Number field 27.1.1543319746516623033280478216838436483146079.1

• Label: 27.1.154...079.1
• Degree: $27$
• Signature: $(1, 13)$
• Discriminant: $-1.543\times 10^{42}$
• Root discriminant: \(36.52\)
• Ramified prime: $1759$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $D_{27}$ (as 27T8)

Normalized defining polynomial

x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1

Invariants

Degree:		$27$
Signature:		$(1, 13)$
Discriminant:		\(-1543319746516623033280478216838436483146079\) \(\medspace = -\,1759^{13}\)
Root discriminant:		\(36.52\)
Galois root discriminant:		$1759^{1/2}\approx 41.94043395102154$
Ramified primes:		\(1759\)
Discriminant root field:		\(\Q(\sqrt{-1759}) \)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{12}-\frac{2}{9}a^{4}$, $\frac{1}{9}a^{21}+\frac{1}{9}a^{13}-\frac{2}{9}a^{5}$, $\frac{1}{27}a^{22}+\frac{1}{27}a^{21}-\frac{1}{27}a^{20}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}+\frac{1}{27}a^{16}+\frac{1}{27}a^{14}+\frac{1}{27}a^{13}-\frac{1}{27}a^{12}-\frac{1}{27}a^{10}-\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{2}{27}a^{6}-\frac{2}{27}a^{5}+\frac{2}{27}a^{4}+\frac{2}{27}a^{2}+\frac{2}{27}a-\frac{2}{27}$, $\frac{1}{27}a^{23}+\frac{1}{27}a^{21}+\frac{1}{27}a^{20}-\frac{1}{27}a^{19}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{1}{27}a^{15}+\frac{1}{27}a^{13}+\frac{1}{27}a^{12}-\frac{1}{27}a^{11}-\frac{1}{27}a^{9}-\frac{1}{27}a^{8}-\frac{2}{27}a^{7}-\frac{2}{27}a^{5}-\frac{2}{27}a^{4}+\frac{2}{27}a^{3}+\frac{2}{27}a+\frac{2}{27}$, $\frac{1}{27}a^{24}-\frac{1}{9}a^{8}+\frac{2}{27}$, $\frac{1}{202797}a^{25}+\frac{254}{22533}a^{24}+\frac{220}{67599}a^{23}+\frac{442}{28971}a^{22}-\frac{1472}{28971}a^{21}+\frac{7313}{202797}a^{20}+\frac{3403}{67599}a^{19}+\frac{10271}{202797}a^{18}+\frac{9236}{202797}a^{17}+\frac{235}{202797}a^{16}+\frac{3856}{67599}a^{15}+\frac{11230}{202797}a^{14}+\frac{766}{5481}a^{13}-\frac{30865}{202797}a^{12}+\frac{6691}{67599}a^{11}-\frac{8143}{202797}a^{10}+\frac{19178}{202797}a^{9}+\frac{31636}{202797}a^{8}+\frac{32230}{67599}a^{7}-\frac{2675}{28971}a^{6}-\frac{20306}{202797}a^{5}-\frac{72379}{202797}a^{4}-\frac{20714}{67599}a^{3}+\frac{5501}{28971}a^{2}+\frac{14401}{28971}a+\frac{17494}{202797}$, $\frac{1}{34\cdots 29}a^{26}+\frac{43\cdots 75}{34\cdots 29}a^{25}-\frac{58\cdots 47}{34\cdots 29}a^{24}+\frac{69\cdots 05}{11\cdots 43}a^{23}-\frac{60\cdots 85}{16\cdots 49}a^{22}-\frac{46\cdots 71}{17\cdots 99}a^{21}+\frac{10\cdots 71}{37\cdots 81}a^{20}-\frac{44\cdots 24}{11\cdots 43}a^{19}+\frac{45\cdots 51}{16\cdots 49}a^{18}-\frac{20\cdots 54}{37\cdots 81}a^{17}+\frac{97\cdots 88}{37\cdots 81}a^{16}+\frac{98\cdots 24}{75\cdots 93}a^{15}+\frac{44\cdots 85}{75\cdots 93}a^{14}-\frac{11\cdots 87}{36\cdots 27}a^{13}-\frac{38\cdots 66}{53\cdots 83}a^{12}-\frac{29\cdots 86}{11\cdots 43}a^{11}+\frac{13\cdots 37}{11\cdots 43}a^{10}-\frac{15\cdots 67}{11\cdots 43}a^{9}+\frac{86\cdots 57}{11\cdots 43}a^{8}-\frac{20\cdots 90}{11\cdots 43}a^{7}-\frac{34\cdots 05}{11\cdots 43}a^{6}-\frac{17\cdots 36}{37\cdots 81}a^{5}+\frac{37\cdots 58}{12\cdots 27}a^{4}+\frac{22\cdots 72}{11\cdots 43}a^{3}+\frac{91\cdots 65}{48\cdots 47}a^{2}+\frac{22\cdots 67}{34\cdots 29}a+\frac{89\cdots 32}{34\cdots 29}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

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:		$13$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{16\cdots 70}{34\cdots 29}a^{26}+\frac{60\cdots 38}{34\cdots 29}a^{25}+\frac{35\cdots 07}{34\cdots 29}a^{24}-\frac{11\cdots 54}{11\cdots 43}a^{23}+\frac{38\cdots 62}{16\cdots 49}a^{22}-\frac{48\cdots 47}{51\cdots 97}a^{21}-\frac{31\cdots 41}{37\cdots 81}a^{20}+\frac{12\cdots 80}{16\cdots 49}a^{19}-\frac{18\cdots 11}{11\cdots 43}a^{18}+\frac{25\cdots 33}{12\cdots 27}a^{17}+\frac{40\cdots 74}{37\cdots 81}a^{16}-\frac{48\cdots 38}{75\cdots 93}a^{15}+\frac{50\cdots 87}{28\cdots 27}a^{14}-\frac{11\cdots 32}{36\cdots 27}a^{13}+\frac{18\cdots 73}{37\cdots 81}a^{12}-\frac{91\cdots 71}{16\cdots 49}a^{11}+\frac{64\cdots 97}{11\cdots 43}a^{10}-\frac{48\cdots 28}{11\cdots 43}a^{9}+\frac{16\cdots 64}{11\cdots 43}a^{8}+\frac{19\cdots 43}{11\cdots 43}a^{7}-\frac{40\cdots 02}{11\cdots 43}a^{6}+\frac{66\cdots 61}{12\cdots 27}a^{5}-\frac{57\cdots 22}{12\cdots 27}a^{4}+\frac{41\cdots 42}{11\cdots 43}a^{3}-\frac{87\cdots 31}{48\cdots 47}a^{2}+\frac{24\cdots 87}{34\cdots 29}a-\frac{97\cdots 70}{91\cdots 17}$, $\frac{14\cdots 23}{34\cdots 29}a^{26}-\frac{84\cdots 62}{34\cdots 29}a^{25}+\frac{43\cdots 28}{34\cdots 29}a^{24}-\frac{35\cdots 97}{11\cdots 43}a^{23}+\frac{66\cdots 70}{16\cdots 49}a^{22}-\frac{42\cdots 54}{51\cdots 97}a^{21}-\frac{41\cdots 74}{37\cdots 81}a^{20}+\frac{36\cdots 30}{11\cdots 43}a^{19}-\frac{19\cdots 84}{39\cdots 67}a^{18}+\frac{13\cdots 82}{53\cdots 83}a^{17}+\frac{35\cdots 19}{37\cdots 81}a^{16}-\frac{28\cdots 71}{75\cdots 93}a^{15}+\frac{66\cdots 03}{75\cdots 93}a^{14}-\frac{19\cdots 98}{12\cdots 09}a^{13}+\frac{31\cdots 20}{12\cdots 27}a^{12}-\frac{37\cdots 99}{11\cdots 43}a^{11}+\frac{44\cdots 16}{11\cdots 43}a^{10}-\frac{46\cdots 20}{11\cdots 43}a^{9}+\frac{44\cdots 27}{11\cdots 43}a^{8}-\frac{36\cdots 27}{11\cdots 43}a^{7}+\frac{26\cdots 86}{11\cdots 43}a^{6}-\frac{56\cdots 01}{37\cdots 81}a^{5}+\frac{30\cdots 89}{37\cdots 81}a^{4}-\frac{37\cdots 62}{11\cdots 43}a^{3}+\frac{49\cdots 97}{48\cdots 47}a^{2}-\frac{53\cdots 72}{34\cdots 29}a-\frac{42\cdots 09}{34\cdots 29}$, $\frac{19\cdots 14}{34\cdots 29}a^{26}-\frac{10\cdots 39}{34\cdots 29}a^{25}+\frac{51\cdots 24}{34\cdots 29}a^{24}-\frac{36\cdots 76}{11\cdots 43}a^{23}+\frac{43\cdots 80}{16\cdots 49}a^{22}+\frac{17\cdots 91}{51\cdots 97}a^{21}-\frac{70\cdots 41}{41\cdots 09}a^{20}+\frac{10\cdots 38}{30\cdots 39}a^{19}-\frac{57\cdots 35}{16\cdots 49}a^{18}-\frac{18\cdots 52}{12\cdots 27}a^{17}+\frac{20\cdots 99}{12\cdots 27}a^{16}-\frac{33\cdots 11}{75\cdots 93}a^{15}+\frac{64\cdots 82}{75\cdots 93}a^{14}-\frac{49\cdots 55}{36\cdots 27}a^{13}+\frac{98\cdots 65}{53\cdots 83}a^{12}-\frac{24\cdots 39}{11\cdots 43}a^{11}+\frac{24\cdots 06}{11\cdots 43}a^{10}-\frac{21\cdots 76}{11\cdots 43}a^{9}+\frac{16\cdots 27}{11\cdots 43}a^{8}-\frac{97\cdots 70}{11\cdots 43}a^{7}+\frac{41\cdots 50}{11\cdots 43}a^{6}-\frac{47\cdots 12}{12\cdots 27}a^{5}-\frac{38\cdots 35}{37\cdots 81}a^{4}+\frac{11\cdots 48}{11\cdots 43}a^{3}-\frac{21\cdots 82}{48\cdots 47}a^{2}+\frac{24\cdots 76}{34\cdots 29}a+\frac{38\cdots 44}{34\cdots 29}$, $\frac{33\cdots 06}{34\cdots 29}a^{26}-\frac{27\cdots 72}{48\cdots 47}a^{25}+\frac{14\cdots 12}{48\cdots 47}a^{24}-\frac{81\cdots 12}{11\cdots 43}a^{23}+\frac{14\cdots 80}{16\cdots 49}a^{22}-\frac{26\cdots 05}{51\cdots 97}a^{21}-\frac{10\cdots 64}{37\cdots 81}a^{20}+\frac{85\cdots 05}{11\cdots 43}a^{19}-\frac{12\cdots 85}{11\cdots 43}a^{18}+\frac{17\cdots 54}{37\cdots 81}a^{17}+\frac{92\cdots 20}{37\cdots 81}a^{16}-\frac{96\cdots 42}{10\cdots 99}a^{15}+\frac{15\cdots 19}{75\cdots 93}a^{14}-\frac{43\cdots 71}{12\cdots 09}a^{13}+\frac{67\cdots 34}{12\cdots 27}a^{12}-\frac{79\cdots 42}{11\cdots 43}a^{11}+\frac{91\cdots 41}{11\cdots 43}a^{10}-\frac{93\cdots 20}{11\cdots 43}a^{9}+\frac{84\cdots 29}{11\cdots 43}a^{8}-\frac{67\cdots 63}{11\cdots 43}a^{7}+\frac{47\cdots 17}{11\cdots 43}a^{6}-\frac{92\cdots 96}{37\cdots 81}a^{5}+\frac{63\cdots 79}{53\cdots 83}a^{4}-\frac{47\cdots 97}{11\cdots 43}a^{3}+\frac{46\cdots 95}{48\cdots 47}a^{2}-\frac{30\cdots 36}{34\cdots 29}a-\frac{37\cdots 90}{34\cdots 29}$, $\frac{34\cdots 99}{34\cdots 29}a^{26}-\frac{19\cdots 28}{34\cdots 29}a^{25}+\frac{10\cdots 50}{34\cdots 29}a^{24}-\frac{85\cdots 72}{11\cdots 43}a^{23}+\frac{58\cdots 38}{55\cdots 81}a^{22}-\frac{16\cdots 09}{51\cdots 97}a^{21}-\frac{15\cdots 03}{59\cdots 87}a^{20}+\frac{87\cdots 20}{11\cdots 43}a^{19}-\frac{13\cdots 16}{11\cdots 43}a^{18}+\frac{28\cdots 60}{37\cdots 81}a^{17}+\frac{26\cdots 65}{12\cdots 27}a^{16}-\frac{68\cdots 04}{75\cdots 93}a^{15}+\frac{16\cdots 80}{75\cdots 93}a^{14}-\frac{77\cdots 58}{19\cdots 43}a^{13}+\frac{23\cdots 62}{37\cdots 81}a^{12}-\frac{94\cdots 16}{11\cdots 43}a^{11}+\frac{11\cdots 99}{11\cdots 43}a^{10}-\frac{17\cdots 98}{16\cdots 49}a^{9}+\frac{16\cdots 24}{16\cdots 49}a^{8}-\frac{97\cdots 28}{11\cdots 43}a^{7}+\frac{72\cdots 39}{11\cdots 43}a^{6}-\frac{15\cdots 37}{37\cdots 81}a^{5}+\frac{29\cdots 94}{13\cdots 89}a^{4}-\frac{11\cdots 44}{11\cdots 43}a^{3}+\frac{15\cdots 50}{48\cdots 47}a^{2}-\frac{20\cdots 10}{34\cdots 29}a+\frac{50\cdots 50}{34\cdots 29}$, $\frac{49\cdots 98}{46\cdots 73}a^{26}-\frac{28\cdots 42}{46\cdots 73}a^{25}+\frac{14\cdots 52}{46\cdots 73}a^{24}-\frac{12\cdots 71}{15\cdots 91}a^{23}+\frac{24\cdots 39}{22\cdots 13}a^{22}-\frac{16\cdots 79}{51\cdots 97}a^{21}-\frac{13\cdots 28}{51\cdots 97}a^{20}+\frac{18\cdots 09}{22\cdots 13}a^{19}-\frac{19\cdots 50}{15\cdots 91}a^{18}+\frac{13\cdots 72}{17\cdots 99}a^{17}+\frac{11\cdots 83}{51\cdots 97}a^{16}-\frac{98\cdots 08}{10\cdots 41}a^{15}+\frac{33\cdots 87}{14\cdots 63}a^{14}-\frac{21\cdots 51}{50\cdots 99}a^{13}+\frac{33\cdots 18}{51\cdots 97}a^{12}-\frac{19\cdots 13}{22\cdots 13}a^{11}+\frac{16\cdots 50}{15\cdots 91}a^{10}-\frac{17\cdots 30}{15\cdots 91}a^{9}+\frac{16\cdots 64}{15\cdots 91}a^{8}-\frac{13\cdots 42}{15\cdots 91}a^{7}+\frac{10\cdots 35}{15\cdots 91}a^{6}-\frac{74\cdots 60}{17\cdots 99}a^{5}+\frac{40\cdots 57}{17\cdots 99}a^{4}-\frac{15\cdots 58}{15\cdots 91}a^{3}+\frac{22\cdots 96}{66\cdots 39}a^{2}-\frac{28\cdots 50}{46\cdots 73}a+\frac{73\cdots 98}{46\cdots 73}$, $\frac{21\cdots 24}{34\cdots 29}a^{26}-\frac{28\cdots 32}{91\cdots 17}a^{25}+\frac{55\cdots 33}{34\cdots 29}a^{24}-\frac{37\cdots 04}{11\cdots 43}a^{23}+\frac{58\cdots 32}{16\cdots 49}a^{22}+\frac{15\cdots 46}{17\cdots 99}a^{21}-\frac{54\cdots 00}{37\cdots 81}a^{20}+\frac{39\cdots 49}{11\cdots 43}a^{19}-\frac{52\cdots 16}{11\cdots 43}a^{18}+\frac{51\cdots 18}{53\cdots 83}a^{17}+\frac{49\cdots 60}{37\cdots 81}a^{16}-\frac{33\cdots 68}{75\cdots 93}a^{15}+\frac{72\cdots 86}{75\cdots 93}a^{14}-\frac{62\cdots 54}{36\cdots 27}a^{13}+\frac{85\cdots 97}{34\cdots 71}a^{12}-\frac{37\cdots 13}{11\cdots 43}a^{11}+\frac{43\cdots 07}{11\cdots 43}a^{10}-\frac{44\cdots 73}{11\cdots 43}a^{9}+\frac{41\cdots 29}{11\cdots 43}a^{8}-\frac{33\cdots 64}{11\cdots 43}a^{7}+\frac{24\cdots 21}{11\cdots 43}a^{6}-\frac{50\cdots 65}{37\cdots 81}a^{5}+\frac{25\cdots 68}{37\cdots 81}a^{4}-\frac{31\cdots 78}{11\cdots 43}a^{3}+\frac{38\cdots 70}{48\cdots 47}a^{2}-\frac{28\cdots 81}{34\cdots 29}a-\frac{10\cdots 91}{11\cdots 01}$, $\frac{38\cdots 67}{34\cdots 29}a^{26}-\frac{21\cdots 40}{34\cdots 29}a^{25}+\frac{10\cdots 95}{34\cdots 29}a^{24}-\frac{84\cdots 47}{11\cdots 43}a^{23}+\frac{15\cdots 22}{16\cdots 49}a^{22}-\frac{86\cdots 91}{17\cdots 99}a^{21}-\frac{46\cdots 66}{16\cdots 51}a^{20}+\frac{88\cdots 39}{11\cdots 43}a^{19}-\frac{12\cdots 13}{11\cdots 43}a^{18}+\frac{17\cdots 97}{37\cdots 81}a^{17}+\frac{32\cdots 25}{12\cdots 27}a^{16}-\frac{70\cdots 51}{75\cdots 93}a^{15}+\frac{16\cdots 67}{75\cdots 93}a^{14}-\frac{20\cdots 22}{52\cdots 61}a^{13}+\frac{21\cdots 25}{37\cdots 81}a^{12}-\frac{85\cdots 00}{11\cdots 43}a^{11}+\frac{10\cdots 13}{11\cdots 43}a^{10}-\frac{14\cdots 68}{16\cdots 49}a^{9}+\frac{13\cdots 36}{16\cdots 49}a^{8}-\frac{78\cdots 30}{11\cdots 43}a^{7}+\frac{19\cdots 96}{39\cdots 67}a^{6}-\frac{11\cdots 83}{37\cdots 81}a^{5}+\frac{58\cdots 14}{37\cdots 81}a^{4}-\frac{69\cdots 41}{11\cdots 43}a^{3}+\frac{81\cdots 30}{48\cdots 47}a^{2}-\frac{14\cdots 44}{91\cdots 17}a-\frac{38\cdots 68}{34\cdots 29}$, $\frac{23\cdots 13}{46\cdots 73}a^{26}-\frac{13\cdots 37}{46\cdots 73}a^{25}+\frac{66\cdots 59}{46\cdots 73}a^{24}-\frac{76\cdots 16}{22\cdots 13}a^{23}+\frac{99\cdots 84}{22\cdots 13}a^{22}-\frac{31\cdots 68}{57\cdots 33}a^{21}-\frac{24\cdots 16}{19\cdots 11}a^{20}+\frac{55\cdots 23}{15\cdots 91}a^{19}-\frac{82\cdots 01}{15\cdots 91}a^{18}+\frac{13\cdots 04}{51\cdots 97}a^{17}+\frac{19\cdots 54}{17\cdots 99}a^{16}-\frac{44\cdots 86}{10\cdots 41}a^{15}+\frac{10\cdots 66}{10\cdots 41}a^{14}-\frac{89\cdots 51}{50\cdots 99}a^{13}+\frac{13\cdots 71}{51\cdots 97}a^{12}-\frac{15\cdots 25}{41\cdots 43}a^{11}+\frac{65\cdots 36}{15\cdots 91}a^{10}-\frac{68\cdots 82}{15\cdots 91}a^{9}+\frac{64\cdots 77}{15\cdots 91}a^{8}-\frac{76\cdots 81}{22\cdots 13}a^{7}+\frac{39\cdots 45}{15\cdots 91}a^{6}-\frac{83\cdots 93}{51\cdots 97}a^{5}+\frac{44\cdots 55}{51\cdots 97}a^{4}-\frac{82\cdots 30}{22\cdots 13}a^{3}+\frac{80\cdots 74}{66\cdots 39}a^{2}-\frac{10\cdots 41}{46\cdots 73}a+\frac{10\cdots 64}{66\cdots 39}$, $\frac{10\cdots 69}{34\cdots 29}a^{26}-\frac{57\cdots 60}{34\cdots 29}a^{25}+\frac{29\cdots 26}{34\cdots 29}a^{24}-\frac{23\cdots 83}{11\cdots 43}a^{23}+\frac{41\cdots 18}{16\cdots 49}a^{22}-\frac{15\cdots 59}{51\cdots 97}a^{21}-\frac{28\cdots 71}{37\cdots 81}a^{20}+\frac{23\cdots 92}{11\cdots 43}a^{19}-\frac{35\cdots 37}{11\cdots 43}a^{18}+\frac{53\cdots 91}{37\cdots 81}a^{17}+\frac{35\cdots 55}{53\cdots 83}a^{16}-\frac{19\cdots 32}{75\cdots 93}a^{15}+\frac{43\cdots 67}{75\cdots 93}a^{14}-\frac{12\cdots 70}{12\cdots 09}a^{13}+\frac{67\cdots 09}{41\cdots 09}a^{12}-\frac{83\cdots 11}{39\cdots 67}a^{11}+\frac{41\cdots 39}{16\cdots 49}a^{10}-\frac{30\cdots 86}{11\cdots 43}a^{9}+\frac{28\cdots 07}{11\cdots 43}a^{8}-\frac{24\cdots 07}{11\cdots 43}a^{7}+\frac{18\cdots 85}{11\cdots 43}a^{6}-\frac{55\cdots 95}{53\cdots 83}a^{5}+\frac{21\cdots 85}{37\cdots 81}a^{4}-\frac{29\cdots 54}{11\cdots 43}a^{3}+\frac{44\cdots 88}{48\cdots 47}a^{2}-\frac{69\cdots 66}{34\cdots 29}a+\frac{98\cdots 04}{34\cdots 29}$, $\frac{10\cdots 67}{32\cdots 67}a^{26}-\frac{43\cdots 09}{22\cdots 69}a^{25}+\frac{22\cdots 50}{22\cdots 69}a^{24}-\frac{18\cdots 23}{75\cdots 23}a^{23}+\frac{37\cdots 90}{10\cdots 89}a^{22}-\frac{589848865894903}{490267873736631}a^{21}-\frac{20\cdots 55}{25\cdots 41}a^{20}+\frac{19\cdots 21}{75\cdots 23}a^{19}-\frac{30\cdots 50}{75\cdots 23}a^{18}+\frac{72\cdots 63}{27\cdots 49}a^{17}+\frac{16\cdots 93}{25\cdots 41}a^{16}-\frac{22\cdots 90}{75\cdots 23}a^{15}+\frac{53\cdots 21}{75\cdots 23}a^{14}-\frac{33\cdots 99}{25\cdots 41}a^{13}+\frac{51\cdots 50}{25\cdots 41}a^{12}-\frac{56\cdots 24}{20\cdots 79}a^{11}+\frac{25\cdots 58}{75\cdots 23}a^{10}-\frac{26\cdots 98}{75\cdots 23}a^{9}+\frac{25\cdots 96}{75\cdots 23}a^{8}-\frac{21\cdots 39}{75\cdots 23}a^{7}+\frac{79\cdots 22}{37\cdots 41}a^{6}-\frac{38\cdots 12}{27\cdots 49}a^{5}+\frac{61\cdots 70}{83\cdots 47}a^{4}-\frac{23\cdots 63}{75\cdots 23}a^{3}+\frac{30\cdots 04}{32\cdots 67}a^{2}-\frac{48\cdots 86}{32\cdots 67}a+\frac{39\cdots 52}{22\cdots 69}$, $a$, $\frac{62\cdots 01}{34\cdots 29}a^{26}-\frac{37\cdots 57}{34\cdots 29}a^{25}+\frac{19\cdots 42}{34\cdots 29}a^{24}-\frac{16\cdots 66}{11\cdots 43}a^{23}+\frac{30\cdots 73}{16\cdots 49}a^{22}-\frac{14\cdots 76}{51\cdots 97}a^{21}-\frac{64\cdots 83}{12\cdots 27}a^{20}+\frac{16\cdots 94}{11\cdots 43}a^{19}-\frac{35\cdots 75}{16\cdots 49}a^{18}+\frac{13\cdots 09}{12\cdots 27}a^{17}+\frac{55\cdots 96}{12\cdots 27}a^{16}-\frac{12\cdots 76}{75\cdots 93}a^{15}+\frac{29\cdots 85}{75\cdots 93}a^{14}-\frac{26\cdots 04}{36\cdots 27}a^{13}+\frac{58\cdots 03}{53\cdots 83}a^{12}-\frac{16\cdots 00}{11\cdots 43}a^{11}+\frac{19\cdots 32}{11\cdots 43}a^{10}-\frac{20\cdots 35}{11\cdots 43}a^{9}+\frac{18\cdots 03}{11\cdots 43}a^{8}-\frac{15\cdots 68}{11\cdots 43}a^{7}+\frac{10\cdots 00}{11\cdots 43}a^{6}-\frac{74\cdots 64}{12\cdots 27}a^{5}+\frac{11\cdots 41}{37\cdots 81}a^{4}-\frac{13\cdots 89}{11\cdots 43}a^{3}+\frac{15\cdots 95}{48\cdots 47}a^{2}-\frac{14\cdots 67}{34\cdots 29}a+\frac{17\cdots 86}{34\cdots 29}$ (assuming GRH)
Regulator:		\( 71844122831.24568 \) (assuming GRH)
Unit signature rank:		\( 1 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 71844122831.24568 \cdot 1}{2\cdot\sqrt{1543319746516623033280478216838436483146079}}\cr\approx \mathstrut & 1.37562981139276 \end{aligned}\] (assuming GRH)

Galois group

$D_{27}$ (as 27T8):

A solvable group of order 54

The 15 conjugacy class representatives for $D_{27}$

Character table for $D_{27}$

Intermediate fields

3.1.1759.1, 9.1.9573337234561.1

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

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.9.0.1}{9} }^{3}$	${\href{/padicField/3.2.0.1}{2} }^{13}{,}\,{\href{/padicField/3.1.0.1}{1} }$	$27$	${\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} }$	$27$	$27$	$27$	${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }$	$27$	${\href{/padicField/29.2.0.1}{2} }^{13}{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.9.0.1}{9} }^{3}$	${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }$	$27$	$27$	$27$	$27$	${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(1759\)	$\Q_{1759}$	$x$	$1$	$1$	$0$	Trivial	$$[\ ]$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*^54	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.1759.2t1.a.a	$1$	$ 1759 $	\(\Q(\sqrt{-1759}) \)	$C_2$ (as 2T1)	$1$	$-1$
*^54	2.1759.3t2.a.a	$2$	$ 1759 $	3.1.1759.1	$S_3$ (as 3T2)	$1$	$0$
*^54	2.1759.9t3.a.a	$2$	$ 1759 $	9.1.9573337234561.1	$D_{9}$ (as 9T3)	$1$	$0$
*^54	2.1759.9t3.a.b	$2$	$ 1759 $	9.1.9573337234561.1	$D_{9}$ (as 9T3)	$1$	$0$
*^54	2.1759.9t3.a.c	$2$	$ 1759 $	9.1.9573337234561.1	$D_{9}$ (as 9T3)	$1$	$0$
*^54	2.1759.27t8.a.h	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.e	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.d	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.g	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.a	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.b	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.f	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.c	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$
*^54	2.1759.27t8.a.i	$2$	$ 1759 $	27.1.1543319746516623033280478216838436483146079.1	$D_{27}$ (as 27T8)	$1$	$0$

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