Normalized defining polynomial
\( x^{18} - 9 x^{16} - 84 x^{15} + 72 x^{14} + 828 x^{13} + 648 x^{12} - 1548 x^{11} - 8766 x^{10} + \cdots - 721709 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[2, 8]$ |
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| Discriminant: |
\(587152195005575401214701453922304\)
\(\medspace = 2^{12}\cdot 3^{37}\cdot 7^{12}\cdot 23\)
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| Root discriminant: | \(66.14\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(23\)
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| Discriminant root field: | \(\Q(\sqrt{69}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{24}a^{9}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a+\frac{1}{24}$, $\frac{1}{24}a^{10}-\frac{1}{8}a^{8}-\frac{3}{8}a^{2}-\frac{1}{3}a+\frac{1}{8}$, $\frac{1}{24}a^{11}-\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}+\frac{1}{6}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{48}a^{12}-\frac{1}{16}a^{8}+\frac{1}{16}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{3}{16}$, $\frac{1}{48}a^{13}-\frac{1}{48}a^{9}-\frac{1}{8}a^{8}-\frac{3}{16}a^{5}+\frac{1}{12}a^{4}+\frac{3}{16}a+\frac{1}{24}$, $\frac{1}{144}a^{14}-\frac{1}{144}a^{13}+\frac{1}{144}a^{12}-\frac{1}{72}a^{11}-\frac{1}{144}a^{10}+\frac{1}{144}a^{9}+\frac{1}{48}a^{8}-\frac{1}{12}a^{7}+\frac{1}{48}a^{6}+\frac{1}{144}a^{5}+\frac{35}{144}a^{4}-\frac{13}{72}a^{3}-\frac{59}{144}a^{2}-\frac{25}{144}a-\frac{47}{144}$, $\frac{1}{288}a^{15}-\frac{1}{288}a^{14}+\frac{1}{288}a^{13}+\frac{1}{288}a^{12}+\frac{5}{288}a^{11}-\frac{5}{288}a^{10}-\frac{1}{96}a^{9}-\frac{1}{96}a^{8}+\frac{1}{96}a^{7}-\frac{35}{288}a^{6}+\frac{35}{288}a^{5}+\frac{19}{288}a^{4}+\frac{7}{288}a^{3}+\frac{89}{288}a^{2}-\frac{17}{288}a-\frac{11}{96}$, $\frac{1}{4032}a^{16}-\frac{1}{2016}a^{15}-\frac{1}{672}a^{14}+\frac{1}{288}a^{13}-\frac{17}{2016}a^{12}-\frac{1}{96}a^{11}+\frac{5}{2016}a^{10}+\frac{5}{288}a^{9}-\frac{3}{28}a^{8}+\frac{35}{288}a^{7}-\frac{193}{2016}a^{6}-\frac{7}{96}a^{5}+\frac{289}{2016}a^{4}+\frac{7}{288}a^{3}+\frac{5}{672}a^{2}-\frac{151}{2016}a+\frac{271}{4032}$, $\frac{1}{24\cdots 48}a^{17}+\frac{99\cdots 67}{83\cdots 16}a^{16}-\frac{13\cdots 43}{77\cdots 64}a^{15}+\frac{20\cdots 87}{62\cdots 12}a^{14}-\frac{11\cdots 11}{20\cdots 04}a^{13}+\frac{16\cdots 27}{15\cdots 28}a^{12}-\frac{51\cdots 71}{31\cdots 56}a^{11}+\frac{10\cdots 03}{69\cdots 68}a^{10}+\frac{84\cdots 51}{12\cdots 24}a^{9}+\frac{58\cdots 53}{12\cdots 24}a^{8}+\frac{18\cdots 55}{25\cdots 88}a^{7}-\frac{53\cdots 67}{62\cdots 12}a^{6}+\frac{76\cdots 39}{62\cdots 12}a^{5}+\frac{62\cdots 45}{10\cdots 52}a^{4}+\frac{51\cdots 03}{31\cdots 56}a^{3}+\frac{19\cdots 37}{62\cdots 12}a^{2}+\frac{14\cdots 93}{27\cdots 72}a-\frac{77\cdots 11}{24\cdots 48}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{76\cdots 69}{41\cdots 56}a^{17}+\frac{17\cdots 67}{41\cdots 56}a^{16}-\frac{35\cdots 33}{26\cdots 16}a^{15}-\frac{18\cdots 17}{10\cdots 64}a^{14}-\frac{23\cdots 25}{10\cdots 64}a^{13}+\frac{38\cdots 31}{26\cdots 16}a^{12}+\frac{19\cdots 59}{52\cdots 32}a^{11}+\frac{22\cdots 67}{10\cdots 64}a^{10}-\frac{77\cdots 31}{69\cdots 76}a^{9}-\frac{13\cdots 25}{20\cdots 28}a^{8}-\frac{12\cdots 65}{26\cdots 16}a^{7}+\frac{13\cdots 33}{10\cdots 64}a^{6}+\frac{26\cdots 31}{10\cdots 64}a^{5}+\frac{22\cdots 85}{52\cdots 32}a^{4}-\frac{19\cdots 67}{52\cdots 32}a^{3}+\frac{34\cdots 17}{10\cdots 64}a^{2}-\frac{10\cdots 95}{41\cdots 56}a-\frac{13\cdots 55}{13\cdots 52}$, $\frac{79\cdots 07}{12\cdots 24}a^{17}+\frac{30\cdots 95}{41\cdots 08}a^{16}-\frac{15\cdots 61}{31\cdots 56}a^{15}-\frac{15\cdots 03}{31\cdots 56}a^{14}-\frac{59\cdots 41}{25\cdots 88}a^{13}+\frac{15\cdots 93}{31\cdots 56}a^{12}+\frac{10\cdots 65}{31\cdots 56}a^{11}-\frac{10\cdots 21}{10\cdots 52}a^{10}-\frac{23\cdots 93}{62\cdots 12}a^{9}-\frac{47\cdots 53}{62\cdots 12}a^{8}+\frac{17\cdots 15}{10\cdots 52}a^{7}+\frac{15\cdots 55}{31\cdots 56}a^{6}-\frac{25\cdots 97}{15\cdots 28}a^{5}-\frac{90\cdots 97}{10\cdots 52}a^{4}-\frac{52\cdots 25}{31\cdots 56}a^{3}+\frac{46\cdots 91}{31\cdots 56}a^{2}+\frac{94\cdots 61}{41\cdots 08}a-\frac{34\cdots 43}{12\cdots 24}$, $\frac{42\cdots 51}{83\cdots 16}a^{17}-\frac{28\cdots 35}{24\cdots 48}a^{16}-\frac{39\cdots 91}{62\cdots 12}a^{15}-\frac{97\cdots 59}{25\cdots 88}a^{14}+\frac{20\cdots 63}{31\cdots 56}a^{13}+\frac{32\cdots 25}{62\cdots 12}a^{12}-\frac{16\cdots 29}{69\cdots 68}a^{11}-\frac{34\cdots 05}{15\cdots 28}a^{10}-\frac{22\cdots 79}{12\cdots 24}a^{9}-\frac{18\cdots 15}{41\cdots 08}a^{8}+\frac{15\cdots 31}{62\cdots 12}a^{7}+\frac{86\cdots 49}{15\cdots 28}a^{6}-\frac{22\cdots 39}{25\cdots 88}a^{5}-\frac{76\cdots 47}{62\cdots 12}a^{4}+\frac{44\cdots 49}{62\cdots 12}a^{3}+\frac{27\cdots 14}{21\cdots 99}a^{2}+\frac{12\cdots 85}{24\cdots 48}a-\frac{82\cdots 11}{24\cdots 48}$, $\frac{15\cdots 43}{15\cdots 28}a^{17}+\frac{61\cdots 61}{20\cdots 04}a^{16}-\frac{33\cdots 65}{31\cdots 56}a^{15}-\frac{26\cdots 21}{31\cdots 56}a^{14}+\frac{31\cdots 49}{51\cdots 76}a^{13}+\frac{29\cdots 57}{31\cdots 56}a^{12}+\frac{15\cdots 09}{31\cdots 56}a^{11}-\frac{77\cdots 07}{34\cdots 84}a^{10}-\frac{47\cdots 17}{77\cdots 64}a^{9}-\frac{30\cdots 29}{15\cdots 28}a^{8}+\frac{24\cdots 71}{10\cdots 52}a^{7}+\frac{28\cdots 61}{31\cdots 56}a^{6}-\frac{15\cdots 87}{15\cdots 28}a^{5}-\frac{87\cdots 51}{10\cdots 52}a^{4}-\frac{29\cdots 97}{31\cdots 56}a^{3}+\frac{41\cdots 79}{31\cdots 56}a^{2}+\frac{19\cdots 63}{17\cdots 92}a-\frac{58\cdots 55}{62\cdots 12}$, $\frac{43\cdots 71}{69\cdots 68}a^{17}+\frac{77\cdots 49}{41\cdots 08}a^{16}-\frac{46\cdots 73}{69\cdots 68}a^{15}-\frac{11\cdots 31}{20\cdots 04}a^{14}+\frac{81\cdots 25}{20\cdots 04}a^{13}+\frac{12\cdots 21}{20\cdots 04}a^{12}+\frac{89\cdots 17}{20\cdots 04}a^{11}-\frac{34\cdots 11}{20\cdots 04}a^{10}-\frac{12\cdots 21}{20\cdots 04}a^{9}-\frac{41\cdots 97}{34\cdots 84}a^{8}+\frac{51\cdots 11}{20\cdots 04}a^{7}+\frac{47\cdots 61}{69\cdots 68}a^{6}+\frac{38\cdots 27}{20\cdots 04}a^{5}-\frac{25\cdots 81}{20\cdots 04}a^{4}-\frac{69\cdots 61}{20\cdots 04}a^{3}+\frac{47\cdots 79}{20\cdots 04}a^{2}+\frac{11\cdots 87}{25\cdots 88}a-\frac{21\cdots 25}{41\cdots 08}$, $\frac{46\cdots 99}{35\cdots 64}a^{17}-\frac{17\cdots 99}{39\cdots 96}a^{16}-\frac{94\cdots 39}{44\cdots 08}a^{15}-\frac{81\cdots 91}{89\cdots 16}a^{14}+\frac{15\cdots 87}{29\cdots 72}a^{13}+\frac{69\cdots 03}{44\cdots 08}a^{12}-\frac{37\cdots 01}{22\cdots 04}a^{11}-\frac{30\cdots 81}{29\cdots 72}a^{10}-\frac{26\cdots 47}{17\cdots 32}a^{9}-\frac{25\cdots 11}{17\cdots 32}a^{8}+\frac{68\cdots 55}{49\cdots 12}a^{7}+\frac{12\cdots 03}{89\cdots 16}a^{6}-\frac{14\cdots 55}{89\cdots 16}a^{5}-\frac{56\cdots 93}{74\cdots 68}a^{4}+\frac{16\cdots 73}{22\cdots 04}a^{3}+\frac{23\cdots 39}{89\cdots 16}a^{2}+\frac{10\cdots 37}{11\cdots 88}a-\frac{27\cdots 11}{35\cdots 64}$, $\frac{19\cdots 49}{83\cdots 16}a^{17}-\frac{59\cdots 81}{27\cdots 72}a^{16}-\frac{16\cdots 83}{69\cdots 68}a^{15}-\frac{17\cdots 83}{10\cdots 52}a^{14}+\frac{41\cdots 31}{10\cdots 52}a^{13}+\frac{43\cdots 67}{20\cdots 04}a^{12}-\frac{43\cdots 25}{69\cdots 68}a^{11}-\frac{31\cdots 17}{34\cdots 84}a^{10}-\frac{96\cdots 59}{41\cdots 08}a^{9}-\frac{98\cdots 13}{41\cdots 08}a^{8}+\frac{12\cdots 59}{69\cdots 68}a^{7}+\frac{96\cdots 55}{34\cdots 84}a^{6}-\frac{47\cdots 11}{25\cdots 88}a^{5}-\frac{22\cdots 13}{20\cdots 04}a^{4}-\frac{38\cdots 33}{20\cdots 04}a^{3}+\frac{46\cdots 79}{34\cdots 84}a^{2}+\frac{87\cdots 39}{27\cdots 72}a+\frac{38\cdots 77}{83\cdots 16}$, $\frac{71\cdots 49}{83\cdots 16}a^{17}-\frac{12\cdots 99}{24\cdots 48}a^{16}-\frac{52\cdots 81}{31\cdots 56}a^{15}-\frac{30\cdots 47}{69\cdots 68}a^{14}+\frac{34\cdots 99}{62\cdots 12}a^{13}+\frac{48\cdots 55}{38\cdots 82}a^{12}-\frac{72\cdots 91}{25\cdots 88}a^{11}-\frac{78\cdots 71}{62\cdots 12}a^{10}-\frac{20\cdots 63}{12\cdots 24}a^{9}+\frac{78\cdots 19}{41\cdots 08}a^{8}+\frac{67\cdots 89}{31\cdots 56}a^{7}+\frac{13\cdots 31}{62\cdots 12}a^{6}-\frac{16\cdots 89}{69\cdots 68}a^{5}-\frac{43\cdots 67}{31\cdots 56}a^{4}-\frac{69\cdots 37}{38\cdots 82}a^{3}+\frac{31\cdots 65}{20\cdots 04}a^{2}+\frac{85\cdots 79}{24\cdots 48}a+\frac{16\cdots 89}{24\cdots 48}$, $\frac{65\cdots 65}{24\cdots 48}a^{17}-\frac{12\cdots 89}{24\cdots 48}a^{16}-\frac{87\cdots 21}{62\cdots 12}a^{15}-\frac{20\cdots 09}{10\cdots 52}a^{14}+\frac{58\cdots 99}{10\cdots 52}a^{13}+\frac{68\cdots 35}{62\cdots 12}a^{12}-\frac{25\cdots 51}{62\cdots 12}a^{11}-\frac{10\cdots 95}{31\cdots 56}a^{10}-\frac{69\cdots 37}{41\cdots 08}a^{9}-\frac{24\cdots 43}{12\cdots 24}a^{8}+\frac{77\cdots 41}{62\cdots 12}a^{7}-\frac{95\cdots 39}{31\cdots 56}a^{6}+\frac{14\cdots 99}{51\cdots 76}a^{5}-\frac{61\cdots 85}{69\cdots 68}a^{4}+\frac{32\cdots 39}{62\cdots 12}a^{3}-\frac{16\cdots 79}{31\cdots 56}a^{2}+\frac{29\cdots 33}{24\cdots 48}a-\frac{82\cdots 03}{83\cdots 16}$
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| Regulator: | \( 4159681971.98 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 4159681971.98 \cdot 2}{2\cdot\sqrt{587152195005575401214701453922304}}\cr\approx \mathstrut & 1.66795266566 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3:A_4^2.S_4$ (as 18T658):
| A solvable group of order 27648 |
| The 96 conjugacy class representatives for $C_2^3:A_4^2.S_4$ |
| Character table for $C_2^3:A_4^2.S_4$ |
Intermediate fields
| 3.3.756.1, 9.9.2917096519063104.1 |
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 24 siblings: | 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 | R | R | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.3.2a1.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ |
| 2.1.3.2a1.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 2.4.3.8a1.2 | $x^{12} + 3 x^{9} + 3 x^{8} + 3 x^{6} + 6 x^{5} + 3 x^{4} + x^{3} + 3 x^{2} + 3 x + 3$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $$[\ ]_{3}^{4}$$ | |
|
\(3\)
| 3.1.18.37c4.17 | $x^{18} + 9 x^{8} + 6 x^{6} + 9 x^{4} + 9 x^{2} + 3$ | $18$ | $1$ | $37$ | not computed | not computed |
|
\(7\)
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 7.2.1.0a1.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 7.1.2.1a1.1 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 7.2.6.10a1.4 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 192996 x^{6} + 266328 x^{5} + 234495 x^{4} + 131220 x^{3} + 45198 x^{2} + 8755 x + 757$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $$[\ ]_{6}^{2}$$ | |
|
\(23\)
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 23.2.1.0a1.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 23.1.2.1a1.1 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 23.12.1.0a1.1 | $x^{12} + 21 x^{7} + 21 x^{6} + 15 x^{5} + 14 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 5$ | $1$ | $12$ | $0$ | $C_{12}$ | $$[\ ]^{12}$$ |