Number field 27.9.17717054310925604811052271173453197606912.1

• Label: 27.9.177...912.1
• Degree: $27$
• Signature: $(9, 9)$
• Discriminant: $-1.772\times 10^{40}$
• Root discriminant: \(30.95\)
• Ramified primes: $2,3$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $S_3\times C_9$ (as 27T12)

Normalized defining polynomial

x^27 - 18*x^25 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53

Invariants

Degree:		$27$
Signature:		$(9, 9)$
Discriminant:		\(-17717054310925604811052271173453197606912\) \(\medspace = -\,2^{18}\cdot 3^{73}\)
Root discriminant:		\(30.95\)
Galois root discriminant:		$2^{2/3}3^{49/18}\approx 31.5876084551639$
Ramified primes:		\(2\), \(3\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:		$C_9$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}+\frac{1}{3}a^{9}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{20}-\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{23}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{24}-\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{3}a^{25}-\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{11\cdots 93}a^{26}-\frac{24\cdots 14}{38\cdots 31}a^{25}-\frac{40\cdots 69}{38\cdots 31}a^{24}-\frac{73\cdots 53}{11\cdots 93}a^{23}-\frac{44\cdots 02}{38\cdots 31}a^{22}+\frac{19\cdots 57}{11\cdots 93}a^{21}+\frac{63\cdots 00}{38\cdots 31}a^{20}-\frac{78\cdots 13}{11\cdots 93}a^{19}+\frac{36\cdots 16}{38\cdots 31}a^{18}-\frac{21\cdots 37}{11\cdots 93}a^{17}+\frac{17\cdots 81}{11\cdots 93}a^{16}+\frac{10\cdots 01}{11\cdots 93}a^{15}+\frac{34\cdots 71}{11\cdots 93}a^{14}-\frac{83\cdots 02}{11\cdots 93}a^{13}-\frac{57\cdots 39}{11\cdots 93}a^{12}-\frac{75\cdots 88}{38\cdots 31}a^{11}-\frac{54\cdots 51}{11\cdots 93}a^{10}-\frac{57\cdots 14}{11\cdots 93}a^{9}-\frac{15\cdots 80}{11\cdots 93}a^{8}-\frac{41\cdots 83}{11\cdots 93}a^{7}+\frac{22\cdots 05}{11\cdots 93}a^{6}-\frac{22\cdots 89}{11\cdots 93}a^{5}+\frac{44\cdots 51}{11\cdots 93}a^{4}-\frac{10\cdots 46}{38\cdots 31}a^{3}+\frac{70\cdots 18}{38\cdots 31}a^{2}-\frac{19\cdots 83}{38\cdots 31}a+\frac{15\cdots 69}{11\cdots 93}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{82\cdots 42}{12\cdots 31}a^{26}-\frac{51\cdots 16}{12\cdots 31}a^{25}-\frac{14\cdots 33}{12\cdots 31}a^{24}+\frac{91\cdots 24}{12\cdots 31}a^{23}+\frac{14\cdots 68}{12\cdots 31}a^{22}-\frac{14\cdots 06}{12\cdots 31}a^{21}-\frac{78\cdots 63}{12\cdots 31}a^{20}+\frac{10\cdots 63}{12\cdots 31}a^{19}+\frac{90\cdots 78}{38\cdots 93}a^{18}-\frac{41\cdots 20}{12\cdots 31}a^{17}-\frac{93\cdots 76}{12\cdots 31}a^{16}+\frac{11\cdots 70}{12\cdots 31}a^{15}+\frac{23\cdots 09}{12\cdots 31}a^{14}-\frac{32\cdots 42}{12\cdots 31}a^{13}-\frac{40\cdots 71}{12\cdots 31}a^{12}+\frac{45\cdots 85}{12\cdots 31}a^{11}+\frac{21\cdots 86}{12\cdots 31}a^{10}-\frac{13\cdots 03}{38\cdots 93}a^{9}+\frac{12\cdots 31}{12\cdots 31}a^{8}+\frac{46\cdots 74}{12\cdots 31}a^{7}+\frac{57\cdots 86}{12\cdots 31}a^{6}-\frac{13\cdots 80}{12\cdots 31}a^{5}-\frac{71\cdots 18}{12\cdots 31}a^{4}-\frac{42\cdots 99}{12\cdots 31}a^{3}-\frac{25\cdots 59}{12\cdots 31}a^{2}-\frac{72\cdots 07}{12\cdots 31}a-\frac{21\cdots 19}{38\cdots 93}$, $\frac{25\cdots 68}{12\cdots 31}a^{26}-\frac{15\cdots 22}{12\cdots 31}a^{25}-\frac{45\cdots 29}{12\cdots 31}a^{24}+\frac{27\cdots 49}{12\cdots 31}a^{23}+\frac{46\cdots 40}{12\cdots 31}a^{22}-\frac{44\cdots 32}{12\cdots 31}a^{21}-\frac{24\cdots 00}{12\cdots 31}a^{20}+\frac{31\cdots 08}{12\cdots 31}a^{19}+\frac{28\cdots 11}{38\cdots 93}a^{18}-\frac{12\cdots 24}{12\cdots 31}a^{17}-\frac{29\cdots 41}{12\cdots 31}a^{16}+\frac{36\cdots 99}{12\cdots 31}a^{15}+\frac{72\cdots 00}{12\cdots 31}a^{14}-\frac{99\cdots 45}{12\cdots 31}a^{13}-\frac{12\cdots 77}{12\cdots 31}a^{12}+\frac{14\cdots 96}{12\cdots 31}a^{11}+\frac{69\cdots 00}{12\cdots 31}a^{10}-\frac{40\cdots 76}{38\cdots 93}a^{9}+\frac{38\cdots 80}{12\cdots 31}a^{8}+\frac{14\cdots 61}{12\cdots 31}a^{7}+\frac{18\cdots 27}{12\cdots 31}a^{6}-\frac{40\cdots 99}{12\cdots 31}a^{5}-\frac{22\cdots 32}{12\cdots 31}a^{4}-\frac{13\cdots 12}{12\cdots 31}a^{3}-\frac{80\cdots 52}{12\cdots 31}a^{2}-\frac{23\cdots 68}{12\cdots 31}a-\frac{67\cdots 00}{38\cdots 93}$, $\frac{11\cdots 24}{11\cdots 93}a^{26}-\frac{87\cdots 58}{11\cdots 93}a^{25}-\frac{67\cdots 99}{38\cdots 31}a^{24}+\frac{15\cdots 86}{11\cdots 93}a^{23}+\frac{20\cdots 56}{11\cdots 93}a^{22}-\frac{76\cdots 10}{38\cdots 31}a^{21}-\frac{10\cdots 83}{11\cdots 93}a^{20}+\frac{15\cdots 78}{11\cdots 93}a^{19}+\frac{39\cdots 82}{11\cdots 93}a^{18}-\frac{61\cdots 50}{11\cdots 93}a^{17}-\frac{12\cdots 43}{11\cdots 93}a^{16}+\frac{17\cdots 22}{11\cdots 93}a^{15}+\frac{29\cdots 50}{11\cdots 93}a^{14}-\frac{47\cdots 88}{11\cdots 93}a^{13}-\frac{50\cdots 91}{11\cdots 93}a^{12}+\frac{66\cdots 68}{11\cdots 93}a^{11}+\frac{20\cdots 10}{11\cdots 93}a^{10}-\frac{19\cdots 36}{38\cdots 31}a^{9}+\frac{24\cdots 83}{11\cdots 93}a^{8}+\frac{59\cdots 88}{11\cdots 93}a^{7}+\frac{35\cdots 64}{11\cdots 93}a^{6}-\frac{16\cdots 10}{11\cdots 93}a^{5}-\frac{86\cdots 91}{11\cdots 93}a^{4}-\frac{54\cdots 32}{11\cdots 93}a^{3}-\frac{31\cdots 82}{11\cdots 93}a^{2}-\frac{83\cdots 14}{11\cdots 93}a-\frac{78\cdots 83}{11\cdots 93}$, $\frac{56\cdots 22}{11\cdots 93}a^{26}-\frac{12\cdots 96}{38\cdots 31}a^{25}-\frac{33\cdots 55}{38\cdots 31}a^{24}+\frac{64\cdots 33}{11\cdots 93}a^{23}+\frac{10\cdots 02}{11\cdots 93}a^{22}-\frac{10\cdots 64}{11\cdots 93}a^{21}-\frac{53\cdots 03}{11\cdots 93}a^{20}+\frac{23\cdots 52}{38\cdots 31}a^{19}+\frac{20\cdots 87}{11\cdots 93}a^{18}-\frac{28\cdots 40}{11\cdots 93}a^{17}-\frac{21\cdots 60}{38\cdots 31}a^{16}+\frac{81\cdots 61}{11\cdots 93}a^{15}+\frac{15\cdots 66}{11\cdots 93}a^{14}-\frac{22\cdots 81}{11\cdots 93}a^{13}-\frac{27\cdots 24}{11\cdots 93}a^{12}+\frac{31\cdots 16}{11\cdots 93}a^{11}+\frac{46\cdots 90}{38\cdots 31}a^{10}-\frac{29\cdots 56}{11\cdots 93}a^{9}+\frac{95\cdots 80}{11\cdots 93}a^{8}+\frac{10\cdots 67}{38\cdots 31}a^{7}+\frac{34\cdots 95}{11\cdots 93}a^{6}-\frac{88\cdots 63}{11\cdots 93}a^{5}-\frac{47\cdots 72}{11\cdots 93}a^{4}-\frac{94\cdots 20}{38\cdots 31}a^{3}-\frac{17\cdots 70}{11\cdots 93}a^{2}-\frac{16\cdots 16}{38\cdots 31}a-\frac{15\cdots 73}{38\cdots 31}$, $\frac{49\cdots 60}{11\cdots 93}a^{26}-\frac{30\cdots 18}{11\cdots 93}a^{25}-\frac{29\cdots 33}{38\cdots 31}a^{24}+\frac{54\cdots 67}{11\cdots 93}a^{23}+\frac{30\cdots 52}{38\cdots 31}a^{22}-\frac{87\cdots 02}{11\cdots 93}a^{21}-\frac{47\cdots 35}{11\cdots 93}a^{20}+\frac{61\cdots 43}{11\cdots 93}a^{19}+\frac{61\cdots 69}{38\cdots 31}a^{18}-\frac{24\cdots 40}{11\cdots 93}a^{17}-\frac{56\cdots 73}{11\cdots 93}a^{16}+\frac{70\cdots 61}{11\cdots 93}a^{15}+\frac{14\cdots 29}{11\cdots 93}a^{14}-\frac{64\cdots 99}{38\cdots 31}a^{13}-\frac{83\cdots 87}{38\cdots 31}a^{12}+\frac{27\cdots 33}{11\cdots 93}a^{11}+\frac{13\cdots 30}{11\cdots 93}a^{10}-\frac{26\cdots 43}{11\cdots 93}a^{9}+\frac{76\cdots 27}{11\cdots 93}a^{8}+\frac{28\cdots 29}{11\cdots 93}a^{7}+\frac{36\cdots 41}{11\cdots 93}a^{6}-\frac{79\cdots 19}{11\cdots 93}a^{5}-\frac{14\cdots 86}{38\cdots 31}a^{4}-\frac{25\cdots 83}{11\cdots 93}a^{3}-\frac{15\cdots 79}{11\cdots 93}a^{2}-\frac{44\cdots 27}{11\cdots 93}a-\frac{43\cdots 39}{11\cdots 93}$, $\frac{98\cdots 30}{11\cdots 93}a^{26}-\frac{68\cdots 82}{11\cdots 93}a^{25}-\frac{17\cdots 60}{11\cdots 93}a^{24}+\frac{11\cdots 17}{11\cdots 93}a^{23}+\frac{17\cdots 70}{11\cdots 93}a^{22}-\frac{18\cdots 66}{11\cdots 93}a^{21}-\frac{91\cdots 66}{11\cdots 93}a^{20}+\frac{42\cdots 56}{38\cdots 31}a^{19}+\frac{34\cdots 07}{11\cdots 93}a^{18}-\frac{50\cdots 66}{11\cdots 93}a^{17}-\frac{10\cdots 67}{11\cdots 93}a^{16}+\frac{14\cdots 61}{11\cdots 93}a^{15}+\frac{26\cdots 14}{11\cdots 93}a^{14}-\frac{39\cdots 27}{11\cdots 93}a^{13}-\frac{15\cdots 59}{38\cdots 31}a^{12}+\frac{54\cdots 66}{11\cdots 93}a^{11}+\frac{71\cdots 04}{38\cdots 31}a^{10}-\frac{16\cdots 66}{38\cdots 31}a^{9}+\frac{17\cdots 42}{11\cdots 93}a^{8}+\frac{52\cdots 55}{11\cdots 93}a^{7}+\frac{18\cdots 63}{38\cdots 31}a^{6}-\frac{14\cdots 55}{11\cdots 93}a^{5}-\frac{81\cdots 55}{11\cdots 93}a^{4}-\frac{48\cdots 46}{11\cdots 93}a^{3}-\frac{28\cdots 10}{11\cdots 93}a^{2}-\frac{27\cdots 76}{38\cdots 31}a-\frac{81\cdots 94}{11\cdots 93}$, $\frac{79\cdots 34}{38\cdots 31}a^{26}-\frac{16\cdots 58}{11\cdots 93}a^{25}-\frac{41\cdots 06}{11\cdots 93}a^{24}+\frac{29\cdots 31}{11\cdots 93}a^{23}+\frac{42\cdots 94}{11\cdots 93}a^{22}-\frac{45\cdots 18}{11\cdots 93}a^{21}-\frac{73\cdots 92}{38\cdots 31}a^{20}+\frac{31\cdots 26}{11\cdots 93}a^{19}+\frac{84\cdots 25}{11\cdots 93}a^{18}-\frac{41\cdots 02}{38\cdots 31}a^{17}-\frac{25\cdots 28}{11\cdots 93}a^{16}+\frac{35\cdots 87}{11\cdots 93}a^{15}+\frac{63\cdots 28}{11\cdots 93}a^{14}-\frac{96\cdots 21}{11\cdots 93}a^{13}-\frac{10\cdots 65}{11\cdots 93}a^{12}+\frac{45\cdots 88}{38\cdots 31}a^{11}+\frac{51\cdots 00}{11\cdots 93}a^{10}-\frac{12\cdots 64}{11\cdots 93}a^{9}+\frac{15\cdots 82}{38\cdots 31}a^{8}+\frac{12\cdots 29}{11\cdots 93}a^{7}+\frac{33\cdots 61}{38\cdots 31}a^{6}-\frac{36\cdots 73}{11\cdots 93}a^{5}-\frac{18\cdots 49}{11\cdots 93}a^{4}-\frac{38\cdots 56}{38\cdots 31}a^{3}-\frac{22\cdots 80}{38\cdots 31}a^{2}-\frac{18\cdots 50}{11\cdots 93}a-\frac{55\cdots 64}{38\cdots 31}$, $\frac{29\cdots 98}{38\cdots 31}a^{26}-\frac{76\cdots 32}{11\cdots 93}a^{25}-\frac{15\cdots 90}{11\cdots 93}a^{24}+\frac{13\cdots 82}{11\cdots 93}a^{23}+\frac{52\cdots 70}{38\cdots 31}a^{22}-\frac{64\cdots 62}{38\cdots 31}a^{21}-\frac{26\cdots 90}{38\cdots 31}a^{20}+\frac{12\cdots 46}{11\cdots 93}a^{19}+\frac{29\cdots 28}{11\cdots 93}a^{18}-\frac{16\cdots 78}{38\cdots 31}a^{17}-\frac{88\cdots 01}{11\cdots 93}a^{16}+\frac{47\cdots 24}{38\cdots 31}a^{15}+\frac{21\cdots 96}{11\cdots 93}a^{14}-\frac{12\cdots 04}{38\cdots 31}a^{13}-\frac{34\cdots 46}{11\cdots 93}a^{12}+\frac{17\cdots 68}{38\cdots 31}a^{11}+\frac{10\cdots 48}{11\cdots 93}a^{10}-\frac{44\cdots 52}{11\cdots 93}a^{9}+\frac{76\cdots 97}{38\cdots 31}a^{8}+\frac{42\cdots 16}{11\cdots 93}a^{7}-\frac{20\cdots 44}{11\cdots 93}a^{6}-\frac{12\cdots 42}{11\cdots 93}a^{5}-\frac{19\cdots 41}{38\cdots 31}a^{4}-\frac{38\cdots 24}{11\cdots 93}a^{3}-\frac{71\cdots 60}{38\cdots 31}a^{2}-\frac{52\cdots 90}{11\cdots 93}a-\frac{13\cdots 22}{38\cdots 31}$, $\frac{83\cdots 16}{38\cdots 31}a^{26}-\frac{15\cdots 73}{11\cdots 93}a^{25}-\frac{44\cdots 08}{11\cdots 93}a^{24}+\frac{27\cdots 57}{11\cdots 93}a^{23}+\frac{45\cdots 64}{11\cdots 93}a^{22}-\frac{44\cdots 89}{11\cdots 93}a^{21}-\frac{23\cdots 01}{11\cdots 93}a^{20}+\frac{31\cdots 07}{11\cdots 93}a^{19}+\frac{30\cdots 75}{38\cdots 31}a^{18}-\frac{41\cdots 64}{38\cdots 31}a^{17}-\frac{28\cdots 22}{11\cdots 93}a^{16}+\frac{11\cdots 88}{38\cdots 31}a^{15}+\frac{70\cdots 09}{11\cdots 93}a^{14}-\frac{32\cdots 25}{38\cdots 31}a^{13}-\frac{41\cdots 58}{38\cdots 31}a^{12}+\frac{13\cdots 29}{11\cdots 93}a^{11}+\frac{67\cdots 21}{11\cdots 93}a^{10}-\frac{13\cdots 46}{11\cdots 93}a^{9}+\frac{13\cdots 75}{38\cdots 31}a^{8}+\frac{47\cdots 55}{38\cdots 31}a^{7}+\frac{16\cdots 81}{11\cdots 93}a^{6}-\frac{40\cdots 53}{11\cdots 93}a^{5}-\frac{21\cdots 65}{11\cdots 93}a^{4}-\frac{12\cdots 92}{11\cdots 93}a^{3}-\frac{77\cdots 82}{11\cdots 93}a^{2}-\frac{72\cdots 36}{38\cdots 31}a-\frac{20\cdots 42}{11\cdots 93}$, $\frac{18\cdots 34}{11\cdots 93}a^{26}-\frac{11\cdots 70}{11\cdots 93}a^{25}-\frac{31\cdots 64}{11\cdots 93}a^{24}+\frac{64\cdots 73}{38\cdots 31}a^{23}+\frac{32\cdots 02}{11\cdots 93}a^{22}-\frac{31\cdots 06}{11\cdots 93}a^{21}-\frac{17\cdots 53}{11\cdots 93}a^{20}+\frac{22\cdots 39}{11\cdots 93}a^{19}+\frac{66\cdots 64}{11\cdots 93}a^{18}-\frac{29\cdots 90}{38\cdots 31}a^{17}-\frac{20\cdots 98}{11\cdots 93}a^{16}+\frac{25\cdots 15}{11\cdots 93}a^{15}+\frac{51\cdots 09}{11\cdots 93}a^{14}-\frac{69\cdots 60}{11\cdots 93}a^{13}-\frac{30\cdots 00}{38\cdots 31}a^{12}+\frac{99\cdots 72}{11\cdots 93}a^{11}+\frac{49\cdots 64}{11\cdots 93}a^{10}-\frac{31\cdots 84}{38\cdots 31}a^{9}+\frac{27\cdots 24}{11\cdots 93}a^{8}+\frac{10\cdots 66}{11\cdots 93}a^{7}+\frac{13\cdots 57}{11\cdots 93}a^{6}-\frac{28\cdots 74}{11\cdots 93}a^{5}-\frac{52\cdots 56}{38\cdots 31}a^{4}-\frac{93\cdots 62}{11\cdots 93}a^{3}-\frac{56\cdots 27}{11\cdots 93}a^{2}-\frac{54\cdots 52}{38\cdots 31}a-\frac{15\cdots 69}{11\cdots 93}$, $\frac{64\cdots 58}{11\cdots 93}a^{26}-\frac{14\cdots 38}{38\cdots 31}a^{25}-\frac{37\cdots 78}{38\cdots 31}a^{24}+\frac{74\cdots 23}{11\cdots 93}a^{23}+\frac{11\cdots 41}{11\cdots 93}a^{22}-\frac{11\cdots 36}{11\cdots 93}a^{21}-\frac{61\cdots 10}{11\cdots 93}a^{20}+\frac{81\cdots 73}{11\cdots 93}a^{19}+\frac{78\cdots 56}{38\cdots 31}a^{18}-\frac{32\cdots 93}{11\cdots 93}a^{17}-\frac{24\cdots 09}{38\cdots 31}a^{16}+\frac{31\cdots 85}{38\cdots 31}a^{15}+\frac{59\cdots 35}{38\cdots 31}a^{14}-\frac{25\cdots 72}{11\cdots 93}a^{13}-\frac{10\cdots 37}{38\cdots 31}a^{12}+\frac{36\cdots 65}{11\cdots 93}a^{11}+\frac{16\cdots 64}{11\cdots 93}a^{10}-\frac{33\cdots 13}{11\cdots 93}a^{9}+\frac{35\cdots 13}{38\cdots 31}a^{8}+\frac{12\cdots 06}{38\cdots 31}a^{7}+\frac{13\cdots 17}{38\cdots 31}a^{6}-\frac{10\cdots 12}{11\cdots 93}a^{5}-\frac{18\cdots 08}{38\cdots 31}a^{4}-\frac{32\cdots 89}{11\cdots 93}a^{3}-\frac{19\cdots 00}{11\cdots 93}a^{2}-\frac{18\cdots 20}{38\cdots 31}a-\frac{54\cdots 65}{11\cdots 93}$, $\frac{21\cdots 23}{11\cdots 93}a^{26}-\frac{14\cdots 43}{11\cdots 93}a^{25}-\frac{12\cdots 73}{38\cdots 31}a^{24}+\frac{83\cdots 38}{38\cdots 31}a^{23}+\frac{12\cdots 09}{38\cdots 31}a^{22}-\frac{39\cdots 40}{11\cdots 93}a^{21}-\frac{19\cdots 48}{11\cdots 93}a^{20}+\frac{27\cdots 45}{11\cdots 93}a^{19}+\frac{25\cdots 77}{38\cdots 31}a^{18}-\frac{36\cdots 38}{38\cdots 31}a^{17}-\frac{77\cdots 22}{38\cdots 31}a^{16}+\frac{30\cdots 35}{11\cdots 93}a^{15}+\frac{57\cdots 71}{11\cdots 93}a^{14}-\frac{28\cdots 03}{38\cdots 31}a^{13}-\frac{10\cdots 25}{11\cdots 93}a^{12}+\frac{11\cdots 59}{11\cdots 93}a^{11}+\frac{49\cdots 45}{11\cdots 93}a^{10}-\frac{10\cdots 04}{11\cdots 93}a^{9}+\frac{37\cdots 92}{11\cdots 93}a^{8}+\frac{11\cdots 96}{11\cdots 93}a^{7}+\frac{12\cdots 36}{11\cdots 93}a^{6}-\frac{32\cdots 54}{11\cdots 93}a^{5}-\frac{58\cdots 32}{38\cdots 31}a^{4}-\frac{10\cdots 67}{11\cdots 93}a^{3}-\frac{63\cdots 64}{11\cdots 93}a^{2}-\frac{59\cdots 50}{38\cdots 31}a-\frac{17\cdots 65}{11\cdots 93}$, $\frac{11\cdots 43}{11\cdots 93}a^{26}-\frac{12\cdots 82}{38\cdots 31}a^{25}-\frac{20\cdots 65}{11\cdots 93}a^{24}+\frac{22\cdots 56}{38\cdots 31}a^{23}+\frac{21\cdots 38}{11\cdots 93}a^{22}-\frac{14\cdots 32}{11\cdots 93}a^{21}-\frac{40\cdots 29}{38\cdots 31}a^{20}+\frac{39\cdots 86}{38\cdots 31}a^{19}+\frac{49\cdots 49}{11\cdots 93}a^{18}-\frac{49\cdots 23}{11\cdots 93}a^{17}-\frac{52\cdots 44}{38\cdots 31}a^{16}+\frac{14\cdots 51}{11\cdots 93}a^{15}+\frac{39\cdots 96}{11\cdots 93}a^{14}-\frac{13\cdots 19}{38\cdots 31}a^{13}-\frac{25\cdots 04}{38\cdots 31}a^{12}+\frac{59\cdots 10}{11\cdots 93}a^{11}+\frac{55\cdots 88}{11\cdots 93}a^{10}-\frac{69\cdots 79}{11\cdots 93}a^{9}+\frac{38\cdots 67}{11\cdots 93}a^{8}+\frac{27\cdots 36}{38\cdots 31}a^{7}+\frac{17\cdots 07}{11\cdots 93}a^{6}-\frac{22\cdots 99}{11\cdots 93}a^{5}-\frac{12\cdots 24}{11\cdots 93}a^{4}-\frac{71\cdots 32}{11\cdots 93}a^{3}-\frac{46\cdots 29}{11\cdots 93}a^{2}-\frac{13\cdots 15}{11\cdots 93}a-\frac{46\cdots 63}{38\cdots 31}$, $\frac{35\cdots 46}{11\cdots 93}a^{26}-\frac{19\cdots 54}{11\cdots 93}a^{25}-\frac{20\cdots 00}{38\cdots 31}a^{24}+\frac{34\cdots 27}{11\cdots 93}a^{23}+\frac{65\cdots 65}{11\cdots 93}a^{22}-\frac{58\cdots 60}{11\cdots 93}a^{21}-\frac{34\cdots 88}{11\cdots 93}a^{20}+\frac{14\cdots 62}{38\cdots 31}a^{19}+\frac{13\cdots 39}{11\cdots 93}a^{18}-\frac{17\cdots 05}{11\cdots 93}a^{17}-\frac{42\cdots 79}{11\cdots 93}a^{16}+\frac{48\cdots 09}{11\cdots 93}a^{15}+\frac{10\cdots 67}{11\cdots 93}a^{14}-\frac{44\cdots 65}{38\cdots 31}a^{13}-\frac{19\cdots 26}{11\cdots 93}a^{12}+\frac{63\cdots 94}{38\cdots 31}a^{11}+\frac{37\cdots 79}{38\cdots 31}a^{10}-\frac{63\cdots 51}{38\cdots 31}a^{9}+\frac{43\cdots 30}{11\cdots 93}a^{8}+\frac{71\cdots 26}{38\cdots 31}a^{7}+\frac{33\cdots 17}{11\cdots 93}a^{6}-\frac{19\cdots 13}{38\cdots 31}a^{5}-\frac{33\cdots 03}{11\cdots 93}a^{4}-\frac{19\cdots 41}{11\cdots 93}a^{3}-\frac{11\cdots 30}{11\cdots 93}a^{2}-\frac{11\cdots 03}{38\cdots 31}a-\frac{34\cdots 71}{11\cdots 93}$, $\frac{31\cdots 25}{11\cdots 93}a^{26}-\frac{22\cdots 28}{11\cdots 93}a^{25}-\frac{55\cdots 92}{11\cdots 93}a^{24}+\frac{39\cdots 60}{11\cdots 93}a^{23}+\frac{56\cdots 93}{11\cdots 93}a^{22}-\frac{20\cdots 84}{38\cdots 31}a^{21}-\frac{29\cdots 04}{11\cdots 93}a^{20}+\frac{13\cdots 44}{38\cdots 31}a^{19}+\frac{11\cdots 02}{11\cdots 93}a^{18}-\frac{16\cdots 51}{11\cdots 93}a^{17}-\frac{34\cdots 92}{11\cdots 93}a^{16}+\frac{46\cdots 36}{11\cdots 93}a^{15}+\frac{83\cdots 13}{11\cdots 93}a^{14}-\frac{12\cdots 12}{11\cdots 93}a^{13}-\frac{14\cdots 08}{11\cdots 93}a^{12}+\frac{60\cdots 49}{38\cdots 31}a^{11}+\frac{65\cdots 88}{11\cdots 93}a^{10}-\frac{16\cdots 44}{11\cdots 93}a^{9}+\frac{20\cdots 66}{38\cdots 31}a^{8}+\frac{55\cdots 11}{38\cdots 31}a^{7}+\frac{40\cdots 03}{38\cdots 31}a^{6}-\frac{46\cdots 85}{11\cdots 93}a^{5}-\frac{24\cdots 59}{11\cdots 93}a^{4}-\frac{15\cdots 88}{11\cdots 93}a^{3}-\frac{88\cdots 70}{11\cdots 93}a^{2}-\frac{24\cdots 00}{11\cdots 93}a-\frac{21\cdots 75}{11\cdots 93}$, $\frac{13\cdots 42}{38\cdots 31}a^{26}-\frac{28\cdots 29}{11\cdots 93}a^{25}-\frac{23\cdots 94}{38\cdots 31}a^{24}+\frac{49\cdots 96}{11\cdots 93}a^{23}+\frac{73\cdots 47}{11\cdots 93}a^{22}-\frac{76\cdots 75}{11\cdots 93}a^{21}-\frac{37\cdots 90}{11\cdots 93}a^{20}+\frac{52\cdots 22}{11\cdots 93}a^{19}+\frac{14\cdots 25}{11\cdots 93}a^{18}-\frac{21\cdots 28}{11\cdots 93}a^{17}-\frac{44\cdots 70}{11\cdots 93}a^{16}+\frac{19\cdots 40}{38\cdots 31}a^{15}+\frac{10\cdots 38}{11\cdots 93}a^{14}-\frac{54\cdots 47}{38\cdots 31}a^{13}-\frac{18\cdots 75}{11\cdots 93}a^{12}+\frac{23\cdots 45}{11\cdots 93}a^{11}+\frac{29\cdots 71}{38\cdots 31}a^{10}-\frac{21\cdots 25}{11\cdots 93}a^{9}+\frac{76\cdots 10}{11\cdots 93}a^{8}+\frac{21\cdots 92}{11\cdots 93}a^{7}+\frac{66\cdots 17}{38\cdots 31}a^{6}-\frac{61\cdots 32}{11\cdots 93}a^{5}-\frac{32\cdots 83}{11\cdots 93}a^{4}-\frac{66\cdots 07}{38\cdots 31}a^{3}-\frac{39\cdots 24}{38\cdots 31}a^{2}-\frac{32\cdots 35}{11\cdots 93}a-\frac{10\cdots 71}{38\cdots 31}$, $\frac{18\cdots 23}{11\cdots 93}a^{26}-\frac{11\cdots 78}{11\cdots 93}a^{25}-\frac{11\cdots 67}{38\cdots 31}a^{24}+\frac{67\cdots 13}{38\cdots 31}a^{23}+\frac{34\cdots 28}{11\cdots 93}a^{22}-\frac{10\cdots 95}{38\cdots 31}a^{21}-\frac{17\cdots 82}{11\cdots 93}a^{20}+\frac{23\cdots 10}{11\cdots 93}a^{19}+\frac{69\cdots 35}{11\cdots 93}a^{18}-\frac{92\cdots 50}{11\cdots 93}a^{17}-\frac{71\cdots 33}{38\cdots 31}a^{16}+\frac{26\cdots 02}{11\cdots 93}a^{15}+\frac{17\cdots 83}{38\cdots 31}a^{14}-\frac{72\cdots 50}{11\cdots 93}a^{13}-\frac{94\cdots 74}{11\cdots 93}a^{12}+\frac{10\cdots 38}{11\cdots 93}a^{11}+\frac{52\cdots 09}{11\cdots 93}a^{10}-\frac{10\cdots 55}{11\cdots 93}a^{9}+\frac{94\cdots 83}{38\cdots 31}a^{8}+\frac{10\cdots 35}{11\cdots 93}a^{7}+\frac{13\cdots 28}{11\cdots 93}a^{6}-\frac{10\cdots 08}{38\cdots 31}a^{5}-\frac{16\cdots 36}{11\cdots 93}a^{4}-\frac{96\cdots 25}{11\cdots 93}a^{3}-\frac{19\cdots 70}{38\cdots 31}a^{2}-\frac{55\cdots 61}{38\cdots 31}a-\frac{15\cdots 30}{11\cdots 93}$ (assuming GRH)
Regulator:		\( 6360080771.530314 \) (assuming GRH)
Unit signature rank:		\( 9 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{9}\cdot 6360080771.530314 \cdot 1}{2\cdot\sqrt{17717054310925604811052271173453197606912}}\cr\approx \mathstrut & 0.186691976425618 \end{aligned}\] (assuming GRH)

Galois group

$S_3\times C_9$ (as 27T12):

A solvable group of order 54

The 27 conjugacy class representatives for $S_3\times C_9$

Character table for $S_3\times C_9$

Intermediate fields

\(\Q(\zeta_{9})^+\), \(\Q(\sqrt[3]{12})\), \(\Q(\zeta_{27})^+\), 9.3.74384733888.5

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:	18.0.12100864846032214829641728.3

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	$18{,}\,{\href{/padicField/5.9.0.1}{9} }$	${\href{/padicField/7.9.0.1}{9} }^{3}$	$18{,}\,{\href{/padicField/11.9.0.1}{9} }$	${\href{/padicField/13.9.0.1}{9} }^{3}$	${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }^{3}$	${\href{/padicField/19.3.0.1}{3} }^{9}$	$18{,}\,{\href{/padicField/23.9.0.1}{9} }$	$18{,}\,{\href{/padicField/29.9.0.1}{9} }$	${\href{/padicField/31.9.0.1}{9} }^{3}$	${\href{/padicField/37.3.0.1}{3} }^{9}$	$18{,}\,{\href{/padicField/41.9.0.1}{9} }$	${\href{/padicField/43.9.0.1}{9} }^{3}$	$18{,}\,{\href{/padicField/47.9.0.1}{9} }$	${\href{/padicField/53.2.0.1}{2} }^{9}{,}\,{\href{/padicField/53.1.0.1}{1} }^{9}$	$18{,}\,{\href{/padicField/59.9.0.1}{9} }$

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
\(2\)		Deg $27$	$3$	$9$	$18$
\(3\)		Deg $27$	$27$	$1$	$73$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*^54	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.3.2t1.a.a	$1$	$ 3 $	\(\Q(\sqrt{-3}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.9.6t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})\)	$C_6$ (as 6T1)	$0$	$-1$
*^54	1.9.3t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
	1.9.6t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})\)	$C_6$ (as 6T1)	$0$	$-1$
*^54	1.9.3t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
*^54	1.27.9t1.a.a	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*^54	1.27.9t1.a.b	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*^54	1.27.9t1.a.c	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
	1.27.18t1.a.a	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
	1.27.18t1.a.b	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
	1.27.18t1.a.c	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*^54	1.27.9t1.a.d	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
*^54	1.27.9t1.a.e	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
	1.27.18t1.a.d	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*^54	1.27.9t1.a.f	$1$	$ 3^{3}$	\(\Q(\zeta_{27})^+\)	$C_9$ (as 9T1)	$0$	$1$
	1.27.18t1.a.e	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
	1.27.18t1.a.f	$1$	$ 3^{3}$	\(\Q(\zeta_{27})\)	$C_{18}$ (as 18T1)	$0$	$-1$
*^54	2.972.3t2.c.a	$2$	$ 2^{2} \cdot 3^{5}$	\(\Q(\sqrt[3]{12})\)	$S_3$ (as 3T2)	$1$	$0$
*^54	2.972.6t5.c.a	$2$	$ 2^{2} \cdot 3^{5}$	6.0.2834352.3	$S_3\times C_3$ (as 6T5)	$0$	$0$
*^54	2.972.6t5.c.b	$2$	$ 2^{2} \cdot 3^{5}$	6.0.2834352.3	$S_3\times C_3$ (as 6T5)	$0$	$0$
*^54	2.2916.18t16.c.a	$2$	$ 2^{2} \cdot 3^{6}$	27.9.17717054310925604811052271173453197606912.1	$S_3\times C_9$ (as 27T12)	$0$	$0$
*^54	2.2916.18t16.c.b	$2$	$ 2^{2} \cdot 3^{6}$	27.9.17717054310925604811052271173453197606912.1	$S_3\times C_9$ (as 27T12)	$0$	$0$
*^54	2.2916.18t16.c.c	$2$	$ 2^{2} \cdot 3^{6}$	27.9.17717054310925604811052271173453197606912.1	$S_3\times C_9$ (as 27T12)	$0$	$0$
*^54	2.2916.18t16.c.d	$2$	$ 2^{2} \cdot 3^{6}$	27.9.17717054310925604811052271173453197606912.1	$S_3\times C_9$ (as 27T12)	$0$	$0$
*^54	2.2916.18t16.c.e	$2$	$ 2^{2} \cdot 3^{6}$	27.9.17717054310925604811052271173453197606912.1	$S_3\times C_9$ (as 27T12)	$0$	$0$
*^54	2.2916.18t16.c.f	$2$	$ 2^{2} \cdot 3^{6}$	27.9.17717054310925604811052271173453197606912.1	$S_3\times C_9$ (as 27T12)	$0$	$0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers