 |
| The 30 conjugacy class representatives for $C_3^2:C_{18}$ |
| Character table for $C_3^2:C_{18}$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\), deg 9 |
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 |
| Minimal sibling: | $ x^{18} - 126 x^{16} - 180 x^{15} + 6399 x^{14} + 17658 x^{13} - 155118 x^{12} - 660636 x^{11} + 1518795 x^{10} + 11367594 x^{9} + 3292002 x^{8} - 83929608 x^{7} - 145724094 x^{6} + 164166912 x^{5} + 665587530 x^{4} + 371026674 x^{3} - 658975581 x^{2} - 941917518 x - 345930423 $ |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.9.0.1}{9} }^{3}$ | R | ${\href{/padicField/5.9.0.1}{9} }^{3}$ | $18{,}\,{\href{/padicField/7.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/11.9.0.1}{9} }$ | ${\href{/padicField/13.9.0.1}{9} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{9}$ | $18{,}\,{\href{/padicField/23.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/29.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/31.9.0.1}{9} }$ | ${\href{/padicField/37.3.0.1}{3} }^{9}$ | $18{,}\,{\href{/padicField/41.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/43.9.0.1}{9} }$ | $18{,}\,{\href{/padicField/47.9.0.1}{9} }$ | ${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $27$ | $27$ | $1$ | $73$ | |||
|
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.2.1.1 | $x^{2} + 214$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |