Normalized defining polynomial
\( x^{18} - 6 x^{17} - 18 x^{16} + 141 x^{15} + 108 x^{14} - 1227 x^{13} - 438 x^{12} + 5220 x^{11} + \cdots - 16 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $[18, 0]$ |
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| Discriminant: |
\(16096057926792443193781494140625\)
\(\medspace = 3^{26}\cdot 5^{12}\cdot 11^{10}\)
|
| |
| Root discriminant: | \(54.16\) |
| |
| Galois root discriminant: | $3^{85/54}5^{3/4}11^{2/3}\approx 93.22094273535318$ | ||
| Ramified primes: |
\(3\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{37\cdots 68}a^{17}+\frac{71\cdots 83}{94\cdots 92}a^{16}+\frac{10\cdots 91}{18\cdots 84}a^{15}+\frac{44\cdots 45}{37\cdots 68}a^{14}-\frac{16\cdots 61}{18\cdots 84}a^{13}-\frac{36\cdots 91}{37\cdots 68}a^{12}-\frac{23\cdots 25}{94\cdots 92}a^{11}-\frac{43\cdots 81}{94\cdots 92}a^{10}+\frac{31\cdots 15}{94\cdots 92}a^{9}+\frac{51\cdots 85}{18\cdots 84}a^{8}+\frac{40\cdots 49}{37\cdots 68}a^{7}+\frac{10\cdots 09}{37\cdots 68}a^{6}+\frac{16\cdots 53}{37\cdots 68}a^{5}-\frac{41\cdots 39}{18\cdots 84}a^{4}-\frac{39\cdots 67}{37\cdots 68}a^{3}+\frac{49\cdots 17}{94\cdots 92}a^{2}+\frac{96\cdots 27}{23\cdots 48}a-\frac{12\cdots 51}{47\cdots 96}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $17$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{210228674823}{4495183640512}a^{17}-\frac{333656980311}{1123795910128}a^{16}-\frac{1568915003739}{2247591820256}a^{15}+\frac{29630734176591}{4495183640512}a^{14}+\frac{4669835956473}{2247591820256}a^{13}-\frac{236079774883209}{4495183640512}a^{12}+\frac{1386225793605}{1123795910128}a^{11}+\frac{221926009061193}{1123795910128}a^{10}+\frac{12345341245681}{1123795910128}a^{9}-\frac{775022287054389}{2247591820256}a^{8}-\frac{349422813396393}{4495183640512}a^{7}+\frac{942031494560691}{4495183640512}a^{6}+\frac{261278007000147}{4495183640512}a^{5}-\frac{3750313132797}{2247591820256}a^{4}+\frac{22479559006527}{4495183640512}a^{3}-\frac{23630818341045}{1123795910128}a^{2}-\frac{1779153925239}{280948977532}a+\frac{126818574567}{561897955064}$, $\frac{17\cdots 53}{37\cdots 68}a^{17}-\frac{12\cdots 49}{94\cdots 92}a^{16}-\frac{32\cdots 13}{18\cdots 84}a^{15}+\frac{14\cdots 13}{37\cdots 68}a^{14}+\frac{47\cdots 63}{18\cdots 84}a^{13}-\frac{15\cdots 19}{37\cdots 68}a^{12}-\frac{17\cdots 77}{94\cdots 92}a^{11}+\frac{17\cdots 91}{94\cdots 92}a^{10}+\frac{70\cdots 31}{94\cdots 92}a^{9}-\frac{48\cdots 99}{18\cdots 84}a^{8}-\frac{56\cdots 71}{37\cdots 68}a^{7}-\frac{63\cdots 03}{37\cdots 68}a^{6}+\frac{44\cdots 77}{37\cdots 68}a^{5}+\frac{71\cdots 17}{18\cdots 84}a^{4}-\frac{12\cdots 63}{37\cdots 68}a^{3}-\frac{11\cdots 47}{94\cdots 92}a^{2}+\frac{34\cdots 49}{23\cdots 48}a+\frac{16\cdots 05}{47\cdots 96}$, $\frac{48\cdots 33}{37\cdots 68}a^{17}-\frac{91\cdots 93}{94\cdots 92}a^{16}-\frac{19\cdots 89}{18\cdots 84}a^{15}+\frac{77\cdots 37}{37\cdots 68}a^{14}-\frac{27\cdots 17}{18\cdots 84}a^{13}-\frac{58\cdots 59}{37\cdots 68}a^{12}+\frac{15\cdots 39}{94\cdots 92}a^{11}+\frac{53\cdots 47}{94\cdots 92}a^{10}-\frac{55\cdots 81}{94\cdots 92}a^{9}-\frac{20\cdots 43}{18\cdots 84}a^{8}+\frac{34\cdots 09}{37\cdots 68}a^{7}+\frac{40\cdots 61}{37\cdots 68}a^{6}-\frac{22\cdots 35}{37\cdots 68}a^{5}-\frac{92\cdots 59}{18\cdots 84}a^{4}+\frac{54\cdots 49}{37\cdots 68}a^{3}+\frac{84\cdots 09}{94\cdots 92}a^{2}-\frac{13\cdots 49}{23\cdots 48}a-\frac{11\cdots 91}{47\cdots 96}$, $\frac{22\cdots 39}{37\cdots 68}a^{17}-\frac{53\cdots 19}{94\cdots 92}a^{16}+\frac{51\cdots 33}{18\cdots 84}a^{15}+\frac{44\cdots 63}{37\cdots 68}a^{14}-\frac{44\cdots 03}{18\cdots 84}a^{13}-\frac{32\cdots 13}{37\cdots 68}a^{12}+\frac{20\cdots 21}{94\cdots 92}a^{11}+\frac{30\cdots 01}{94\cdots 92}a^{10}-\frac{80\cdots 43}{94\cdots 92}a^{9}-\frac{14\cdots 85}{18\cdots 84}a^{8}+\frac{58\cdots 51}{37\cdots 68}a^{7}+\frac{43\cdots 87}{37\cdots 68}a^{6}-\frac{43\cdots 93}{37\cdots 68}a^{5}-\frac{14\cdots 69}{18\cdots 84}a^{4}+\frac{10\cdots 79}{37\cdots 68}a^{3}+\frac{16\cdots 79}{94\cdots 92}a^{2}-\frac{19\cdots 43}{23\cdots 48}a-\frac{25\cdots 93}{47\cdots 96}$, $\frac{76\cdots 09}{18\cdots 84}a^{17}-\frac{12\cdots 97}{47\cdots 96}a^{16}-\frac{57\cdots 89}{94\cdots 92}a^{15}+\frac{10\cdots 85}{18\cdots 84}a^{14}+\frac{16\cdots 55}{94\cdots 92}a^{13}-\frac{83\cdots 75}{18\cdots 84}a^{12}+\frac{47\cdots 95}{47\cdots 96}a^{11}+\frac{73\cdots 99}{47\cdots 96}a^{10}+\frac{52\cdots 35}{47\cdots 96}a^{9}-\frac{21\cdots 15}{94\cdots 92}a^{8}-\frac{12\cdots 47}{18\cdots 84}a^{7}+\frac{80\cdots 33}{18\cdots 84}a^{6}+\frac{51\cdots 93}{18\cdots 84}a^{5}+\frac{11\cdots 69}{94\cdots 92}a^{4}+\frac{73\cdots 25}{18\cdots 84}a^{3}-\frac{26\cdots 95}{47\cdots 96}a^{2}-\frac{10\cdots 90}{58\cdots 87}a+\frac{46\cdots 49}{23\cdots 48}$, $\frac{64\cdots 87}{37\cdots 68}a^{17}-\frac{10\cdots 75}{94\cdots 92}a^{16}-\frac{47\cdots 79}{18\cdots 84}a^{15}+\frac{94\cdots 03}{37\cdots 68}a^{14}+\frac{11\cdots 37}{18\cdots 84}a^{13}-\frac{78\cdots 29}{37\cdots 68}a^{12}+\frac{22\cdots 73}{94\cdots 92}a^{11}+\frac{79\cdots 33}{94\cdots 92}a^{10}-\frac{49\cdots 55}{94\cdots 92}a^{9}-\frac{32\cdots 25}{18\cdots 84}a^{8}-\frac{54\cdots 09}{37\cdots 68}a^{7}+\frac{64\cdots 99}{37\cdots 68}a^{6}+\frac{10\cdots 59}{37\cdots 68}a^{5}-\frac{14\cdots 45}{18\cdots 84}a^{4}-\frac{51\cdots 37}{37\cdots 68}a^{3}+\frac{11\cdots 99}{94\cdots 92}a^{2}+\frac{50\cdots 19}{23\cdots 48}a-\frac{12\cdots 89}{47\cdots 96}$, $\frac{58\cdots 03}{37\cdots 68}a^{17}-\frac{15\cdots 83}{94\cdots 92}a^{16}+\frac{30\cdots 81}{18\cdots 84}a^{15}+\frac{13\cdots 67}{37\cdots 68}a^{14}-\frac{16\cdots 67}{18\cdots 84}a^{13}-\frac{11\cdots 85}{37\cdots 68}a^{12}+\frac{76\cdots 61}{94\cdots 92}a^{11}+\frac{12\cdots 01}{94\cdots 92}a^{10}-\frac{30\cdots 63}{94\cdots 92}a^{9}-\frac{74\cdots 29}{18\cdots 84}a^{8}+\frac{22\cdots 95}{37\cdots 68}a^{7}+\frac{26\cdots 31}{37\cdots 68}a^{6}-\frac{14\cdots 49}{37\cdots 68}a^{5}-\frac{10\cdots 37}{18\cdots 84}a^{4}+\frac{20\cdots 19}{37\cdots 68}a^{3}+\frac{13\cdots 39}{94\cdots 92}a^{2}+\frac{27\cdots 69}{23\cdots 48}a-\frac{32\cdots 97}{47\cdots 96}$, $\frac{23\cdots 01}{37\cdots 68}a^{17}-\frac{32\cdots 69}{94\cdots 92}a^{16}-\frac{26\cdots 33}{18\cdots 84}a^{15}+\frac{32\cdots 61}{37\cdots 68}a^{14}+\frac{23\cdots 39}{18\cdots 84}a^{13}-\frac{29\cdots 51}{37\cdots 68}a^{12}-\frac{65\cdots 05}{94\cdots 92}a^{11}+\frac{32\cdots 11}{94\cdots 92}a^{10}+\frac{26\cdots 59}{94\cdots 92}a^{9}-\frac{14\cdots 63}{18\cdots 84}a^{8}-\frac{23\cdots 35}{37\cdots 68}a^{7}+\frac{28\cdots 45}{37\cdots 68}a^{6}+\frac{23\cdots 21}{37\cdots 68}a^{5}-\frac{63\cdots 11}{18\cdots 84}a^{4}-\frac{94\cdots 75}{37\cdots 68}a^{3}+\frac{64\cdots 01}{94\cdots 92}a^{2}+\frac{76\cdots 41}{23\cdots 48}a-\frac{26\cdots 39}{47\cdots 96}$, $\frac{14\cdots 61}{18\cdots 84}a^{17}-\frac{24\cdots 25}{47\cdots 96}a^{16}-\frac{97\cdots 05}{94\cdots 92}a^{15}+\frac{21\cdots 57}{18\cdots 84}a^{14}+\frac{47\cdots 75}{94\cdots 92}a^{13}-\frac{17\cdots 87}{18\cdots 84}a^{12}+\frac{13\cdots 87}{47\cdots 96}a^{11}+\frac{17\cdots 31}{47\cdots 96}a^{10}-\frac{43\cdots 05}{47\cdots 96}a^{9}-\frac{71\cdots 55}{94\cdots 92}a^{8}+\frac{13\cdots 57}{18\cdots 84}a^{7}+\frac{14\cdots 69}{18\cdots 84}a^{6}-\frac{43\cdots 39}{18\cdots 84}a^{5}-\frac{33\cdots 55}{94\cdots 92}a^{4}-\frac{38\cdots 55}{18\cdots 84}a^{3}+\frac{33\cdots 77}{47\cdots 96}a^{2}+\frac{85\cdots 97}{11\cdots 74}a-\frac{67\cdots 11}{23\cdots 48}$, $\frac{40\cdots 27}{18\cdots 84}a^{17}-\frac{12\cdots 59}{47\cdots 96}a^{16}-\frac{97\cdots 51}{94\cdots 92}a^{15}+\frac{25\cdots 99}{18\cdots 84}a^{14}+\frac{15\cdots 61}{94\cdots 92}a^{13}-\frac{35\cdots 41}{18\cdots 84}a^{12}-\frac{60\cdots 87}{47\cdots 96}a^{11}+\frac{41\cdots 13}{47\cdots 96}a^{10}+\frac{23\cdots 57}{47\cdots 96}a^{9}-\frac{10\cdots 53}{94\cdots 92}a^{8}-\frac{18\cdots 61}{18\cdots 84}a^{7}-\frac{33\cdots 89}{18\cdots 84}a^{6}+\frac{13\cdots 99}{18\cdots 84}a^{5}+\frac{28\cdots 11}{94\cdots 92}a^{4}-\frac{32\cdots 73}{18\cdots 84}a^{3}-\frac{53\cdots 41}{47\cdots 96}a^{2}-\frac{29\cdots 65}{58\cdots 87}a+\frac{11\cdots 71}{23\cdots 48}$, $\frac{16\cdots 63}{37\cdots 68}a^{17}-\frac{25\cdots 07}{94\cdots 92}a^{16}-\frac{12\cdots 31}{18\cdots 84}a^{15}+\frac{22\cdots 91}{37\cdots 68}a^{14}+\frac{53\cdots 69}{18\cdots 84}a^{13}-\frac{16\cdots 13}{37\cdots 68}a^{12}-\frac{77\cdots 03}{94\cdots 92}a^{11}+\frac{13\cdots 21}{94\cdots 92}a^{10}+\frac{50\cdots 93}{94\cdots 92}a^{9}-\frac{30\cdots 57}{18\cdots 84}a^{8}-\frac{54\cdots 85}{37\cdots 68}a^{7}-\frac{46\cdots 41}{37\cdots 68}a^{6}+\frac{19\cdots 55}{37\cdots 68}a^{5}+\frac{48\cdots 39}{18\cdots 84}a^{4}+\frac{22\cdots 43}{37\cdots 68}a^{3}-\frac{86\cdots 53}{94\cdots 92}a^{2}-\frac{56\cdots 57}{23\cdots 48}a+\frac{20\cdots 63}{47\cdots 96}$, $\frac{12\cdots 07}{18\cdots 84}a^{17}-\frac{17\cdots 29}{47\cdots 96}a^{16}-\frac{12\cdots 87}{94\cdots 92}a^{15}+\frac{16\cdots 75}{18\cdots 84}a^{14}+\frac{94\cdots 57}{94\cdots 92}a^{13}-\frac{14\cdots 77}{18\cdots 84}a^{12}-\frac{25\cdots 57}{47\cdots 96}a^{11}+\frac{14\cdots 41}{47\cdots 96}a^{10}+\frac{10\cdots 01}{47\cdots 96}a^{9}-\frac{53\cdots 05}{94\cdots 92}a^{8}-\frac{99\cdots 85}{18\cdots 84}a^{7}+\frac{69\cdots 31}{18\cdots 84}a^{6}+\frac{84\cdots 91}{18\cdots 84}a^{5}-\frac{29\cdots 73}{94\cdots 92}a^{4}-\frac{23\cdots 09}{18\cdots 84}a^{3}-\frac{10\cdots 83}{47\cdots 96}a^{2}+\frac{55\cdots 83}{11\cdots 74}a+\frac{99\cdots 27}{23\cdots 48}$, $\frac{39\cdots 49}{18\cdots 84}a^{17}-\frac{73\cdots 23}{47\cdots 96}a^{16}-\frac{15\cdots 97}{94\cdots 92}a^{15}+\frac{63\cdots 37}{18\cdots 84}a^{14}-\frac{21\cdots 45}{94\cdots 92}a^{13}-\frac{47\cdots 71}{18\cdots 84}a^{12}+\frac{12\cdots 73}{47\cdots 96}a^{11}+\frac{44\cdots 39}{47\cdots 96}a^{10}-\frac{45\cdots 65}{47\cdots 96}a^{9}-\frac{18\cdots 87}{94\cdots 92}a^{8}+\frac{28\cdots 29}{18\cdots 84}a^{7}+\frac{38\cdots 65}{18\cdots 84}a^{6}-\frac{19\cdots 07}{18\cdots 84}a^{5}-\frac{10\cdots 03}{94\cdots 92}a^{4}+\frac{47\cdots 49}{18\cdots 84}a^{3}+\frac{10\cdots 23}{47\cdots 96}a^{2}-\frac{78\cdots 61}{11\cdots 74}a-\frac{22\cdots 59}{23\cdots 48}$, $\frac{125920666707489}{37\cdots 68}a^{17}+\frac{15\cdots 79}{94\cdots 92}a^{16}-\frac{21\cdots 77}{18\cdots 84}a^{15}-\frac{11\cdots 43}{37\cdots 68}a^{14}+\frac{49\cdots 39}{18\cdots 84}a^{13}+\frac{72\cdots 77}{37\cdots 68}a^{12}-\frac{22\cdots 09}{94\cdots 92}a^{11}-\frac{75\cdots 81}{94\cdots 92}a^{10}+\frac{10\cdots 59}{94\cdots 92}a^{9}+\frac{64\cdots 45}{18\cdots 84}a^{8}-\frac{91\cdots 51}{37\cdots 68}a^{7}-\frac{35\cdots 11}{37\cdots 68}a^{6}+\frac{97\cdots 97}{37\cdots 68}a^{5}+\frac{21\cdots 53}{18\cdots 84}a^{4}-\frac{42\cdots 07}{37\cdots 68}a^{3}-\frac{46\cdots 11}{94\cdots 92}a^{2}+\frac{38\cdots 81}{23\cdots 48}a+\frac{33\cdots 57}{47\cdots 96}$, $\frac{34\cdots 39}{37\cdots 68}a^{17}-\frac{40\cdots 55}{94\cdots 92}a^{16}-\frac{45\cdots 71}{18\cdots 84}a^{15}+\frac{40\cdots 63}{37\cdots 68}a^{14}+\frac{50\cdots 53}{18\cdots 84}a^{13}-\frac{36\cdots 53}{37\cdots 68}a^{12}-\frac{17\cdots 35}{94\cdots 92}a^{11}+\frac{37\cdots 65}{94\cdots 92}a^{10}+\frac{70\cdots 13}{94\cdots 92}a^{9}-\frac{12\cdots 61}{18\cdots 84}a^{8}-\frac{58\cdots 65}{37\cdots 68}a^{7}+\frac{84\cdots 27}{37\cdots 68}a^{6}+\frac{47\cdots 27}{37\cdots 68}a^{5}+\frac{40\cdots 47}{18\cdots 84}a^{4}-\frac{12\cdots 65}{37\cdots 68}a^{3}-\frac{92\cdots 29}{94\cdots 92}a^{2}+\frac{34\cdots 17}{23\cdots 48}a+\frac{21\cdots 79}{47\cdots 96}$, $\frac{31\cdots 95}{37\cdots 68}a^{17}-\frac{50\cdots 51}{94\cdots 92}a^{16}-\frac{22\cdots 51}{18\cdots 84}a^{15}+\frac{45\cdots 07}{37\cdots 68}a^{14}+\frac{57\cdots 73}{18\cdots 84}a^{13}-\frac{37\cdots 53}{37\cdots 68}a^{12}+\frac{85\cdots 17}{94\cdots 92}a^{11}+\frac{36\cdots 93}{94\cdots 92}a^{10}-\frac{51\cdots 43}{94\cdots 92}a^{9}-\frac{14\cdots 17}{18\cdots 84}a^{8}-\frac{52\cdots 25}{37\cdots 68}a^{7}+\frac{26\cdots 43}{37\cdots 68}a^{6}+\frac{91\cdots 83}{37\cdots 68}a^{5}-\frac{51\cdots 81}{18\cdots 84}a^{4}-\frac{55\cdots 25}{37\cdots 68}a^{3}+\frac{28\cdots 31}{94\cdots 92}a^{2}+\frac{74\cdots 07}{23\cdots 48}a+\frac{13\cdots 51}{47\cdots 96}$, $\frac{14\cdots 29}{94\cdots 92}a^{17}-\frac{27\cdots 67}{23\cdots 48}a^{16}-\frac{60\cdots 85}{47\cdots 96}a^{15}+\frac{23\cdots 17}{94\cdots 92}a^{14}-\frac{76\cdots 13}{47\cdots 96}a^{13}-\frac{17\cdots 35}{94\cdots 92}a^{12}+\frac{45\cdots 57}{23\cdots 48}a^{11}+\frac{16\cdots 27}{23\cdots 48}a^{10}-\frac{16\cdots 45}{23\cdots 48}a^{9}-\frac{67\cdots 99}{47\cdots 96}a^{8}+\frac{10\cdots 01}{94\cdots 92}a^{7}+\frac{14\cdots 85}{94\cdots 92}a^{6}-\frac{67\cdots 07}{94\cdots 92}a^{5}-\frac{37\cdots 83}{47\cdots 96}a^{4}+\frac{15\cdots 65}{94\cdots 92}a^{3}+\frac{40\cdots 93}{23\cdots 48}a^{2}-\frac{502934546769420}{58\cdots 87}a-\frac{74\cdots 37}{11\cdots 74}$
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| Regulator: | \( 14711932756.8 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 14711932756.8 \cdot 1}{2\cdot\sqrt{16096057926792443193781494140625}}\cr\approx \mathstrut & 0.480639981584 \end{aligned}\] (assuming GRH)
Galois group
$\He_3:C_4$ (as 18T49):
| A solvable group of order 108 |
| The 14 conjugacy class representatives for $\He_3:C_4$ |
| Character table for $\He_3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 6.6.55130625.1 |
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 |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | 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 | ${\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.2.0.1}{2} }^{3}$ | R | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | R | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.9.26b15.1 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4176 x^{14} + 14112 x^{13} + 38304 x^{12} + 85251 x^{11} + 157572 x^{10} + 243605 x^{9} + 315879 x^{8} + 343158 x^{7} + 310740 x^{6} + 232260 x^{5} + 140952 x^{4} + 67560 x^{3} + 24336 x^{2} + 5952 x + 755$ | $9$ | $2$ | $26$ | not computed | not computed |
|
\(5\)
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 5.3.4.9a1.1 | $x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 491 x^{2} + 324 x + 81$ | $4$ | $3$ | $9$ | $C_{12}$ | $$[\ ]_{4}^{3}$$ | |
|
\(11\)
| 11.3.1.0a1.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 11.2.3.4a1.1 | $x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 95 x + 8$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ | |
| 11.3.3.6a1.1 | $x^{9} + 6 x^{7} + 27 x^{6} + 12 x^{5} + 108 x^{4} + 251 x^{3} + 108 x^{2} + 486 x + 740$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $$[\ ]_{3}^{6}$$ |