Normalized defining polynomial
\( x^{18} - 9 x^{17} + 39 x^{16} + 45 x^{15} - 876 x^{14} + 3330 x^{13} - 20 x^{12} - 18045 x^{11} + \cdots + 451250391 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $[0, 9]$ |
| |
| Discriminant: |
\(-7168555859200162677766007466667663950714918483\)
\(\medspace = -\,3^{33}\cdot 17^{12}\cdot 19^{12}\)
|
| |
| Root discriminant: | \(352.80\) |
| |
| Galois root discriminant: | $3^{13/6}17^{2/3}19^{2/3}\approx 508.82026534963563$ | ||
| Ramified primes: |
\(3\), \(17\), \(19\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_3^2$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{10547258813}a^{15}+\frac{2324979178}{10547258813}a^{14}+\frac{3578809376}{10547258813}a^{13}+\frac{2730171935}{10547258813}a^{12}+\frac{4110903137}{10547258813}a^{11}-\frac{1448458}{88632427}a^{10}-\frac{774024388}{10547258813}a^{9}-\frac{690084}{3132539}a^{8}+\frac{254772118}{620426989}a^{7}-\frac{2440695069}{10547258813}a^{6}+\frac{242869400}{811327601}a^{5}+\frac{1027737824}{10547258813}a^{4}-\frac{1043064941}{10547258813}a^{3}-\frac{5229406996}{10547258813}a^{2}-\frac{3073689043}{10547258813}a-\frac{4705476940}{10547258813}$, $\frac{1}{44\cdots 13}a^{16}-\frac{16\cdots 90}{44\cdots 13}a^{15}+\frac{72\cdots 46}{44\cdots 13}a^{14}+\frac{12\cdots 59}{26\cdots 89}a^{13}-\frac{13\cdots 80}{44\cdots 13}a^{12}-\frac{19\cdots 99}{44\cdots 13}a^{11}-\frac{17\cdots 28}{44\cdots 13}a^{10}+\frac{18\cdots 67}{44\cdots 13}a^{9}-\frac{21\cdots 45}{44\cdots 13}a^{8}+\frac{21\cdots 60}{44\cdots 13}a^{7}+\frac{10\cdots 26}{44\cdots 13}a^{6}-\frac{89\cdots 93}{44\cdots 13}a^{5}+\frac{23\cdots 67}{44\cdots 13}a^{4}+\frac{18\cdots 97}{63\cdots 59}a^{3}-\frac{43\cdots 05}{44\cdots 13}a^{2}+\frac{12\cdots 03}{44\cdots 13}a-\frac{44\cdots 40}{44\cdots 13}$, $\frac{1}{92\cdots 19}a^{17}+\frac{355}{92\cdots 19}a^{16}+\frac{19\cdots 96}{92\cdots 19}a^{15}+\frac{17\cdots 70}{92\cdots 19}a^{14}+\frac{26\cdots 99}{54\cdots 07}a^{13}-\frac{16\cdots 26}{92\cdots 19}a^{12}-\frac{14\cdots 53}{13\cdots 17}a^{11}-\frac{45\cdots 74}{92\cdots 19}a^{10}+\frac{21\cdots 29}{92\cdots 19}a^{9}+\frac{10\cdots 71}{29\cdots 49}a^{8}-\frac{87\cdots 93}{92\cdots 19}a^{7}-\frac{34\cdots 09}{92\cdots 19}a^{6}+\frac{47\cdots 09}{13\cdots 17}a^{5}+\frac{23\cdots 55}{71\cdots 63}a^{4}-\frac{15\cdots 91}{92\cdots 19}a^{3}+\frac{11\cdots 04}{71\cdots 63}a^{2}+\frac{90\cdots 44}{13\cdots 17}a+\frac{31\cdots 83}{71\cdots 63}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{18}\times C_{18}$, which has order $26244$ (assuming GRH) |
| |
| Narrow class group: | $C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{18}\times C_{18}$, which has order $26244$ (assuming GRH) |
|
Unit group
| Rank: | $8$ |
| |
| Torsion generator: |
\( -\frac{85652246508}{7957781197910448463} a^{17} + \frac{53957699292}{468104776347673439} a^{16} - \frac{4681232368692}{7957781197910448463} a^{15} + \frac{2383271790810}{7957781197910448463} a^{14} + \frac{77436815281224}{7957781197910448463} a^{13} - \frac{23445927891963}{468104776347673439} a^{12} + \frac{506923560041328}{7957781197910448463} a^{11} + \frac{1197001442277462}{7957781197910448463} a^{10} - \frac{4062008220800641}{7957781197910448463} a^{9} - \frac{22284256000813080}{7957781197910448463} a^{8} + \frac{6345365462330304}{468104776347673439} a^{7} - \frac{136779849957013695}{7957781197910448463} a^{6} - \frac{1694080604608791450}{7957781197910448463} a^{5} + \frac{7794831425435136648}{7957781197910448463} a^{4} - \frac{14591997765187381049}{7957781197910448463} a^{3} + \frac{6263420755008827502}{7957781197910448463} a^{2} + \frac{20268926623431909342}{7957781197910448463} a - \frac{25865148080628118729}{7957781197910448463} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{19\cdots 81}{13\cdots 17}a^{17}-\frac{65\cdots 69}{54\cdots 07}a^{16}+\frac{42\cdots 58}{92\cdots 19}a^{15}+\frac{10\cdots 68}{92\cdots 19}a^{14}-\frac{11\cdots 41}{92\cdots 19}a^{13}+\frac{19\cdots 51}{54\cdots 07}a^{12}+\frac{31\cdots 45}{71\cdots 63}a^{11}-\frac{22\cdots 24}{92\cdots 19}a^{10}+\frac{50\cdots 90}{92\cdots 19}a^{9}+\frac{14\cdots 68}{29\cdots 49}a^{8}-\frac{39\cdots 52}{54\cdots 07}a^{7}-\frac{85\cdots 50}{92\cdots 19}a^{6}+\frac{27\cdots 95}{92\cdots 19}a^{5}-\frac{50\cdots 36}{92\cdots 19}a^{4}+\frac{69\cdots 72}{24\cdots 87}a^{3}+\frac{16\cdots 22}{92\cdots 19}a^{2}-\frac{13\cdots 51}{92\cdots 19}a-\frac{22\cdots 18}{92\cdots 19}$, $\frac{96\cdots 97}{23\cdots 51}a^{17}+\frac{14\cdots 61}{33\cdots 93}a^{16}-\frac{50\cdots 02}{23\cdots 51}a^{15}+\frac{21\cdots 60}{23\cdots 51}a^{14}+\frac{81\cdots 49}{23\cdots 51}a^{13}-\frac{43\cdots 33}{23\cdots 51}a^{12}+\frac{57\cdots 57}{23\cdots 51}a^{11}+\frac{96\cdots 10}{23\cdots 51}a^{10}-\frac{46\cdots 70}{23\cdots 51}a^{9}-\frac{75\cdots 40}{75\cdots 21}a^{8}+\frac{10\cdots 82}{23\cdots 51}a^{7}-\frac{17\cdots 78}{23\cdots 51}a^{6}-\frac{18\cdots 09}{23\cdots 51}a^{5}+\frac{80\cdots 20}{23\cdots 51}a^{4}-\frac{17\cdots 40}{23\cdots 51}a^{3}+\frac{99\cdots 14}{23\cdots 51}a^{2}+\frac{62\cdots 47}{76\cdots 93}a-\frac{40\cdots 14}{23\cdots 51}$, $\frac{47\cdots 14}{92\cdots 19}a^{17}+\frac{21\cdots 32}{13\cdots 17}a^{16}-\frac{92\cdots 18}{92\cdots 19}a^{15}-\frac{74\cdots 40}{92\cdots 19}a^{14}+\frac{10\cdots 12}{92\cdots 19}a^{13}+\frac{11\cdots 37}{92\cdots 19}a^{12}-\frac{35\cdots 02}{92\cdots 19}a^{11}-\frac{51\cdots 32}{92\cdots 19}a^{10}+\frac{26\cdots 69}{92\cdots 19}a^{9}-\frac{43\cdots 81}{29\cdots 49}a^{8}-\frac{42\cdots 02}{92\cdots 19}a^{7}-\frac{25\cdots 59}{92\cdots 19}a^{6}-\frac{92\cdots 00}{92\cdots 19}a^{5}-\frac{25\cdots 97}{92\cdots 19}a^{4}-\frac{39\cdots 21}{24\cdots 87}a^{3}-\frac{79\cdots 21}{92\cdots 19}a^{2}-\frac{20\cdots 71}{92\cdots 19}a-\frac{12\cdots 53}{92\cdots 19}$, $\frac{81\cdots 40}{13\cdots 17}a^{17}-\frac{58\cdots 77}{92\cdots 19}a^{16}+\frac{29\cdots 84}{92\cdots 19}a^{15}-\frac{25\cdots 62}{24\cdots 87}a^{14}-\frac{48\cdots 88}{92\cdots 19}a^{13}+\frac{24\cdots 67}{92\cdots 19}a^{12}-\frac{79\cdots 00}{24\cdots 87}a^{11}-\frac{69\cdots 28}{92\cdots 19}a^{10}+\frac{25\cdots 41}{92\cdots 19}a^{9}+\frac{47\cdots 85}{29\cdots 49}a^{8}-\frac{64\cdots 64}{92\cdots 19}a^{7}+\frac{90\cdots 87}{92\cdots 19}a^{6}+\frac{11\cdots 52}{92\cdots 19}a^{5}-\frac{46\cdots 09}{92\cdots 19}a^{4}+\frac{93\cdots 05}{92\cdots 19}a^{3}-\frac{19\cdots 03}{54\cdots 07}a^{2}-\frac{77\cdots 46}{54\cdots 07}a+\frac{23\cdots 58}{92\cdots 19}$, $\frac{14\cdots 34}{92\cdots 19}a^{17}+\frac{11\cdots 86}{10\cdots 09}a^{16}-\frac{32\cdots 07}{92\cdots 19}a^{15}-\frac{16\cdots 86}{92\cdots 19}a^{14}+\frac{10\cdots 82}{92\cdots 19}a^{13}-\frac{31\cdots 36}{13\cdots 17}a^{12}-\frac{79\cdots 74}{92\cdots 19}a^{11}+\frac{17\cdots 06}{92\cdots 19}a^{10}+\frac{26\cdots 33}{92\cdots 19}a^{9}-\frac{14\cdots 81}{29\cdots 49}a^{8}+\frac{13\cdots 44}{92\cdots 19}a^{7}+\frac{19\cdots 46}{92\cdots 19}a^{6}-\frac{28\cdots 34}{92\cdots 19}a^{5}+\frac{24\cdots 55}{92\cdots 19}a^{4}+\frac{62\cdots 14}{92\cdots 19}a^{3}-\frac{13\cdots 92}{92\cdots 19}a^{2}-\frac{99\cdots 19}{92\cdots 19}a+\frac{51\cdots 24}{92\cdots 19}$, $\frac{16\cdots 48}{13\cdots 17}a^{17}-\frac{12\cdots 33}{92\cdots 19}a^{16}+\frac{60\cdots 81}{92\cdots 19}a^{15}-\frac{23\cdots 52}{92\cdots 19}a^{14}-\frac{10\cdots 48}{92\cdots 19}a^{13}+\frac{51\cdots 68}{92\cdots 19}a^{12}-\frac{62\cdots 88}{89\cdots 73}a^{11}-\frac{13\cdots 08}{92\cdots 19}a^{10}+\frac{52\cdots 57}{92\cdots 19}a^{9}+\frac{74\cdots 16}{22\cdots 73}a^{8}-\frac{13\cdots 74}{92\cdots 19}a^{7}+\frac{18\cdots 15}{92\cdots 19}a^{6}+\frac{23\cdots 62}{92\cdots 19}a^{5}-\frac{97\cdots 96}{92\cdots 19}a^{4}+\frac{20\cdots 59}{92\cdots 19}a^{3}-\frac{98\cdots 97}{92\cdots 19}a^{2}-\frac{31\cdots 41}{13\cdots 17}a+\frac{42\cdots 79}{92\cdots 19}$, $\frac{11\cdots 35}{92\cdots 19}a^{17}+\frac{12\cdots 91}{92\cdots 19}a^{16}-\frac{50\cdots 90}{54\cdots 07}a^{15}+\frac{27\cdots 46}{92\cdots 19}a^{14}-\frac{34\cdots 53}{13\cdots 17}a^{13}-\frac{21\cdots 56}{92\cdots 19}a^{12}+\frac{83\cdots 11}{92\cdots 19}a^{11}-\frac{17\cdots 78}{92\cdots 19}a^{10}+\frac{30\cdots 47}{92\cdots 19}a^{9}-\frac{21\cdots 89}{42\cdots 07}a^{8}+\frac{23\cdots 68}{92\cdots 19}a^{7}-\frac{10\cdots 69}{10\cdots 09}a^{6}+\frac{21\cdots 67}{13\cdots 17}a^{5}-\frac{52\cdots 40}{92\cdots 19}a^{4}-\frac{50\cdots 99}{92\cdots 19}a^{3}+\frac{74\cdots 18}{92\cdots 19}a^{2}-\frac{10\cdots 95}{92\cdots 19}a-\frac{17\cdots 50}{13\cdots 17}$, $\frac{79\cdots 15}{92\cdots 19}a^{17}-\frac{41\cdots 75}{71\cdots 63}a^{16}+\frac{15\cdots 94}{92\cdots 19}a^{15}+\frac{56\cdots 42}{54\cdots 07}a^{14}-\frac{71\cdots 81}{13\cdots 17}a^{13}+\frac{41\cdots 64}{71\cdots 63}a^{12}+\frac{42\cdots 15}{92\cdots 19}a^{11}+\frac{93\cdots 46}{92\cdots 19}a^{10}-\frac{85\cdots 05}{92\cdots 19}a^{9}+\frac{19\cdots 57}{42\cdots 07}a^{8}+\frac{14\cdots 20}{92\cdots 19}a^{7}-\frac{25\cdots 97}{77\cdots 01}a^{6}+\frac{74\cdots 11}{13\cdots 17}a^{5}+\frac{13\cdots 38}{92\cdots 19}a^{4}-\frac{86\cdots 23}{71\cdots 63}a^{3}-\frac{12\cdots 46}{92\cdots 19}a^{2}+\frac{18\cdots 81}{54\cdots 07}a+\frac{36\cdots 58}{13\cdots 17}$
|
| |
| Regulator: | \( 815207056414.5446 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 815207056414.5446 \cdot 26244}{6\cdot\sqrt{7168555859200162677766007466667663950714918483}}\cr\approx \mathstrut & 0.642761250966627 \end{aligned}\] (assuming GRH)
Galois group
$C_3^2:C_6$ (as 18T23):
| A solvable group of order 54 |
| The 18 conjugacy class representatives for $C_3^2:C_6$ |
| Character table for $C_3^2:C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 6.0.1928163650072427.6, 6.0.14795494587.2, 6.0.1928163650072427.5, 6.0.3518667.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
| 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 | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | R | R | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.6.11a1.7 | $x^{6} + 9 x + 3$ | $6$ | $1$ | $11$ | $S_3\times C_3$ | $$[2, \frac{5}{2}]_{2}$$ |
| 3.1.6.11a1.7 | $x^{6} + 9 x + 3$ | $6$ | $1$ | $11$ | $S_3\times C_3$ | $$[2, \frac{5}{2}]_{2}$$ | |
| 3.1.6.11a1.7 | $x^{6} + 9 x + 3$ | $6$ | $1$ | $11$ | $S_3\times C_3$ | $$[2, \frac{5}{2}]_{2}$$ | |
|
\(17\)
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 17.2.3.4a1.2 | $x^{6} + 48 x^{5} + 777 x^{4} + 4384 x^{3} + 2331 x^{2} + 432 x + 44$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(19\)
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |