Normalized defining polynomial
\( x^{18} - 3 x^{17} + 10 x^{16} - 9 x^{15} - 7 x^{14} + 118 x^{13} + 42 x^{12} - 536 x^{11} + 2032 x^{10} + \cdots + 4096 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-200930163501792205662554161152\)
\(\medspace = -\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}\)
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| Root discriminant: | \(42.46\) |
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| Galois root discriminant: | $2^{3/2}3^{1/2}7^{5/6}13^{5/6}\approx 210.20358299362027$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{8}a^{9}+\frac{1}{16}a^{8}-\frac{1}{16}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{1}{32}a^{9}+\frac{3}{32}a^{8}+\frac{1}{16}a^{7}+\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{9}-\frac{1}{32}a^{8}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{64}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{11}-\frac{1}{64}a^{10}+\frac{1}{16}a^{9}-\frac{3}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{3}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{15}+\frac{1}{256}a^{14}+\frac{1}{128}a^{13}+\frac{3}{256}a^{12}-\frac{3}{256}a^{11}+\frac{3}{128}a^{10}-\frac{5}{128}a^{9}+\frac{1}{32}a^{8}+\frac{1}{32}a^{7}+\frac{5}{32}a^{6}-\frac{1}{32}a^{5}-\frac{3}{16}a^{4}+\frac{5}{16}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{123392}a^{16}-\frac{5}{123392}a^{15}-\frac{109}{30848}a^{14}-\frac{713}{123392}a^{13}-\frac{277}{123392}a^{12}-\frac{245}{15424}a^{11}-\frac{2343}{61696}a^{10}+\frac{2001}{30848}a^{9}+\frac{737}{15424}a^{8}-\frac{277}{15424}a^{7}+\frac{2393}{15424}a^{6}-\frac{26}{241}a^{5}-\frac{849}{7712}a^{4}+\frac{181}{3856}a^{3}-\frac{311}{1928}a^{2}+\frac{145}{964}a-\frac{93}{241}$, $\frac{1}{15\cdots 28}a^{17}-\frac{57\cdots 23}{15\cdots 28}a^{16}-\frac{33\cdots 33}{79\cdots 64}a^{15}-\frac{11\cdots 25}{15\cdots 28}a^{14}+\frac{12\cdots 49}{15\cdots 28}a^{13}-\frac{11\cdots 39}{79\cdots 64}a^{12}-\frac{15\cdots 27}{79\cdots 64}a^{11}+\frac{12\cdots 13}{24\cdots 52}a^{10}-\frac{63\cdots 69}{99\cdots 08}a^{9}+\frac{11\cdots 33}{19\cdots 16}a^{8}-\frac{23\cdots 89}{19\cdots 16}a^{7}+\frac{20\cdots 11}{99\cdots 08}a^{6}-\frac{10\cdots 45}{99\cdots 08}a^{5}+\frac{20\cdots 11}{24\cdots 52}a^{4}-\frac{15\cdots 77}{17\cdots 68}a^{3}+\frac{23\cdots 65}{62\cdots 88}a^{2}+\frac{25\cdots 53}{62\cdots 88}a-\frac{76\cdots 63}{15\cdots 72}$
| 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: | $8$ |
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| Torsion generator: |
\( -\frac{33791063290749}{3853273527406592} a^{17} + \frac{83163500008499}{3853273527406592} a^{16} - \frac{146255895426307}{1926636763703296} a^{15} + \frac{145556417080237}{3853273527406592} a^{14} + \frac{319135754802959}{3853273527406592} a^{13} - \frac{1906863789803913}{1926636763703296} a^{12} - \frac{1740131002464181}{1926636763703296} a^{11} + \frac{1018760688311745}{240829595462912} a^{10} - \frac{3735376769984855}{240829595462912} a^{9} + \frac{12000190567123727}{481659190925824} a^{8} - \frac{10690311906789047}{481659190925824} a^{7} - \frac{6344304316505175}{240829595462912} a^{6} + \frac{17048333286350749}{240829595462912} a^{5} - \frac{6245044816945993}{60207398865728} a^{4} - \frac{1135028298646069}{30103699432864} a^{3} + \frac{4071134319790741}{15051849716432} a^{2} - \frac{3625977850620357}{15051849716432} a + \frac{243921580809807}{3762962429108} \)
(order $6$)
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| Fundamental units: |
$\frac{48\cdots 01}{15\cdots 28}a^{17}-\frac{95\cdots 11}{15\cdots 28}a^{16}+\frac{20\cdots 87}{79\cdots 64}a^{15}-\frac{78\cdots 53}{15\cdots 28}a^{14}-\frac{18\cdots 27}{15\cdots 28}a^{13}+\frac{27\cdots 61}{79\cdots 64}a^{12}+\frac{35\cdots 89}{79\cdots 64}a^{11}-\frac{11\cdots 45}{12\cdots 76}a^{10}+\frac{54\cdots 67}{99\cdots 08}a^{9}-\frac{13\cdots 55}{19\cdots 16}a^{8}+\frac{16\cdots 07}{19\cdots 16}a^{7}+\frac{90\cdots 79}{99\cdots 08}a^{6}-\frac{18\cdots 21}{99\cdots 08}a^{5}+\frac{10\cdots 71}{24\cdots 52}a^{4}+\frac{29\cdots 75}{17\cdots 68}a^{3}-\frac{39\cdots 75}{62\cdots 88}a^{2}+\frac{54\cdots 09}{62\cdots 88}a-\frac{85\cdots 55}{15\cdots 72}$, $\frac{91\cdots 03}{79\cdots 64}a^{17}-\frac{16\cdots 01}{79\cdots 64}a^{16}+\frac{33\cdots 99}{39\cdots 32}a^{15}+\frac{90\cdots 33}{79\cdots 64}a^{14}-\frac{83\cdots 77}{79\cdots 64}a^{13}+\frac{48\cdots 69}{39\cdots 32}a^{12}+\frac{81\cdots 35}{39\cdots 32}a^{11}-\frac{41\cdots 19}{99\cdots 08}a^{10}+\frac{86\cdots 83}{49\cdots 04}a^{9}-\frac{20\cdots 61}{99\cdots 08}a^{8}+\frac{13\cdots 29}{99\cdots 08}a^{7}+\frac{22\cdots 95}{49\cdots 04}a^{6}-\frac{31\cdots 71}{49\cdots 04}a^{5}+\frac{28\cdots 07}{31\cdots 44}a^{4}+\frac{10\cdots 25}{89\cdots 84}a^{3}-\frac{44\cdots 09}{15\cdots 72}a^{2}+\frac{35\cdots 55}{31\cdots 44}a-\frac{49\cdots 43}{77\cdots 86}$, $\frac{49\cdots 61}{15\cdots 28}a^{17}-\frac{67\cdots 67}{15\cdots 28}a^{16}+\frac{16\cdots 11}{79\cdots 64}a^{15}+\frac{19\cdots 03}{15\cdots 28}a^{14}-\frac{42\cdots 39}{15\cdots 28}a^{13}+\frac{25\cdots 69}{79\cdots 64}a^{12}+\frac{55\cdots 61}{79\cdots 64}a^{11}-\frac{90\cdots 81}{99\cdots 08}a^{10}+\frac{42\cdots 35}{99\cdots 08}a^{9}-\frac{71\cdots 43}{19\cdots 16}a^{8}+\frac{32\cdots 95}{19\cdots 16}a^{7}+\frac{13\cdots 63}{99\cdots 08}a^{6}-\frac{11\cdots 01}{99\cdots 08}a^{5}+\frac{41\cdots 65}{24\cdots 52}a^{4}+\frac{75\cdots 15}{17\cdots 68}a^{3}-\frac{39\cdots 61}{62\cdots 88}a^{2}+\frac{22\cdots 77}{62\cdots 88}a+\frac{31\cdots 73}{15\cdots 72}$, $\frac{94\cdots 61}{49\cdots 04}a^{17}-\frac{91\cdots 41}{19\cdots 16}a^{16}+\frac{32\cdots 89}{19\cdots 16}a^{15}-\frac{19\cdots 05}{24\cdots 52}a^{14}-\frac{37\cdots 95}{19\cdots 16}a^{13}+\frac{42\cdots 41}{19\cdots 16}a^{12}+\frac{49\cdots 37}{24\cdots 52}a^{11}-\frac{91\cdots 77}{99\cdots 08}a^{10}+\frac{16\cdots 27}{49\cdots 04}a^{9}-\frac{13\cdots 91}{24\cdots 52}a^{8}+\frac{11\cdots 29}{24\cdots 52}a^{7}+\frac{14\cdots 59}{24\cdots 52}a^{6}-\frac{59\cdots 81}{38\cdots 43}a^{5}+\frac{26\cdots 45}{12\cdots 76}a^{4}+\frac{77\cdots 49}{89\cdots 84}a^{3}-\frac{18\cdots 15}{31\cdots 44}a^{2}+\frac{75\cdots 95}{15\cdots 72}a-\frac{45\cdots 18}{38\cdots 43}$, $\frac{10\cdots 93}{17\cdots 08}a^{17}-\frac{22\cdots 51}{17\cdots 08}a^{16}+\frac{42\cdots 23}{87\cdots 04}a^{15}-\frac{22\cdots 09}{17\cdots 08}a^{14}-\frac{90\cdots 15}{17\cdots 08}a^{13}+\frac{57\cdots 21}{87\cdots 04}a^{12}+\frac{70\cdots 89}{87\cdots 04}a^{11}-\frac{13\cdots 89}{54\cdots 44}a^{10}+\frac{10\cdots 47}{10\cdots 88}a^{9}-\frac{31\cdots 39}{21\cdots 76}a^{8}+\frac{26\cdots 55}{21\cdots 76}a^{7}+\frac{20\cdots 91}{10\cdots 88}a^{6}-\frac{44\cdots 25}{10\cdots 88}a^{5}+\frac{16\cdots 55}{27\cdots 72}a^{4}+\frac{49\cdots 37}{13\cdots 36}a^{3}-\frac{11\cdots 07}{68\cdots 68}a^{2}+\frac{81\cdots 73}{68\cdots 68}a-\frac{44\cdots 75}{17\cdots 92}$, $\frac{34\cdots 03}{87\cdots 04}a^{17}-\frac{82\cdots 05}{87\cdots 04}a^{16}+\frac{14\cdots 87}{43\cdots 52}a^{15}-\frac{13\cdots 35}{87\cdots 04}a^{14}-\frac{31\cdots 37}{87\cdots 04}a^{13}+\frac{19\cdots 09}{43\cdots 52}a^{12}+\frac{18\cdots 59}{43\cdots 52}a^{11}-\frac{20\cdots 01}{10\cdots 88}a^{10}+\frac{37\cdots 33}{54\cdots 44}a^{9}-\frac{11\cdots 73}{10\cdots 88}a^{8}+\frac{10\cdots 13}{10\cdots 88}a^{7}+\frac{65\cdots 79}{54\cdots 44}a^{6}-\frac{16\cdots 75}{54\cdots 44}a^{5}+\frac{31\cdots 19}{68\cdots 68}a^{4}+\frac{12\cdots 53}{68\cdots 68}a^{3}-\frac{20\cdots 23}{17\cdots 92}a^{2}+\frac{35\cdots 43}{34\cdots 84}a-\frac{23\cdots 63}{85\cdots 46}$, $\frac{18\cdots 81}{28\cdots 88}a^{17}-\frac{21\cdots 35}{14\cdots 44}a^{16}+\frac{15\cdots 07}{28\cdots 88}a^{15}-\frac{75\cdots 31}{28\cdots 88}a^{14}-\frac{52\cdots 53}{89\cdots 84}a^{13}+\frac{20\cdots 79}{28\cdots 88}a^{12}+\frac{47\cdots 95}{71\cdots 72}a^{11}-\frac{42\cdots 69}{14\cdots 44}a^{10}+\frac{19\cdots 89}{17\cdots 68}a^{9}-\frac{31\cdots 23}{17\cdots 68}a^{8}+\frac{14\cdots 89}{89\cdots 84}a^{7}+\frac{69\cdots 69}{35\cdots 36}a^{6}-\frac{45\cdots 47}{89\cdots 84}a^{5}+\frac{13\cdots 01}{17\cdots 68}a^{4}+\frac{13\cdots 97}{44\cdots 92}a^{3}-\frac{86\cdots 41}{44\cdots 92}a^{2}+\frac{95\cdots 16}{55\cdots 49}a-\frac{25\cdots 75}{55\cdots 49}$, $\frac{75\cdots 89}{22\cdots 04}a^{17}-\frac{15\cdots 19}{22\cdots 04}a^{16}+\frac{29\cdots 07}{11\cdots 52}a^{15}-\frac{14\cdots 09}{22\cdots 04}a^{14}-\frac{67\cdots 19}{22\cdots 04}a^{13}+\frac{37\cdots 17}{11\cdots 52}a^{12}+\frac{53\cdots 81}{11\cdots 52}a^{11}-\frac{11\cdots 85}{89\cdots 84}a^{10}+\frac{70\cdots 45}{14\cdots 44}a^{9}-\frac{19\cdots 15}{28\cdots 88}a^{8}+\frac{14\cdots 51}{28\cdots 88}a^{7}+\frac{15\cdots 07}{14\cdots 44}a^{6}-\frac{28\cdots 29}{14\cdots 44}a^{5}+\frac{99\cdots 39}{35\cdots 36}a^{4}+\frac{45\cdots 75}{17\cdots 68}a^{3}-\frac{71\cdots 47}{89\cdots 84}a^{2}+\frac{45\cdots 85}{89\cdots 84}a-\frac{23\cdots 23}{22\cdots 96}$
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| Regulator: | \( 297602831.7716659 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 297602831.7716659 \cdot 1}{6\cdot\sqrt{200930163501792205662554161152}}\cr\approx \mathstrut & 1.68881491303713 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3^2$ (as 18T46):
| A solvable group of order 108 |
| The 27 conjugacy class representatives for $C_3\times S_3^2$ |
| Character table for $C_3\times S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.728.1, 6.0.14309568.1, 6.0.223587.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 27 sibling: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.898666574987777490026496.5 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ | |
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ | |
| 2.2.2.6a1.5 | $x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$ | $2$ | $2$ | $6$ | $C_2^2$ | $$[3]^{2}$$ | |
|
\(3\)
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 3.3.2.3a1.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 4 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(7\)
| 7.3.1.0a1.1 | $x^{3} + 6 x^{2} + 4$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 7.1.3.2a1.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.1.6.5a1.3 | $x^{6} + 21$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
| 7.3.2.3a1.1 | $x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 7 x + 16$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
|
\(13\)
| 13.3.1.0a1.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |
| 13.1.3.2a1.2 | $x^{3} + 26$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 13.3.2.3a1.2 | $x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 13.1.6.5a1.3 | $x^{6} + 39$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |