Normalized defining polynomial
\( x^{18} - 12 x^{15} + 36 x^{14} - 33 x^{12} + 117 x^{11} - 171 x^{10} - 2 x^{9} + 135 x^{8} - 81 x^{7} + \cdots + 44 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[6, 6]$ |
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| Discriminant: |
\(36644198070556426025390625\)
\(\medspace = 3^{36}\cdot 5^{12}\)
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| Root discriminant: | \(26.32\) |
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| Galois root discriminant: | $3^{58/27}5^{3/4}\approx 35.412341444574025$ | ||
| Ramified primes: |
\(3\), \(5\)
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{13}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{952}a^{16}+\frac{9}{952}a^{15}+\frac{19}{238}a^{14}+\frac{151}{952}a^{13}+\frac{9}{136}a^{12}-\frac{47}{238}a^{11}-\frac{1}{7}a^{10}+\frac{309}{952}a^{9}+\frac{31}{68}a^{8}+\frac{65}{136}a^{7}-\frac{40}{119}a^{6}-\frac{167}{476}a^{4}+\frac{181}{476}a^{3}+\frac{249}{952}a^{2}-\frac{219}{476}a-\frac{93}{238}$, $\frac{1}{31\cdots 12}a^{17}-\frac{105841414556345}{31\cdots 12}a^{16}-\frac{46\cdots 19}{78\cdots 78}a^{15}+\frac{49\cdots 29}{31\cdots 12}a^{14}-\frac{31\cdots 77}{31\cdots 12}a^{13}-\frac{17\cdots 95}{78\cdots 78}a^{12}+\frac{59\cdots 36}{39\cdots 39}a^{11}+\frac{84\cdots 69}{31\cdots 12}a^{10}-\frac{38\cdots 25}{78\cdots 78}a^{9}+\frac{15\cdots 87}{45\cdots 16}a^{8}+\frac{42\cdots 25}{15\cdots 56}a^{7}-\frac{24\cdots 31}{78\cdots 78}a^{6}+\frac{29\cdots 03}{78\cdots 78}a^{5}-\frac{16\cdots 23}{39\cdots 39}a^{4}+\frac{11\cdots 09}{31\cdots 12}a^{3}+\frac{68\cdots 09}{15\cdots 56}a^{2}+\frac{68\cdots 94}{39\cdots 39}a+\frac{14\cdots 92}{39\cdots 39}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{641937521115}{10860908935042}a^{17}+\frac{79116760827}{5430454467521}a^{16}+\frac{26764657449}{5430454467521}a^{15}-\frac{3931760030940}{5430454467521}a^{14}+\frac{10570081998636}{5430454467521}a^{13}+\frac{2349750710280}{5430454467521}a^{12}-\frac{17287431279039}{10860908935042}a^{11}+\frac{64571716853325}{10860908935042}a^{10}-\frac{91019866630781}{10860908935042}a^{9}-\frac{15425530973811}{5430454467521}a^{8}+\frac{67883497525053}{10860908935042}a^{7}-\frac{18927509618457}{10860908935042}a^{6}-\frac{17173191890457}{5430454467521}a^{5}+\frac{16438189858146}{5430454467521}a^{4}+\frac{41382068045949}{10860908935042}a^{3}-\frac{216890839459953}{10860908935042}a^{2}+\frac{250464765196773}{10860908935042}a-\frac{33149262805525}{5430454467521}$, $\frac{190182269157885}{92\cdots 68}a^{17}+\frac{39\cdots 75}{15\cdots 56}a^{16}+\frac{52\cdots 07}{15\cdots 56}a^{15}-\frac{92\cdots 03}{39\cdots 39}a^{14}+\frac{18\cdots 29}{39\cdots 39}a^{13}+\frac{76\cdots 67}{15\cdots 56}a^{12}+\frac{32\cdots 87}{78\cdots 78}a^{11}+\frac{13\cdots 15}{92\cdots 68}a^{10}-\frac{66\cdots 01}{39\cdots 39}a^{9}-\frac{19\cdots 13}{11\cdots 54}a^{8}-\frac{85\cdots 35}{78\cdots 78}a^{7}+\frac{11\cdots 09}{78\cdots 78}a^{6}-\frac{28\cdots 43}{92\cdots 68}a^{5}+\frac{78\cdots 17}{22\cdots 08}a^{4}-\frac{63\cdots 07}{15\cdots 56}a^{3}-\frac{54\cdots 51}{22\cdots 08}a^{2}-\frac{22\cdots 19}{78\cdots 78}a+\frac{99\cdots 42}{39\cdots 39}$, $\frac{21\cdots 45}{31\cdots 12}a^{17}+\frac{15\cdots 27}{31\cdots 12}a^{16}+\frac{32\cdots 53}{78\cdots 78}a^{15}-\frac{24\cdots 83}{31\cdots 12}a^{14}+\frac{57\cdots 23}{31\cdots 12}a^{13}+\frac{51\cdots 82}{39\cdots 39}a^{12}-\frac{82\cdots 29}{78\cdots 78}a^{11}+\frac{21\cdots 29}{31\cdots 12}a^{10}-\frac{29\cdots 95}{46\cdots 34}a^{9}-\frac{20\cdots 49}{45\cdots 16}a^{8}+\frac{77\cdots 75}{15\cdots 56}a^{7}-\frac{13\cdots 71}{78\cdots 78}a^{6}-\frac{27\cdots 33}{78\cdots 78}a^{5}+\frac{24\cdots 35}{78\cdots 78}a^{4}+\frac{22\cdots 75}{45\cdots 16}a^{3}-\frac{34\cdots 77}{15\cdots 56}a^{2}+\frac{74\cdots 19}{39\cdots 39}a-\frac{18\cdots 74}{39\cdots 39}$, $\frac{21\cdots 45}{31\cdots 12}a^{17}+\frac{15\cdots 27}{31\cdots 12}a^{16}+\frac{32\cdots 53}{78\cdots 78}a^{15}-\frac{24\cdots 83}{31\cdots 12}a^{14}+\frac{57\cdots 23}{31\cdots 12}a^{13}+\frac{51\cdots 82}{39\cdots 39}a^{12}-\frac{82\cdots 29}{78\cdots 78}a^{11}+\frac{21\cdots 29}{31\cdots 12}a^{10}-\frac{29\cdots 95}{46\cdots 34}a^{9}-\frac{20\cdots 49}{45\cdots 16}a^{8}+\frac{77\cdots 75}{15\cdots 56}a^{7}-\frac{13\cdots 71}{78\cdots 78}a^{6}-\frac{27\cdots 33}{78\cdots 78}a^{5}+\frac{24\cdots 35}{78\cdots 78}a^{4}+\frac{22\cdots 75}{45\cdots 16}a^{3}-\frac{34\cdots 77}{15\cdots 56}a^{2}+\frac{74\cdots 19}{39\cdots 39}a-\frac{14\cdots 35}{39\cdots 39}$, $\frac{312641150013499}{92\cdots 68}a^{17}+\frac{14\cdots 97}{31\cdots 12}a^{16}+\frac{14\cdots 75}{31\cdots 12}a^{15}-\frac{14\cdots 50}{39\cdots 39}a^{14}+\frac{21\cdots 51}{31\cdots 12}a^{13}+\frac{35\cdots 47}{31\cdots 12}a^{12}+\frac{21\cdots 53}{78\cdots 78}a^{11}+\frac{28\cdots 83}{92\cdots 68}a^{10}-\frac{20\cdots 17}{31\cdots 12}a^{9}-\frac{19\cdots 74}{56\cdots 77}a^{8}+\frac{82\cdots 39}{31\cdots 12}a^{7}+\frac{27\cdots 77}{15\cdots 56}a^{6}-\frac{16\cdots 25}{92\cdots 68}a^{5}+\frac{63\cdots 77}{22\cdots 08}a^{4}+\frac{23\cdots 03}{15\cdots 56}a^{3}-\frac{35\cdots 09}{45\cdots 16}a^{2}+\frac{10\cdots 29}{15\cdots 56}a+\frac{36\cdots 15}{78\cdots 78}$, $\frac{12\cdots 21}{31\cdots 12}a^{17}-\frac{11\cdots 77}{15\cdots 56}a^{16}-\frac{52\cdots 83}{31\cdots 12}a^{15}-\frac{22\cdots 49}{31\cdots 12}a^{14}+\frac{33\cdots 29}{15\cdots 56}a^{13}-\frac{23\cdots 97}{31\cdots 12}a^{12}-\frac{18\cdots 31}{39\cdots 39}a^{11}+\frac{44\cdots 69}{31\cdots 12}a^{10}-\frac{50\cdots 49}{31\cdots 12}a^{9}-\frac{14\cdots 93}{45\cdots 16}a^{8}+\frac{50\cdots 23}{31\cdots 12}a^{7}-\frac{98\cdots 49}{39\cdots 39}a^{6}+\frac{45\cdots 69}{15\cdots 56}a^{5}+\frac{950116934760115}{56\cdots 77}a^{4}-\frac{45\cdots 37}{31\cdots 12}a^{3}+\frac{54\cdots 65}{45\cdots 16}a^{2}+\frac{62\cdots 43}{15\cdots 56}a-\frac{81\cdots 13}{78\cdots 78}$, $\frac{41\cdots 63}{31\cdots 12}a^{17}+\frac{25\cdots 33}{31\cdots 12}a^{16}+\frac{770774712031579}{11\cdots 54}a^{15}-\frac{67\cdots 89}{45\cdots 16}a^{14}+\frac{11\cdots 69}{31\cdots 12}a^{13}+\frac{34\cdots 67}{15\cdots 56}a^{12}-\frac{19\cdots 03}{78\cdots 78}a^{11}+\frac{43\cdots 63}{31\cdots 12}a^{10}-\frac{10\cdots 65}{78\cdots 78}a^{9}-\frac{33\cdots 87}{45\cdots 16}a^{8}+\frac{53\cdots 35}{46\cdots 34}a^{7}-\frac{76\cdots 45}{15\cdots 56}a^{6}-\frac{11\cdots 53}{15\cdots 56}a^{5}+\frac{19\cdots 85}{39\cdots 39}a^{4}+\frac{33\cdots 61}{31\cdots 12}a^{3}-\frac{16\cdots 01}{39\cdots 39}a^{2}+\frac{33\cdots 09}{78\cdots 78}a-\frac{44\cdots 13}{39\cdots 39}$, $\frac{27\cdots 51}{15\cdots 56}a^{17}-\frac{51\cdots 15}{15\cdots 56}a^{16}+\frac{598508537524152}{39\cdots 39}a^{15}-\frac{41\cdots 55}{15\cdots 56}a^{14}+\frac{16\cdots 09}{15\cdots 56}a^{13}-\frac{11\cdots 31}{78\cdots 78}a^{12}+\frac{59\cdots 47}{78\cdots 78}a^{11}+\frac{46\cdots 45}{15\cdots 56}a^{10}-\frac{31\cdots 35}{78\cdots 78}a^{9}+\frac{86\cdots 93}{22\cdots 08}a^{8}-\frac{10\cdots 95}{78\cdots 78}a^{7}+\frac{84\cdots 85}{39\cdots 39}a^{6}-\frac{42\cdots 25}{78\cdots 78}a^{5}+\frac{30\cdots 09}{39\cdots 39}a^{4}-\frac{13\cdots 53}{15\cdots 56}a^{3}+\frac{22\cdots 86}{39\cdots 39}a^{2}+\frac{18\cdots 87}{78\cdots 78}a-\frac{56\cdots 98}{39\cdots 39}$, $\frac{10\cdots 87}{78\cdots 78}a^{17}+\frac{25\cdots 31}{31\cdots 12}a^{16}+\frac{25\cdots 87}{31\cdots 12}a^{15}-\frac{12\cdots 73}{78\cdots 78}a^{14}+\frac{12\cdots 25}{31\cdots 12}a^{13}+\frac{60\cdots 25}{31\cdots 12}a^{12}-\frac{10\cdots 07}{56\cdots 77}a^{11}+\frac{51\cdots 94}{39\cdots 39}a^{10}-\frac{41\cdots 13}{31\cdots 12}a^{9}-\frac{15\cdots 29}{22\cdots 08}a^{8}+\frac{31\cdots 17}{31\cdots 12}a^{7}-\frac{39\cdots 37}{39\cdots 39}a^{6}-\frac{63\cdots 49}{78\cdots 78}a^{5}+\frac{56\cdots 11}{15\cdots 56}a^{4}+\frac{93\cdots 29}{15\cdots 56}a^{3}-\frac{12\cdots 85}{31\cdots 12}a^{2}+\frac{57\cdots 55}{15\cdots 56}a-\frac{61\cdots 93}{78\cdots 78}$, $\frac{24\cdots 49}{31\cdots 12}a^{17}+\frac{20\cdots 65}{31\cdots 12}a^{16}+\frac{66\cdots 37}{15\cdots 56}a^{15}-\frac{28\cdots 85}{31\cdots 12}a^{14}+\frac{64\cdots 29}{31\cdots 12}a^{13}+\frac{14\cdots 71}{78\cdots 78}a^{12}-\frac{60\cdots 36}{39\cdots 39}a^{11}+\frac{26\cdots 05}{31\cdots 12}a^{10}-\frac{11\cdots 71}{15\cdots 56}a^{9}-\frac{30\cdots 33}{45\cdots 16}a^{8}+\frac{10\cdots 55}{15\cdots 56}a^{7}-\frac{31\cdots 61}{15\cdots 56}a^{6}-\frac{39\cdots 93}{15\cdots 56}a^{5}-\frac{10\cdots 65}{56\cdots 77}a^{4}+\frac{24\cdots 63}{31\cdots 12}a^{3}-\frac{63\cdots 85}{22\cdots 08}a^{2}+\frac{92\cdots 69}{39\cdots 39}a-\frac{13\cdots 19}{39\cdots 39}$, $\frac{35\cdots 74}{39\cdots 39}a^{17}+\frac{353064137604767}{92\cdots 68}a^{16}+\frac{17\cdots 83}{78\cdots 78}a^{15}-\frac{42\cdots 78}{39\cdots 39}a^{14}+\frac{22\cdots 29}{78\cdots 78}a^{13}+\frac{17\cdots 99}{15\cdots 56}a^{12}-\frac{93\cdots 94}{39\cdots 39}a^{11}+\frac{38\cdots 67}{39\cdots 39}a^{10}-\frac{17\cdots 91}{15\cdots 56}a^{9}-\frac{63\cdots 07}{13\cdots 24}a^{8}+\frac{40\cdots 60}{39\cdots 39}a^{7}-\frac{66\cdots 03}{15\cdots 56}a^{6}-\frac{33\cdots 75}{78\cdots 78}a^{5}+\frac{41\cdots 47}{15\cdots 56}a^{4}+\frac{85\cdots 27}{13\cdots 24}a^{3}-\frac{48\cdots 21}{15\cdots 56}a^{2}+\frac{76\cdots 85}{23\cdots 67}a-\frac{37\cdots 27}{39\cdots 39}$
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| Regulator: | \( 1783677.91642 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 1783677.91642 \cdot 1}{2\cdot\sqrt{36644198070556426025390625}}\cr\approx \mathstrut & 0.580153628567 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), 6.2.4100625.1 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{5}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.9.36c15.10 | $x^{18} + 18 x^{17} + 162 x^{16} + 966 x^{15} + 4266 x^{14} + 14787 x^{13} + 41616 x^{12} + 97056 x^{11} + 189792 x^{10} + 312932 x^{9} + 435312 x^{8} + 508896 x^{7} + 495648 x^{6} + 396432 x^{5} + 254601 x^{4} + 126765 x^{3} + 46188 x^{2} + 11070 x + 1355$ | $9$ | $2$ | $36$ | 18T49 | not computed |
|
\(5\)
| 5.3.2.3a1.2 | $x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 5.3.4.9a1.3 | $x^{12} + 12 x^{10} + 12 x^{9} + 54 x^{8} + 108 x^{7} + 162 x^{6} + 324 x^{5} + 405 x^{4} + 432 x^{3} + 486 x^{2} + 324 x + 86$ | $4$ | $3$ | $9$ | $C_{12}$ | $$[\ ]_{4}^{3}$$ |