Number field 27.27.2519933803975014472273213858760468555951745833259300216067364409.3

• Label: 27.27.251...409.3
• Degree: $27$
• Signature: $(27, 0)$
• Discriminant: $2.520\times 10^{63}$
• Root discriminant: \(222.95\)
• Ramified primes: $3,37$
• Class number: $3$ (GRH)
• Class group: [3] (GRH)
• Galois group: $C_3.\He_3$ (as 27T20)

Normalized defining polynomial

x^27 - 333*x^25 - 333*x^24 + 35298*x^23 + 53946*x^22 - 1819068*x^21 - 3588075*x^20 + 52704576*x^19 + 127734434*x^18 - 907552539*x^17 - 2673287370*x^16 + 9278172984*x^15 + 34236633762*x^14 - 51653439189*x^13 - 270237162441*x^12 + 93428282529*x^11 + 1280114382555*x^10 + 527095739526*x^9 - 3356233487919*x^8 - 3330019648665*x^7 + 3845500135071*x^6 + 6646340649357*x^5 + 111023372826*x^4 - 4196847578151*x^3 - 2127671869662*x^2 - 113904515466*x + 74779074553

Invariants

Degree:		$27$
Signature:		$(27, 0)$
Discriminant:		\(2519933803975014472273213858760468555951745833259300216067364409\) \(\medspace = 3^{54}\cdot 37^{24}\)
Root discriminant:		\(222.95\)
Galois root discriminant:		$3^{58/27}37^{8/9}\approx 262.35190848297987$
Ramified primes:		\(3\), \(37\)
Discriminant root field:		\(\Q\)
$\Aut(K/\Q)$:		$C_3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{37}a^{9}$, $\frac{1}{37}a^{10}$, $\frac{1}{37}a^{11}$, $\frac{1}{37}a^{12}$, $\frac{1}{37}a^{13}$, $\frac{1}{37}a^{14}$, $\frac{1}{37}a^{15}$, $\frac{1}{37}a^{16}$, $\frac{1}{37}a^{17}$, $\frac{1}{1369}a^{18}$, $\frac{1}{1369}a^{19}$, $\frac{1}{1369}a^{20}$, $\frac{1}{1369}a^{21}$, $\frac{1}{1369}a^{22}$, $\frac{1}{1369}a^{23}$, $\frac{1}{1369}a^{24}$, $\frac{1}{1369}a^{25}$, $\frac{1}{14\cdots 93}a^{26}+\frac{26\cdots 68}{14\cdots 93}a^{25}+\frac{25\cdots 38}{14\cdots 93}a^{24}-\frac{63\cdots 32}{14\cdots 93}a^{23}+\frac{18\cdots 64}{14\cdots 93}a^{22}+\frac{30\cdots 11}{14\cdots 93}a^{21}+\frac{49\cdots 88}{14\cdots 93}a^{20}+\frac{16\cdots 91}{14\cdots 93}a^{19}+\frac{39\cdots 86}{14\cdots 93}a^{18}-\frac{38\cdots 25}{39\cdots 89}a^{17}+\frac{12\cdots 04}{39\cdots 89}a^{16}-\frac{30\cdots 74}{39\cdots 89}a^{15}+\frac{32\cdots 48}{39\cdots 89}a^{14}-\frac{16\cdots 63}{39\cdots 89}a^{13}-\frac{43\cdots 13}{39\cdots 89}a^{12}-\frac{15\cdots 17}{10\cdots 97}a^{11}+\frac{36\cdots 74}{39\cdots 89}a^{10}+\frac{39\cdots 00}{39\cdots 89}a^{9}+\frac{41\cdots 81}{10\cdots 97}a^{8}+\frac{39\cdots 75}{10\cdots 97}a^{7}-\frac{33\cdots 18}{10\cdots 97}a^{6}-\frac{47\cdots 21}{10\cdots 97}a^{5}+\frac{29\cdots 12}{10\cdots 97}a^{4}+\frac{41\cdots 77}{10\cdots 97}a^{3}-\frac{18\cdots 74}{10\cdots 97}a^{2}+\frac{48\cdots 30}{10\cdots 97}a+\frac{14\cdots 82}{10\cdots 97}$

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:		$C_{6}\times C_{2}$, which has order $12$ (assuming GRH)

Unit group

Rank:		$26$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{16\cdots 27}{95\cdots 01}a^{26}+\frac{32\cdots 36}{95\cdots 01}a^{25}+\frac{55\cdots 09}{95\cdots 01}a^{24}-\frac{50\cdots 44}{95\cdots 01}a^{23}-\frac{58\cdots 56}{95\cdots 01}a^{22}+\frac{21\cdots 23}{95\cdots 01}a^{21}+\frac{81\cdots 19}{25\cdots 73}a^{20}+\frac{60\cdots 39}{25\cdots 73}a^{19}-\frac{23\cdots 77}{25\cdots 73}a^{18}-\frac{11\cdots 67}{25\cdots 73}a^{17}+\frac{43\cdots 62}{25\cdots 73}a^{16}+\frac{37\cdots 76}{25\cdots 73}a^{15}-\frac{49\cdots 81}{25\cdots 73}a^{14}-\frac{60\cdots 93}{25\cdots 73}a^{13}+\frac{35\cdots 99}{25\cdots 73}a^{12}+\frac{14\cdots 64}{69\cdots 29}a^{11}-\frac{40\cdots 30}{69\cdots 29}a^{10}-\frac{79\cdots 78}{69\cdots 29}a^{9}+\frac{88\cdots 73}{69\cdots 29}a^{8}+\frac{23\cdots 55}{69\cdots 29}a^{7}-\frac{55\cdots 04}{69\cdots 29}a^{6}-\frac{36\cdots 70}{69\cdots 29}a^{5}-\frac{11\cdots 22}{69\cdots 29}a^{4}+\frac{19\cdots 86}{69\cdots 29}a^{3}+\frac{12\cdots 45}{69\cdots 29}a^{2}+\frac{97\cdots 34}{69\cdots 29}a-\frac{47\cdots 12}{69\cdots 29}$, $\frac{44\cdots 02}{95\cdots 01}a^{26}+\frac{94\cdots 62}{95\cdots 01}a^{25}+\frac{14\cdots 58}{95\cdots 01}a^{24}-\frac{16\cdots 29}{95\cdots 01}a^{23}-\frac{15\cdots 34}{95\cdots 01}a^{22}+\frac{83\cdots 98}{95\cdots 01}a^{21}+\frac{21\cdots 35}{25\cdots 73}a^{20}-\frac{18\cdots 69}{25\cdots 73}a^{19}-\frac{63\cdots 36}{25\cdots 73}a^{18}-\frac{21\cdots 80}{25\cdots 73}a^{17}+\frac{11\cdots 38}{25\cdots 73}a^{16}+\frac{84\cdots 54}{25\cdots 73}a^{15}-\frac{13\cdots 31}{25\cdots 73}a^{14}-\frac{14\cdots 88}{25\cdots 73}a^{13}+\frac{92\cdots 32}{25\cdots 73}a^{12}+\frac{36\cdots 34}{69\cdots 29}a^{11}-\frac{10\cdots 60}{69\cdots 29}a^{10}-\frac{19\cdots 74}{69\cdots 29}a^{9}+\frac{23\cdots 45}{69\cdots 29}a^{8}+\frac{60\cdots 70}{69\cdots 29}a^{7}-\frac{16\cdots 83}{69\cdots 29}a^{6}-\frac{91\cdots 97}{69\cdots 29}a^{5}-\frac{25\cdots 22}{69\cdots 29}a^{4}+\frac{50\cdots 89}{69\cdots 29}a^{3}+\frac{32\cdots 19}{69\cdots 29}a^{2}+\frac{23\cdots 06}{69\cdots 29}a-\frac{11\cdots 69}{69\cdots 29}$, $\frac{12\cdots 37}{39\cdots 63}a^{26}+\frac{24\cdots 28}{39\cdots 63}a^{25}+\frac{40\cdots 56}{39\cdots 63}a^{24}-\frac{40\cdots 21}{39\cdots 63}a^{23}-\frac{42\cdots 88}{39\cdots 63}a^{22}+\frac{19\cdots 86}{39\cdots 63}a^{21}+\frac{22\cdots 30}{39\cdots 63}a^{20}-\frac{67\cdots 46}{39\cdots 63}a^{19}-\frac{65\cdots 24}{39\cdots 63}a^{18}-\frac{73\cdots 58}{10\cdots 99}a^{17}+\frac{86\cdots 77}{28\cdots 27}a^{16}+\frac{25\cdots 11}{10\cdots 99}a^{15}-\frac{36\cdots 74}{10\cdots 99}a^{14}-\frac{41\cdots 83}{10\cdots 99}a^{13}+\frac{25\cdots 82}{10\cdots 99}a^{12}+\frac{38\cdots 63}{10\cdots 99}a^{11}-\frac{10\cdots 00}{10\cdots 99}a^{10}-\frac{20\cdots 38}{10\cdots 99}a^{9}+\frac{65\cdots 69}{28\cdots 27}a^{8}+\frac{17\cdots 21}{28\cdots 27}a^{7}-\frac{43\cdots 76}{28\cdots 27}a^{6}-\frac{26\cdots 00}{28\cdots 27}a^{5}-\frac{76\cdots 66}{28\cdots 27}a^{4}+\frac{14\cdots 90}{28\cdots 27}a^{3}+\frac{92\cdots 41}{28\cdots 27}a^{2}+\frac{67\cdots 61}{28\cdots 27}a-\frac{33\cdots 98}{28\cdots 27}$, $\frac{30\cdots 25}{39\cdots 63}a^{26}+\frac{47\cdots 73}{39\cdots 63}a^{25}+\frac{99\cdots 11}{39\cdots 63}a^{24}-\frac{58\cdots 42}{39\cdots 63}a^{23}-\frac{10\cdots 75}{39\cdots 63}a^{22}+\frac{60\cdots 32}{39\cdots 63}a^{21}+\frac{55\cdots 27}{39\cdots 63}a^{20}+\frac{20\cdots 99}{39\cdots 63}a^{19}-\frac{16\cdots 47}{39\cdots 63}a^{18}-\frac{33\cdots 84}{10\cdots 99}a^{17}+\frac{80\cdots 69}{10\cdots 99}a^{16}+\frac{88\cdots 16}{10\cdots 99}a^{15}-\frac{91\cdots 13}{10\cdots 99}a^{14}-\frac{13\cdots 74}{10\cdots 99}a^{13}+\frac{65\cdots 42}{10\cdots 99}a^{12}+\frac{11\cdots 75}{10\cdots 99}a^{11}-\frac{27\cdots 66}{10\cdots 99}a^{10}-\frac{60\cdots 50}{10\cdots 99}a^{9}+\frac{16\cdots 34}{28\cdots 27}a^{8}+\frac{48\cdots 39}{28\cdots 27}a^{7}-\frac{83\cdots 30}{28\cdots 27}a^{6}-\frac{72\cdots 22}{28\cdots 27}a^{5}-\frac{24\cdots 35}{28\cdots 27}a^{4}+\frac{39\cdots 82}{28\cdots 27}a^{3}+\frac{26\cdots 76}{28\cdots 27}a^{2}+\frac{20\cdots 14}{28\cdots 27}a-\frac{96\cdots 31}{28\cdots 27}$, $\frac{25\cdots 92}{39\cdots 63}a^{26}+\frac{56\cdots 05}{39\cdots 63}a^{25}+\frac{83\cdots 70}{39\cdots 63}a^{24}-\frac{99\cdots 47}{39\cdots 63}a^{23}-\frac{87\cdots 00}{39\cdots 63}a^{22}+\frac{56\cdots 32}{39\cdots 63}a^{21}+\frac{44\cdots 70}{39\cdots 63}a^{20}-\frac{83\cdots 59}{39\cdots 63}a^{19}-\frac{13\cdots 59}{39\cdots 63}a^{18}-\frac{86\cdots 88}{10\cdots 99}a^{17}+\frac{64\cdots 62}{10\cdots 99}a^{16}+\frac{41\cdots 37}{10\cdots 99}a^{15}-\frac{72\cdots 77}{10\cdots 99}a^{14}-\frac{73\cdots 10}{10\cdots 99}a^{13}+\frac{51\cdots 25}{10\cdots 99}a^{12}+\frac{70\cdots 10}{10\cdots 99}a^{11}-\frac{22\cdots 85}{10\cdots 99}a^{10}-\frac{38\cdots 04}{10\cdots 99}a^{9}+\frac{13\cdots 60}{28\cdots 27}a^{8}+\frac{32\cdots 18}{28\cdots 27}a^{7}-\frac{97\cdots 39}{28\cdots 27}a^{6}-\frac{49\cdots 62}{28\cdots 27}a^{5}-\frac{13\cdots 12}{28\cdots 27}a^{4}+\frac{27\cdots 27}{28\cdots 27}a^{3}+\frac{17\cdots 65}{28\cdots 27}a^{2}+\frac{12\cdots 80}{28\cdots 27}a-\frac{62\cdots 17}{28\cdots 27}$, $\frac{52\cdots 14}{39\cdots 63}a^{26}-\frac{13\cdots 55}{39\cdots 63}a^{25}-\frac{17\cdots 00}{39\cdots 63}a^{24}+\frac{26\cdots 10}{39\cdots 63}a^{23}+\frac{18\cdots 71}{39\cdots 63}a^{22}-\frac{17\cdots 70}{39\cdots 63}a^{21}-\frac{92\cdots 92}{39\cdots 63}a^{20}+\frac{45\cdots 56}{39\cdots 63}a^{19}+\frac{26\cdots 11}{39\cdots 63}a^{18}-\frac{28\cdots 09}{10\cdots 99}a^{17}-\frac{13\cdots 47}{10\cdots 99}a^{16}-\frac{49\cdots 23}{10\cdots 99}a^{15}+\frac{14\cdots 76}{10\cdots 99}a^{14}+\frac{11\cdots 75}{10\cdots 99}a^{13}-\frac{10\cdots 30}{10\cdots 99}a^{12}-\frac{11\cdots 96}{10\cdots 99}a^{11}+\frac{44\cdots 74}{10\cdots 99}a^{10}+\frac{68\cdots 47}{10\cdots 99}a^{9}-\frac{28\cdots 42}{28\cdots 27}a^{8}-\frac{58\cdots 64}{28\cdots 27}a^{7}+\frac{23\cdots 23}{28\cdots 27}a^{6}+\frac{90\cdots 71}{28\cdots 27}a^{5}+\frac{20\cdots 31}{28\cdots 27}a^{4}-\frac{50\cdots 36}{28\cdots 27}a^{3}-\frac{30\cdots 95}{28\cdots 27}a^{2}-\frac{20\cdots 45}{28\cdots 27}a+\frac{11\cdots 72}{28\cdots 27}$, $\frac{53\cdots 25}{39\cdots 63}a^{26}+\frac{10\cdots 17}{39\cdots 63}a^{25}+\frac{17\cdots 42}{39\cdots 63}a^{24}-\frac{17\cdots 28}{39\cdots 63}a^{23}-\frac{18\cdots 82}{39\cdots 63}a^{22}+\frac{81\cdots 79}{39\cdots 63}a^{21}+\frac{96\cdots 46}{39\cdots 63}a^{20}+\frac{12\cdots 05}{39\cdots 63}a^{19}-\frac{28\cdots 32}{39\cdots 63}a^{18}-\frac{32\cdots 55}{10\cdots 99}a^{17}+\frac{13\cdots 05}{10\cdots 99}a^{16}+\frac{11\cdots 63}{10\cdots 99}a^{15}-\frac{15\cdots 31}{10\cdots 99}a^{14}-\frac{18\cdots 37}{10\cdots 99}a^{13}+\frac{11\cdots 90}{10\cdots 99}a^{12}+\frac{17\cdots 72}{10\cdots 99}a^{11}-\frac{47\cdots 78}{10\cdots 99}a^{10}-\frac{91\cdots 63}{10\cdots 99}a^{9}+\frac{28\cdots 36}{28\cdots 27}a^{8}+\frac{75\cdots 03}{28\cdots 27}a^{7}-\frac{18\cdots 68}{28\cdots 27}a^{6}-\frac{11\cdots 43}{28\cdots 27}a^{5}-\frac{33\cdots 72}{28\cdots 27}a^{4}+\frac{62\cdots 93}{28\cdots 27}a^{3}+\frac{40\cdots 94}{28\cdots 27}a^{2}+\frac{29\cdots 08}{28\cdots 27}a-\frac{14\cdots 52}{28\cdots 27}$, $\frac{15\cdots 41}{39\cdots 63}a^{26}-\frac{33\cdots 64}{39\cdots 63}a^{25}-\frac{49\cdots 10}{39\cdots 63}a^{24}+\frac{60\cdots 56}{39\cdots 63}a^{23}+\frac{51\cdots 46}{39\cdots 63}a^{22}-\frac{34\cdots 28}{39\cdots 63}a^{21}-\frac{26\cdots 65}{39\cdots 63}a^{20}+\frac{54\cdots 21}{39\cdots 63}a^{19}+\frac{78\cdots 94}{39\cdots 63}a^{18}+\frac{47\cdots 65}{10\cdots 99}a^{17}-\frac{38\cdots 08}{10\cdots 99}a^{16}-\frac{23\cdots 49}{10\cdots 99}a^{15}+\frac{43\cdots 45}{10\cdots 99}a^{14}+\frac{42\cdots 37}{10\cdots 99}a^{13}-\frac{30\cdots 62}{10\cdots 99}a^{12}-\frac{41\cdots 37}{10\cdots 99}a^{11}+\frac{13\cdots 97}{10\cdots 99}a^{10}+\frac{22\cdots 77}{10\cdots 99}a^{9}-\frac{80\cdots 26}{28\cdots 27}a^{8}-\frac{18\cdots 03}{28\cdots 27}a^{7}+\frac{58\cdots 91}{28\cdots 27}a^{6}+\frac{29\cdots 77}{28\cdots 27}a^{5}+\frac{76\cdots 50}{28\cdots 27}a^{4}-\frac{16\cdots 91}{28\cdots 27}a^{3}-\frac{10\cdots 99}{28\cdots 27}a^{2}-\frac{71\cdots 21}{28\cdots 27}a+\frac{36\cdots 69}{28\cdots 27}$, $\frac{52\cdots 64}{14\cdots 93}a^{26}-\frac{10\cdots 66}{14\cdots 93}a^{25}-\frac{17\cdots 25}{14\cdots 93}a^{24}+\frac{16\cdots 08}{14\cdots 93}a^{23}+\frac{18\cdots 79}{14\cdots 93}a^{22}-\frac{78\cdots 51}{14\cdots 93}a^{21}-\frac{94\cdots 64}{14\cdots 93}a^{20}-\frac{20\cdots 90}{14\cdots 93}a^{19}+\frac{27\cdots 31}{14\cdots 93}a^{18}+\frac{32\cdots 49}{39\cdots 89}a^{17}-\frac{13\cdots 66}{39\cdots 89}a^{16}-\frac{11\cdots 15}{39\cdots 89}a^{15}+\frac{15\cdots 93}{39\cdots 89}a^{14}+\frac{18\cdots 51}{39\cdots 89}a^{13}-\frac{11\cdots 91}{39\cdots 89}a^{12}-\frac{45\cdots 28}{10\cdots 97}a^{11}+\frac{46\cdots 33}{39\cdots 89}a^{10}+\frac{90\cdots 31}{39\cdots 89}a^{9}-\frac{28\cdots 14}{10\cdots 97}a^{8}-\frac{74\cdots 42}{10\cdots 97}a^{7}+\frac{18\cdots 64}{10\cdots 97}a^{6}+\frac{11\cdots 30}{10\cdots 97}a^{5}+\frac{33\cdots 43}{10\cdots 97}a^{4}-\frac{61\cdots 63}{10\cdots 97}a^{3}-\frac{40\cdots 71}{10\cdots 97}a^{2}-\frac{29\cdots 53}{10\cdots 97}a+\frac{14\cdots 81}{10\cdots 97}$, $\frac{14\cdots 58}{14\cdots 93}a^{26}-\frac{72\cdots 83}{14\cdots 93}a^{25}-\frac{47\cdots 24}{14\cdots 93}a^{24}-\frac{21\cdots 71}{14\cdots 93}a^{23}+\frac{51\cdots 25}{14\cdots 93}a^{22}+\frac{47\cdots 13}{14\cdots 93}a^{21}-\frac{27\cdots 25}{14\cdots 93}a^{20}-\frac{34\cdots 90}{14\cdots 93}a^{19}+\frac{81\cdots 04}{14\cdots 93}a^{18}+\frac{34\cdots 60}{39\cdots 89}a^{17}-\frac{40\cdots 51}{39\cdots 89}a^{16}-\frac{75\cdots 49}{39\cdots 89}a^{15}+\frac{46\cdots 57}{39\cdots 89}a^{14}+\frac{99\cdots 12}{39\cdots 89}a^{13}-\frac{33\cdots 47}{39\cdots 89}a^{12}-\frac{81\cdots 49}{39\cdots 89}a^{11}+\frac{13\cdots 51}{39\cdots 89}a^{10}+\frac{40\cdots 60}{39\cdots 89}a^{9}-\frac{71\cdots 95}{10\cdots 97}a^{8}-\frac{31\cdots 52}{10\cdots 97}a^{7}+\frac{97\cdots 36}{10\cdots 97}a^{6}+\frac{46\cdots 61}{10\cdots 97}a^{5}+\frac{19\cdots 83}{10\cdots 97}a^{4}-\frac{24\cdots 51}{10\cdots 97}a^{3}-\frac{18\cdots 19}{10\cdots 97}a^{2}-\frac{16\cdots 06}{10\cdots 97}a+\frac{69\cdots 82}{10\cdots 97}$, $\frac{47\cdots 86}{14\cdots 93}a^{26}-\frac{10\cdots 59}{14\cdots 93}a^{25}-\frac{42\cdots 88}{39\cdots 89}a^{24}+\frac{18\cdots 78}{14\cdots 93}a^{23}+\frac{16\cdots 00}{14\cdots 93}a^{22}-\frac{10\cdots 18}{14\cdots 93}a^{21}-\frac{84\cdots 95}{14\cdots 93}a^{20}+\frac{16\cdots 79}{14\cdots 93}a^{19}+\frac{24\cdots 33}{14\cdots 93}a^{18}+\frac{15\cdots 62}{39\cdots 89}a^{17}-\frac{12\cdots 77}{39\cdots 89}a^{16}-\frac{76\cdots 10}{39\cdots 89}a^{15}+\frac{13\cdots 63}{39\cdots 89}a^{14}+\frac{13\cdots 88}{39\cdots 89}a^{13}-\frac{97\cdots 39}{39\cdots 89}a^{12}-\frac{13\cdots 41}{39\cdots 89}a^{11}+\frac{41\cdots 12}{39\cdots 89}a^{10}+\frac{72\cdots 81}{39\cdots 89}a^{9}-\frac{25\cdots 58}{10\cdots 97}a^{8}-\frac{60\cdots 89}{10\cdots 97}a^{7}+\frac{18\cdots 85}{10\cdots 97}a^{6}+\frac{93\cdots 30}{10\cdots 97}a^{5}+\frac{24\cdots 88}{10\cdots 97}a^{4}-\frac{51\cdots 46}{10\cdots 97}a^{3}-\frac{32\cdots 42}{10\cdots 97}a^{2}-\frac{23\cdots 90}{10\cdots 97}a+\frac{11\cdots 36}{10\cdots 97}$, $\frac{25\cdots 73}{14\cdots 93}a^{26}-\frac{56\cdots 13}{14\cdots 93}a^{25}-\frac{83\cdots 72}{14\cdots 93}a^{24}+\frac{10\cdots 21}{14\cdots 93}a^{23}+\frac{87\cdots 75}{14\cdots 93}a^{22}-\frac{58\cdots 22}{14\cdots 93}a^{21}-\frac{44\cdots 13}{14\cdots 93}a^{20}+\frac{93\cdots 91}{14\cdots 93}a^{19}+\frac{13\cdots 30}{14\cdots 93}a^{18}+\frac{77\cdots 45}{39\cdots 89}a^{17}-\frac{64\cdots 24}{39\cdots 89}a^{16}-\frac{39\cdots 93}{39\cdots 89}a^{15}+\frac{73\cdots 10}{39\cdots 89}a^{14}+\frac{71\cdots 77}{39\cdots 89}a^{13}-\frac{52\cdots 34}{39\cdots 89}a^{12}-\frac{68\cdots 74}{39\cdots 89}a^{11}+\frac{22\cdots 14}{39\cdots 89}a^{10}+\frac{37\cdots 20}{39\cdots 89}a^{9}-\frac{14\cdots 84}{10\cdots 97}a^{8}-\frac{31\cdots 56}{10\cdots 97}a^{7}+\frac{11\cdots 68}{10\cdots 97}a^{6}+\frac{48\cdots 34}{10\cdots 97}a^{5}+\frac{10\cdots 93}{10\cdots 97}a^{4}-\frac{26\cdots 93}{10\cdots 97}a^{3}-\frac{15\cdots 99}{10\cdots 97}a^{2}-\frac{11\cdots 90}{10\cdots 97}a+\frac{47\cdots 48}{10\cdots 97}$, $\frac{25\cdots 95}{14\cdots 93}a^{26}-\frac{51\cdots 15}{14\cdots 93}a^{25}-\frac{83\cdots 87}{14\cdots 93}a^{24}+\frac{85\cdots 54}{14\cdots 93}a^{23}+\frac{87\cdots 29}{14\cdots 93}a^{22}-\frac{42\cdots 52}{14\cdots 93}a^{21}-\frac{45\cdots 41}{14\cdots 93}a^{20}+\frac{13\cdots 48}{14\cdots 93}a^{19}+\frac{13\cdots 82}{14\cdots 93}a^{18}+\frac{14\cdots 35}{39\cdots 89}a^{17}-\frac{65\cdots 51}{39\cdots 89}a^{16}-\frac{50\cdots 10}{39\cdots 89}a^{15}+\frac{73\cdots 43}{39\cdots 89}a^{14}+\frac{83\cdots 14}{39\cdots 89}a^{13}-\frac{52\cdots 76}{39\cdots 89}a^{12}-\frac{78\cdots 03}{39\cdots 89}a^{11}+\frac{60\cdots 41}{10\cdots 97}a^{10}+\frac{42\cdots 68}{39\cdots 89}a^{9}-\frac{13\cdots 21}{10\cdots 97}a^{8}-\frac{34\cdots 49}{10\cdots 97}a^{7}+\frac{90\cdots 74}{10\cdots 97}a^{6}+\frac{52\cdots 94}{10\cdots 97}a^{5}+\frac{15\cdots 58}{10\cdots 97}a^{4}-\frac{28\cdots 12}{10\cdots 97}a^{3}-\frac{18\cdots 90}{10\cdots 97}a^{2}-\frac{13\cdots 01}{10\cdots 97}a+\frac{67\cdots 96}{10\cdots 97}$, $\frac{39\cdots 77}{14\cdots 93}a^{26}+\frac{61\cdots 63}{14\cdots 93}a^{25}+\frac{13\cdots 18}{14\cdots 93}a^{24}-\frac{71\cdots 17}{14\cdots 93}a^{23}-\frac{13\cdots 80}{14\cdots 93}a^{22}+\frac{11\cdots 07}{14\cdots 93}a^{21}+\frac{72\cdots 65}{14\cdots 93}a^{20}+\frac{30\cdots 55}{14\cdots 93}a^{19}-\frac{21\cdots 39}{14\cdots 93}a^{18}-\frac{47\cdots 33}{39\cdots 89}a^{17}+\frac{10\cdots 84}{39\cdots 89}a^{16}+\frac{12\cdots 26}{39\cdots 89}a^{15}-\frac{11\cdots 94}{39\cdots 89}a^{14}-\frac{18\cdots 88}{39\cdots 89}a^{13}+\frac{84\cdots 04}{39\cdots 89}a^{12}+\frac{16\cdots 82}{39\cdots 89}a^{11}-\frac{35\cdots 98}{39\cdots 89}a^{10}-\frac{83\cdots 03}{39\cdots 89}a^{9}+\frac{19\cdots 74}{10\cdots 97}a^{8}+\frac{67\cdots 51}{10\cdots 97}a^{7}-\frac{72\cdots 25}{10\cdots 97}a^{6}-\frac{10\cdots 38}{10\cdots 97}a^{5}-\frac{36\cdots 00}{10\cdots 97}a^{4}+\frac{54\cdots 87}{10\cdots 97}a^{3}+\frac{37\cdots 66}{10\cdots 97}a^{2}+\frac{30\cdots 73}{10\cdots 97}a-\frac{13\cdots 70}{10\cdots 97}$, $\frac{23\cdots 40}{14\cdots 93}a^{26}+\frac{50\cdots 75}{14\cdots 93}a^{25}+\frac{76\cdots 70}{14\cdots 93}a^{24}-\frac{87\cdots 45}{14\cdots 93}a^{23}-\frac{80\cdots 35}{14\cdots 93}a^{22}+\frac{47\cdots 17}{14\cdots 93}a^{21}+\frac{41\cdots 56}{14\cdots 93}a^{20}-\frac{54\cdots 09}{14\cdots 93}a^{19}-\frac{33\cdots 31}{39\cdots 89}a^{18}-\frac{97\cdots 71}{39\cdots 89}a^{17}+\frac{59\cdots 30}{39\cdots 89}a^{16}+\frac{11\cdots 15}{10\cdots 97}a^{15}-\frac{67\cdots 76}{39\cdots 89}a^{14}-\frac{71\cdots 58}{39\cdots 89}a^{13}+\frac{12\cdots 69}{10\cdots 97}a^{12}+\frac{67\cdots 25}{39\cdots 89}a^{11}-\frac{20\cdots 46}{39\cdots 89}a^{10}-\frac{37\cdots 84}{39\cdots 89}a^{9}+\frac{12\cdots 94}{10\cdots 97}a^{8}+\frac{30\cdots 63}{10\cdots 97}a^{7}-\frac{88\cdots 94}{10\cdots 97}a^{6}-\frac{46\cdots 78}{10\cdots 97}a^{5}-\frac{12\cdots 49}{10\cdots 97}a^{4}+\frac{25\cdots 18}{10\cdots 97}a^{3}+\frac{16\cdots 15}{10\cdots 97}a^{2}+\frac{11\cdots 93}{10\cdots 97}a-\frac{59\cdots 41}{10\cdots 97}$, $\frac{52\cdots 75}{14\cdots 93}a^{26}-\frac{10\cdots 12}{14\cdots 93}a^{25}-\frac{17\cdots 45}{14\cdots 93}a^{24}+\frac{17\cdots 53}{14\cdots 93}a^{23}+\frac{18\cdots 70}{14\cdots 93}a^{22}-\frac{85\cdots 22}{14\cdots 93}a^{21}-\frac{94\cdots 05}{14\cdots 93}a^{20}+\frac{15\cdots 17}{14\cdots 93}a^{19}+\frac{75\cdots 20}{39\cdots 89}a^{18}+\frac{30\cdots 65}{39\cdots 89}a^{17}-\frac{13\cdots 30}{39\cdots 89}a^{16}-\frac{10\cdots 10}{39\cdots 89}a^{15}+\frac{15\cdots 02}{39\cdots 89}a^{14}+\frac{17\cdots 71}{39\cdots 89}a^{13}-\frac{29\cdots 68}{10\cdots 97}a^{12}-\frac{16\cdots 91}{39\cdots 89}a^{11}+\frac{46\cdots 17}{39\cdots 89}a^{10}+\frac{88\cdots 28}{39\cdots 89}a^{9}-\frac{28\cdots 56}{10\cdots 97}a^{8}-\frac{72\cdots 85}{10\cdots 97}a^{7}+\frac{18\cdots 68}{10\cdots 97}a^{6}+\frac{11\cdots 87}{10\cdots 97}a^{5}+\frac{31\cdots 98}{10\cdots 97}a^{4}-\frac{60\cdots 23}{10\cdots 97}a^{3}-\frac{38\cdots 67}{10\cdots 97}a^{2}-\frac{28\cdots 89}{10\cdots 97}a+\frac{14\cdots 89}{10\cdots 97}$, $\frac{18\cdots 32}{14\cdots 93}a^{26}-\frac{39\cdots 79}{14\cdots 93}a^{25}-\frac{60\cdots 79}{14\cdots 93}a^{24}+\frac{67\cdots 36}{14\cdots 93}a^{23}+\frac{63\cdots 21}{14\cdots 93}a^{22}-\frac{35\cdots 68}{14\cdots 93}a^{21}-\frac{32\cdots 88}{14\cdots 93}a^{20}+\frac{33\cdots 74}{14\cdots 93}a^{19}+\frac{96\cdots 29}{14\cdots 93}a^{18}+\frac{84\cdots 20}{39\cdots 89}a^{17}-\frac{46\cdots 18}{39\cdots 89}a^{16}-\frac{33\cdots 62}{39\cdots 89}a^{15}+\frac{53\cdots 84}{39\cdots 89}a^{14}+\frac{57\cdots 11}{39\cdots 89}a^{13}-\frac{37\cdots 49}{39\cdots 89}a^{12}-\frac{54\cdots 22}{39\cdots 89}a^{11}+\frac{16\cdots 98}{39\cdots 89}a^{10}+\frac{29\cdots 50}{39\cdots 89}a^{9}-\frac{97\cdots 47}{10\cdots 97}a^{8}-\frac{24\cdots 13}{10\cdots 97}a^{7}+\frac{66\cdots 77}{10\cdots 97}a^{6}+\frac{37\cdots 60}{10\cdots 97}a^{5}+\frac{10\cdots 13}{10\cdots 97}a^{4}-\frac{20\cdots 56}{10\cdots 97}a^{3}-\frac{13\cdots 35}{10\cdots 97}a^{2}-\frac{94\cdots 68}{10\cdots 97}a+\frac{47\cdots 63}{10\cdots 97}$, $\frac{27\cdots 98}{14\cdots 93}a^{26}+\frac{52\cdots 79}{14\cdots 93}a^{25}+\frac{90\cdots 05}{14\cdots 93}a^{24}-\frac{81\cdots 65}{14\cdots 93}a^{23}-\frac{95\cdots 10}{14\cdots 93}a^{22}+\frac{33\cdots 53}{14\cdots 93}a^{21}+\frac{49\cdots 30}{14\cdots 93}a^{20}+\frac{43\cdots 05}{14\cdots 93}a^{19}-\frac{14\cdots 15}{14\cdots 93}a^{18}-\frac{19\cdots 81}{39\cdots 89}a^{17}+\frac{71\cdots 32}{39\cdots 89}a^{16}+\frac{62\cdots 80}{39\cdots 89}a^{15}-\frac{80\cdots 29}{39\cdots 89}a^{14}-\frac{99\cdots 41}{39\cdots 89}a^{13}+\frac{57\cdots 04}{39\cdots 89}a^{12}+\frac{90\cdots 12}{39\cdots 89}a^{11}-\frac{24\cdots 47}{39\cdots 89}a^{10}-\frac{48\cdots 76}{39\cdots 89}a^{9}+\frac{14\cdots 98}{10\cdots 97}a^{8}+\frac{39\cdots 00}{10\cdots 97}a^{7}-\frac{88\cdots 58}{10\cdots 97}a^{6}-\frac{59\cdots 60}{10\cdots 97}a^{5}-\frac{18\cdots 13}{10\cdots 97}a^{4}+\frac{32\cdots 84}{10\cdots 97}a^{3}+\frac{21\cdots 20}{10\cdots 97}a^{2}+\frac{16\cdots 91}{10\cdots 97}a-\frac{80\cdots 97}{10\cdots 97}$, $\frac{56\cdots 58}{14\cdots 93}a^{26}+\frac{11\cdots 28}{14\cdots 93}a^{25}+\frac{50\cdots 67}{39\cdots 89}a^{24}-\frac{18\cdots 74}{14\cdots 93}a^{23}-\frac{53\cdots 05}{39\cdots 89}a^{22}+\frac{87\cdots 31}{14\cdots 93}a^{21}+\frac{10\cdots 84}{14\cdots 93}a^{20}+\frac{34\cdots 35}{14\cdots 93}a^{19}-\frac{29\cdots 36}{14\cdots 93}a^{18}-\frac{33\cdots 43}{39\cdots 89}a^{17}+\frac{14\cdots 97}{39\cdots 89}a^{16}+\frac{11\cdots 16}{39\cdots 89}a^{15}-\frac{16\cdots 31}{39\cdots 89}a^{14}-\frac{19\cdots 84}{39\cdots 89}a^{13}+\frac{11\cdots 75}{39\cdots 89}a^{12}+\frac{17\cdots 46}{39\cdots 89}a^{11}-\frac{49\cdots 41}{39\cdots 89}a^{10}-\frac{95\cdots 72}{39\cdots 89}a^{9}+\frac{30\cdots 60}{10\cdots 97}a^{8}+\frac{78\cdots 64}{10\cdots 97}a^{7}-\frac{19\cdots 94}{10\cdots 97}a^{6}-\frac{11\cdots 22}{10\cdots 97}a^{5}-\frac{35\cdots 38}{10\cdots 97}a^{4}+\frac{65\cdots 14}{10\cdots 97}a^{3}+\frac{42\cdots 83}{10\cdots 97}a^{2}+\frac{31\cdots 66}{10\cdots 97}a-\frac{15\cdots 42}{10\cdots 97}$, $\frac{42\cdots 45}{14\cdots 93}a^{26}-\frac{95\cdots 89}{14\cdots 93}a^{25}-\frac{14\cdots 72}{14\cdots 93}a^{24}+\frac{16\cdots 42}{14\cdots 93}a^{23}+\frac{14\cdots 09}{14\cdots 93}a^{22}-\frac{96\cdots 96}{14\cdots 93}a^{21}-\frac{75\cdots 45}{14\cdots 93}a^{20}+\frac{14\cdots 80}{14\cdots 93}a^{19}+\frac{22\cdots 03}{14\cdots 93}a^{18}+\frac{14\cdots 30}{39\cdots 89}a^{17}-\frac{10\cdots 14}{39\cdots 89}a^{16}-\frac{18\cdots 62}{10\cdots 97}a^{15}+\frac{12\cdots 35}{39\cdots 89}a^{14}+\frac{12\cdots 14}{39\cdots 89}a^{13}-\frac{87\cdots 26}{39\cdots 89}a^{12}-\frac{32\cdots 47}{10\cdots 97}a^{11}+\frac{37\cdots 13}{39\cdots 89}a^{10}+\frac{65\cdots 15}{39\cdots 89}a^{9}-\frac{22\cdots 30}{10\cdots 97}a^{8}-\frac{54\cdots 33}{10\cdots 97}a^{7}+\frac{16\cdots 28}{10\cdots 97}a^{6}+\frac{83\cdots 71}{10\cdots 97}a^{5}+\frac{22\cdots 86}{10\cdots 97}a^{4}-\frac{45\cdots 02}{10\cdots 97}a^{3}-\frac{28\cdots 49}{10\cdots 97}a^{2}-\frac{20\cdots 46}{10\cdots 97}a+\frac{10\cdots 70}{10\cdots 97}$, $\frac{48\cdots 75}{14\cdots 93}a^{26}-\frac{25\cdots 79}{39\cdots 89}a^{25}-\frac{15\cdots 10}{14\cdots 93}a^{24}+\frac{14\cdots 19}{14\cdots 93}a^{23}+\frac{16\cdots 56}{14\cdots 93}a^{22}-\frac{59\cdots 57}{14\cdots 93}a^{21}-\frac{87\cdots 98}{14\cdots 93}a^{20}-\frac{77\cdots 44}{14\cdots 93}a^{19}+\frac{25\cdots 87}{14\cdots 93}a^{18}+\frac{34\cdots 43}{39\cdots 89}a^{17}-\frac{12\cdots 73}{39\cdots 89}a^{16}-\frac{11\cdots 69}{39\cdots 89}a^{15}+\frac{14\cdots 05}{39\cdots 89}a^{14}+\frac{17\cdots 56}{39\cdots 89}a^{13}-\frac{10\cdots 09}{39\cdots 89}a^{12}-\frac{16\cdots 27}{39\cdots 89}a^{11}+\frac{42\cdots 56}{39\cdots 89}a^{10}+\frac{85\cdots 51}{39\cdots 89}a^{9}-\frac{25\cdots 96}{10\cdots 97}a^{8}-\frac{69\cdots 65}{10\cdots 97}a^{7}+\frac{15\cdots 88}{10\cdots 97}a^{6}+\frac{10\cdots 84}{10\cdots 97}a^{5}+\frac{32\cdots 38}{10\cdots 97}a^{4}-\frac{57\cdots 12}{10\cdots 97}a^{3}-\frac{37\cdots 50}{10\cdots 97}a^{2}-\frac{28\cdots 65}{10\cdots 97}a+\frac{13\cdots 44}{10\cdots 97}$, $\frac{22\cdots 19}{14\cdots 93}a^{26}-\frac{70\cdots 29}{14\cdots 93}a^{25}-\frac{73\cdots 10}{14\cdots 93}a^{24}+\frac{15\cdots 08}{14\cdots 93}a^{23}+\frac{75\cdots 19}{14\cdots 93}a^{22}-\frac{11\cdots 74}{14\cdots 93}a^{21}-\frac{38\cdots 98}{14\cdots 93}a^{20}+\frac{39\cdots 13}{14\cdots 93}a^{19}+\frac{11\cdots 41}{14\cdots 93}a^{18}-\frac{16\cdots 71}{39\cdots 89}a^{17}-\frac{52\cdots 51}{39\cdots 89}a^{16}+\frac{48\cdots 85}{39\cdots 89}a^{15}+\frac{59\cdots 84}{39\cdots 89}a^{14}+\frac{19\cdots 65}{39\cdots 89}a^{13}-\frac{42\cdots 04}{39\cdots 89}a^{12}-\frac{30\cdots 02}{39\cdots 89}a^{11}+\frac{50\cdots 62}{10\cdots 97}a^{10}+\frac{19\cdots 97}{39\cdots 89}a^{9}-\frac{12\cdots 67}{10\cdots 97}a^{8}-\frac{18\cdots 37}{10\cdots 97}a^{7}+\frac{12\cdots 38}{10\cdots 97}a^{6}+\frac{29\cdots 96}{10\cdots 97}a^{5}+\frac{24\cdots 38}{10\cdots 97}a^{4}-\frac{16\cdots 51}{10\cdots 97}a^{3}-\frac{87\cdots 28}{10\cdots 97}a^{2}-\frac{45\cdots 51}{10\cdots 97}a+\frac{30\cdots 32}{10\cdots 97}$, $\frac{55\cdots 07}{14\cdots 93}a^{26}-\frac{12\cdots 54}{14\cdots 93}a^{25}-\frac{18\cdots 39}{14\cdots 93}a^{24}+\frac{21\cdots 96}{14\cdots 93}a^{23}+\frac{19\cdots 64}{14\cdots 93}a^{22}-\frac{11\cdots 30}{14\cdots 93}a^{21}-\frac{97\cdots 59}{14\cdots 93}a^{20}+\frac{14\cdots 29}{14\cdots 93}a^{19}+\frac{28\cdots 35}{14\cdots 93}a^{18}+\frac{21\cdots 06}{39\cdots 89}a^{17}-\frac{13\cdots 58}{39\cdots 89}a^{16}-\frac{94\cdots 78}{39\cdots 89}a^{15}+\frac{15\cdots 51}{39\cdots 89}a^{14}+\frac{16\cdots 32}{39\cdots 89}a^{13}-\frac{11\cdots 05}{39\cdots 89}a^{12}-\frac{15\cdots 54}{39\cdots 89}a^{11}+\frac{47\cdots 02}{39\cdots 89}a^{10}+\frac{86\cdots 14}{39\cdots 89}a^{9}-\frac{28\cdots 01}{10\cdots 97}a^{8}-\frac{71\cdots 00}{10\cdots 97}a^{7}+\frac{19\cdots 46}{10\cdots 97}a^{6}+\frac{10\cdots 72}{10\cdots 97}a^{5}+\frac{31\cdots 45}{10\cdots 97}a^{4}-\frac{60\cdots 39}{10\cdots 97}a^{3}-\frac{39\cdots 40}{10\cdots 97}a^{2}-\frac{29\cdots 44}{10\cdots 97}a+\frac{14\cdots 45}{10\cdots 97}$, $\frac{99\cdots 47}{14\cdots 93}a^{26}+\frac{22\cdots 70}{14\cdots 93}a^{25}+\frac{32\cdots 01}{14\cdots 93}a^{24}-\frac{42\cdots 29}{14\cdots 93}a^{23}-\frac{34\cdots 87}{14\cdots 93}a^{22}+\frac{25\cdots 65}{14\cdots 93}a^{21}+\frac{17\cdots 21}{14\cdots 93}a^{20}-\frac{47\cdots 30}{14\cdots 93}a^{19}-\frac{51\cdots 75}{14\cdots 93}a^{18}-\frac{23\cdots 56}{39\cdots 89}a^{17}+\frac{24\cdots 75}{39\cdots 89}a^{16}+\frac{14\cdots 84}{39\cdots 89}a^{15}-\frac{28\cdots 41}{39\cdots 89}a^{14}-\frac{26\cdots 59}{39\cdots 89}a^{13}+\frac{20\cdots 69}{39\cdots 89}a^{12}+\frac{26\cdots 68}{39\cdots 89}a^{11}-\frac{85\cdots 04}{39\cdots 89}a^{10}-\frac{14\cdots 82}{39\cdots 89}a^{9}+\frac{52\cdots 96}{10\cdots 97}a^{8}+\frac{12\cdots 70}{10\cdots 97}a^{7}-\frac{39\cdots 50}{10\cdots 97}a^{6}-\frac{18\cdots 94}{10\cdots 97}a^{5}-\frac{48\cdots 44}{10\cdots 97}a^{4}+\frac{10\cdots 77}{10\cdots 97}a^{3}+\frac{64\cdots 01}{10\cdots 97}a^{2}+\frac{45\cdots 37}{10\cdots 97}a-\frac{23\cdots 31}{10\cdots 97}$, $\frac{11\cdots 28}{14\cdots 93}a^{26}+\frac{25\cdots 50}{14\cdots 93}a^{25}+\frac{38\cdots 10}{14\cdots 93}a^{24}-\frac{43\cdots 88}{14\cdots 93}a^{23}-\frac{40\cdots 94}{14\cdots 93}a^{22}+\frac{23\cdots 43}{14\cdots 93}a^{21}+\frac{21\cdots 17}{14\cdots 93}a^{20}-\frac{23\cdots 62}{14\cdots 93}a^{19}-\frac{62\cdots 22}{14\cdots 93}a^{18}-\frac{52\cdots 28}{39\cdots 89}a^{17}+\frac{30\cdots 47}{39\cdots 89}a^{16}+\frac{21\cdots 51}{39\cdots 89}a^{15}-\frac{34\cdots 00}{39\cdots 89}a^{14}-\frac{36\cdots 12}{39\cdots 89}a^{13}+\frac{24\cdots 69}{39\cdots 89}a^{12}+\frac{34\cdots 25}{39\cdots 89}a^{11}-\frac{10\cdots 02}{39\cdots 89}a^{10}-\frac{18\cdots 20}{39\cdots 89}a^{9}+\frac{63\cdots 21}{10\cdots 97}a^{8}+\frac{15\cdots 20}{10\cdots 97}a^{7}-\frac{44\cdots 76}{10\cdots 97}a^{6}-\frac{23\cdots 60}{10\cdots 97}a^{5}-\frac{66\cdots 42}{10\cdots 97}a^{4}+\frac{13\cdots 05}{10\cdots 97}a^{3}+\frac{83\cdots 28}{10\cdots 97}a^{2}+\frac{60\cdots 45}{10\cdots 97}a-\frac{30\cdots 96}{10\cdots 97}$, $\frac{13\cdots 77}{14\cdots 93}a^{26}+\frac{26\cdots 77}{14\cdots 93}a^{25}+\frac{44\cdots 72}{14\cdots 93}a^{24}-\frac{40\cdots 41}{14\cdots 93}a^{23}-\frac{46\cdots 05}{14\cdots 93}a^{22}+\frac{17\cdots 09}{14\cdots 93}a^{21}+\frac{24\cdots 70}{14\cdots 93}a^{20}+\frac{16\cdots 48}{14\cdots 93}a^{19}-\frac{71\cdots 13}{14\cdots 93}a^{18}-\frac{93\cdots 27}{39\cdots 89}a^{17}+\frac{34\cdots 55}{39\cdots 89}a^{16}+\frac{30\cdots 97}{39\cdots 89}a^{15}-\frac{39\cdots 36}{39\cdots 89}a^{14}-\frac{48\cdots 28}{39\cdots 89}a^{13}+\frac{28\cdots 32}{39\cdots 89}a^{12}+\frac{44\cdots 68}{39\cdots 89}a^{11}-\frac{11\cdots 33}{39\cdots 89}a^{10}-\frac{23\cdots 20}{39\cdots 89}a^{9}+\frac{71\cdots 04}{10\cdots 97}a^{8}+\frac{19\cdots 73}{10\cdots 97}a^{7}-\frac{44\cdots 87}{10\cdots 97}a^{6}-\frac{29\cdots 16}{10\cdots 97}a^{5}-\frac{89\cdots 26}{10\cdots 97}a^{4}+\frac{15\cdots 96}{10\cdots 97}a^{3}+\frac{10\cdots 53}{10\cdots 97}a^{2}+\frac{79\cdots 13}{10\cdots 97}a-\frac{38\cdots 71}{10\cdots 97}$ (assuming GRH)
Regulator:		\( 69535904668008080000000 \) (assuming GRH)
Unit signature rank:		\( 25 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{27}\cdot(2\pi)^{0}\cdot 69535904668008080000000 \cdot 3}{2\cdot\sqrt{2519933803975014472273213858760468555951745833259300216067364409}}\cr\approx \mathstrut & 0.278878918126825 \end{aligned}\] (assuming GRH)

Galois group

$C_3.\He_3$ (as 27T20):

A solvable group of order 81

The 17 conjugacy class representatives for $C_3.\He_3$

Character table for $C_3.\He_3$

Intermediate fields

3.3.1369.1, 9.9.1363532208525369.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 27 sibling:	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	${\href{/padicField/2.9.0.1}{9} }^{3}$	R	${\href{/padicField/5.9.0.1}{9} }^{3}$	${\href{/padicField/7.9.0.1}{9} }^{3}$	${\href{/padicField/11.3.0.1}{3} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$	${\href{/padicField/13.9.0.1}{9} }^{3}$	${\href{/padicField/17.9.0.1}{9} }^{3}$	${\href{/padicField/19.9.0.1}{9} }^{3}$	${\href{/padicField/23.3.0.1}{3} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$	${\href{/padicField/29.3.0.1}{3} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$	${\href{/padicField/31.3.0.1}{3} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$	R	${\href{/padicField/41.9.0.1}{9} }^{3}$	${\href{/padicField/43.3.0.1}{3} }^{8}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$	${\href{/padicField/47.3.0.1}{3} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$	${\href{/padicField/53.9.0.1}{9} }^{3}$	${\href{/padicField/59.9.0.1}{9} }^{3}$

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 $27$	$9$	$3$	$54$
\(37\)		Deg $27$	$9$	$3$	$24$

Spectrum of ring of integers