Normalized defining polynomial
\( x^{18} - 6 x^{16} - 36 x^{15} + 156 x^{13} + 456 x^{12} + 252 x^{11} - 627 x^{10} - 1148 x^{9} + \cdots + 4 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $[4, 7]$ |
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| Discriminant: |
\(-231812806445087701493115518976\)
\(\medspace = -\,2^{50}\cdot 3^{30}\)
|
| |
| Root discriminant: | \(42.80\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\)
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| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{2}$, $\frac{1}{6}a^{12}-\frac{1}{2}a^{8}+\frac{1}{3}a^{3}$, $\frac{1}{6}a^{13}-\frac{1}{6}a^{9}+\frac{1}{3}a^{4}-\frac{1}{3}$, $\frac{1}{6}a^{14}-\frac{1}{6}a^{10}+\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{18}a^{15}+\frac{1}{18}a^{12}-\frac{1}{6}a^{11}-\frac{1}{9}a^{9}-\frac{1}{2}a^{8}-\frac{2}{9}a^{6}-\frac{2}{9}a^{3}-\frac{1}{3}a^{2}+\frac{4}{9}$, $\frac{1}{72}a^{16}-\frac{1}{36}a^{15}+\frac{1}{24}a^{14}+\frac{1}{18}a^{13}-\frac{5}{72}a^{12}-\frac{1}{12}a^{11}+\frac{7}{72}a^{10}-\frac{1}{9}a^{9}-\frac{1}{2}a^{8}-\frac{1}{18}a^{7}-\frac{5}{36}a^{6}+\frac{1}{3}a^{5}+\frac{1}{36}a^{4}+\frac{5}{18}a^{3}-\frac{5}{12}a^{2}+\frac{1}{9}a-\frac{1}{18}$, $\frac{1}{25\cdots 08}a^{17}-\frac{39546489564797}{85\cdots 36}a^{16}+\frac{63015230678389}{25\cdots 08}a^{15}-\frac{10\cdots 31}{25\cdots 08}a^{14}+\frac{363161113739201}{85\cdots 36}a^{13}-\frac{17\cdots 85}{25\cdots 08}a^{12}+\frac{17\cdots 33}{25\cdots 08}a^{11}+\frac{654287031738455}{85\cdots 36}a^{10}+\frac{274947075337133}{32\cdots 26}a^{9}+\frac{30\cdots 71}{64\cdots 52}a^{8}+\frac{15\cdots 39}{42\cdots 68}a^{7}-\frac{479077759247171}{12\cdots 04}a^{6}+\frac{18\cdots 93}{12\cdots 04}a^{5}-\frac{960576320963017}{42\cdots 68}a^{4}-\frac{51\cdots 65}{12\cdots 04}a^{3}-\frac{49\cdots 85}{12\cdots 04}a^{2}-\frac{193544423348663}{21\cdots 84}a+\frac{668440285904189}{64\cdots 52}$
| 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: | $10$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{3071063}{30605768}a^{17}-\frac{1280075}{30605768}a^{16}-\frac{166214353}{275451912}a^{15}-\frac{309221141}{91817304}a^{14}+\frac{139107491}{91817304}a^{13}+\frac{4335060875}{275451912}a^{12}+\frac{3622896493}{91817304}a^{11}+\frac{182943905}{30605768}a^{10}-\frac{5163632851}{68862978}a^{9}-\frac{704854885}{7651442}a^{8}-\frac{519476377}{15302884}a^{7}+\frac{17601585065}{137725956}a^{6}+\frac{4118295421}{45908652}a^{5}-\frac{4045669249}{45908652}a^{4}-\frac{3887766919}{137725956}a^{3}+\frac{455955253}{45908652}a^{2}+\frac{29988171}{7651442}a+\frac{50826319}{68862978}$, $\frac{38\cdots 69}{25\cdots 08}a^{17}-\frac{201304717691303}{28\cdots 12}a^{16}-\frac{22\cdots 51}{25\cdots 08}a^{15}-\frac{12\cdots 51}{25\cdots 08}a^{14}+\frac{20\cdots 49}{85\cdots 36}a^{13}+\frac{58\cdots 27}{25\cdots 08}a^{12}+\frac{14\cdots 45}{25\cdots 08}a^{11}+\frac{27\cdots 37}{28\cdots 12}a^{10}-\frac{33\cdots 39}{32\cdots 26}a^{9}-\frac{80\cdots 11}{64\cdots 52}a^{8}-\frac{80\cdots 19}{14\cdots 56}a^{7}+\frac{21\cdots 97}{12\cdots 04}a^{6}+\frac{13\cdots 77}{12\cdots 04}a^{5}-\frac{54\cdots 81}{42\cdots 68}a^{4}-\frac{81\cdots 49}{12\cdots 04}a^{3}+\frac{16\cdots 11}{12\cdots 04}a^{2}-\frac{44\cdots 81}{715305262393628}a+\frac{23\cdots 33}{64\cdots 52}$, $\frac{642055187733723}{28\cdots 12}a^{17}-\frac{18\cdots 17}{25\cdots 08}a^{16}-\frac{34\cdots 17}{25\cdots 08}a^{15}-\frac{65\cdots 87}{85\cdots 36}a^{14}+\frac{64\cdots 33}{25\cdots 08}a^{13}+\frac{88\cdots 25}{25\cdots 08}a^{12}+\frac{26\cdots 47}{28\cdots 12}a^{11}+\frac{67\cdots 31}{25\cdots 08}a^{10}-\frac{24\cdots 70}{16\cdots 63}a^{9}-\frac{14\cdots 33}{715305262393628}a^{8}-\frac{14\cdots 09}{12\cdots 04}a^{7}+\frac{30\cdots 27}{12\cdots 04}a^{6}+\frac{81\cdots 77}{42\cdots 68}a^{5}-\frac{21\cdots 45}{12\cdots 04}a^{4}-\frac{47\cdots 43}{12\cdots 04}a^{3}+\frac{17\cdots 37}{14\cdots 56}a^{2}-\frac{20\cdots 31}{64\cdots 52}a+\frac{33\cdots 87}{64\cdots 52}$, $\frac{50\cdots 45}{25\cdots 08}a^{17}-\frac{143655934448845}{25\cdots 08}a^{16}-\frac{30\cdots 59}{25\cdots 08}a^{15}-\frac{17\cdots 37}{25\cdots 08}a^{14}+\frac{81\cdots 71}{25\cdots 08}a^{13}+\frac{79\cdots 05}{25\cdots 08}a^{12}+\frac{22\cdots 25}{25\cdots 08}a^{11}+\frac{11\cdots 31}{25\cdots 08}a^{10}-\frac{42\cdots 65}{32\cdots 26}a^{9}-\frac{14\cdots 91}{64\cdots 52}a^{8}-\frac{18\cdots 31}{12\cdots 04}a^{7}+\frac{25\cdots 71}{12\cdots 04}a^{6}+\frac{30\cdots 37}{12\cdots 04}a^{5}-\frac{13\cdots 21}{12\cdots 04}a^{4}-\frac{11\cdots 13}{12\cdots 04}a^{3}+\frac{14\cdots 53}{12\cdots 04}a^{2}+\frac{58\cdots 87}{64\cdots 52}a+\frac{67\cdots 99}{64\cdots 52}$, $\frac{127092900341065}{64\cdots 52}a^{17}-\frac{8610064227115}{21\cdots 84}a^{16}-\frac{690002230459493}{64\cdots 52}a^{15}-\frac{44\cdots 85}{64\cdots 52}a^{14}+\frac{180784319518523}{21\cdots 84}a^{13}+\frac{17\cdots 37}{64\cdots 52}a^{12}+\frac{54\cdots 11}{64\cdots 52}a^{11}+\frac{99\cdots 81}{21\cdots 84}a^{10}-\frac{27\cdots 55}{32\cdots 26}a^{9}-\frac{29\cdots 07}{16\cdots 63}a^{8}-\frac{17\cdots 41}{10\cdots 42}a^{7}+\frac{32\cdots 57}{32\cdots 26}a^{6}+\frac{39\cdots 63}{32\cdots 26}a^{5}-\frac{54\cdots 59}{10\cdots 42}a^{4}-\frac{16\cdots 87}{32\cdots 26}a^{3}-\frac{72\cdots 19}{32\cdots 26}a^{2}-\frac{227811286238318}{536478946795221}a+\frac{14581431824894}{16\cdots 63}$, $\frac{577536178763947}{25\cdots 08}a^{17}-\frac{315729611270975}{25\cdots 08}a^{16}-\frac{417991587926397}{28\cdots 12}a^{15}-\frac{19\cdots 03}{25\cdots 08}a^{14}+\frac{12\cdots 05}{25\cdots 08}a^{13}+\frac{11\cdots 27}{28\cdots 12}a^{12}+\frac{23\cdots 99}{25\cdots 08}a^{11}-\frac{32\cdots 71}{25\cdots 08}a^{10}-\frac{13\cdots 54}{536478946795221}a^{9}-\frac{20\cdots 93}{64\cdots 52}a^{8}-\frac{11\cdots 89}{12\cdots 04}a^{7}+\frac{65\cdots 13}{14\cdots 56}a^{6}+\frac{63\cdots 67}{12\cdots 04}a^{5}-\frac{11\cdots 67}{12\cdots 04}a^{4}-\frac{12\cdots 83}{14\cdots 56}a^{3}-\frac{66\cdots 57}{12\cdots 04}a^{2}-\frac{94\cdots 11}{64\cdots 52}a-\frac{275591047987733}{21\cdots 84}$, $\frac{390473010914107}{25\cdots 08}a^{17}+\frac{375061467055249}{85\cdots 36}a^{16}-\frac{29\cdots 57}{25\cdots 08}a^{15}-\frac{20\cdots 93}{25\cdots 08}a^{14}-\frac{12\cdots 21}{85\cdots 36}a^{13}+\frac{81\cdots 93}{25\cdots 08}a^{12}+\frac{34\cdots 75}{25\cdots 08}a^{11}+\frac{58\cdots 83}{28\cdots 12}a^{10}-\frac{25\cdots 15}{32\cdots 26}a^{9}-\frac{30\cdots 17}{64\cdots 52}a^{8}-\frac{20\cdots 91}{42\cdots 68}a^{7}+\frac{99\cdots 39}{12\cdots 04}a^{6}+\frac{92\cdots 19}{12\cdots 04}a^{5}+\frac{65\cdots 33}{42\cdots 68}a^{4}-\frac{55\cdots 99}{12\cdots 04}a^{3}+\frac{23\cdots 09}{12\cdots 04}a^{2}-\frac{706555892450371}{715305262393628}a+\frac{16\cdots 15}{64\cdots 52}$, $\frac{60\cdots 43}{25\cdots 08}a^{17}-\frac{409790231259563}{25\cdots 08}a^{16}-\frac{36\cdots 77}{25\cdots 08}a^{15}-\frac{21\cdots 63}{25\cdots 08}a^{14}+\frac{15\cdots 57}{25\cdots 08}a^{13}+\frac{94\cdots 79}{25\cdots 08}a^{12}+\frac{26\cdots 47}{25\cdots 08}a^{11}+\frac{13\cdots 45}{25\cdots 08}a^{10}-\frac{24\cdots 12}{16\cdots 63}a^{9}-\frac{16\cdots 01}{64\cdots 52}a^{8}-\frac{21\cdots 37}{12\cdots 04}a^{7}+\frac{28\cdots 69}{12\cdots 04}a^{6}+\frac{33\cdots 27}{12\cdots 04}a^{5}-\frac{16\cdots 07}{12\cdots 04}a^{4}-\frac{11\cdots 43}{12\cdots 04}a^{3}+\frac{10\cdots 63}{12\cdots 04}a^{2}+\frac{70\cdots 21}{64\cdots 52}a+\frac{35\cdots 29}{64\cdots 52}$, $\frac{31\cdots 67}{12\cdots 04}a^{17}+\frac{10\cdots 53}{12\cdots 04}a^{16}-\frac{18\cdots 77}{12\cdots 04}a^{15}-\frac{11\cdots 59}{12\cdots 04}a^{14}-\frac{42\cdots 31}{12\cdots 04}a^{13}+\frac{45\cdots 99}{12\cdots 04}a^{12}+\frac{15\cdots 19}{12\cdots 04}a^{11}+\frac{13\cdots 77}{12\cdots 04}a^{10}-\frac{31\cdots 99}{32\cdots 26}a^{9}-\frac{96\cdots 91}{32\cdots 26}a^{8}-\frac{20\cdots 01}{64\cdots 52}a^{7}+\frac{42\cdots 09}{64\cdots 52}a^{6}+\frac{17\cdots 35}{64\cdots 52}a^{5}+\frac{52\cdots 61}{64\cdots 52}a^{4}-\frac{32\cdots 87}{64\cdots 52}a^{3}-\frac{19\cdots 89}{64\cdots 52}a^{2}-\frac{68\cdots 57}{32\cdots 26}a-\frac{99\cdots 65}{32\cdots 26}$, $\frac{70\cdots 63}{25\cdots 08}a^{17}-\frac{43\cdots 17}{25\cdots 08}a^{16}-\frac{13\cdots 51}{85\cdots 36}a^{15}-\frac{22\cdots 33}{25\cdots 08}a^{14}+\frac{14\cdots 61}{25\cdots 08}a^{13}+\frac{35\cdots 11}{85\cdots 36}a^{12}+\frac{25\cdots 11}{25\cdots 08}a^{11}-\frac{94\cdots 45}{25\cdots 08}a^{10}-\frac{10\cdots 31}{536478946795221}a^{9}-\frac{12\cdots 61}{64\cdots 52}a^{8}-\frac{74\cdots 49}{12\cdots 04}a^{7}+\frac{14\cdots 45}{42\cdots 68}a^{6}+\frac{18\cdots 91}{12\cdots 04}a^{5}-\frac{35\cdots 93}{12\cdots 04}a^{4}+\frac{30\cdots 59}{42\cdots 68}a^{3}+\frac{50\cdots 01}{12\cdots 04}a^{2}-\frac{14\cdots 55}{64\cdots 52}a-\frac{29\cdots 07}{21\cdots 84}$
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| Regulator: | \( 631164608.461 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{7}\cdot 631164608.461 \cdot 1}{2\cdot\sqrt{231812806445087701493115518976}}\cr\approx \mathstrut & 4.05436637724 \end{aligned}\] (assuming GRH)
Galois group
$C_3^4:(C_6^2:C_4)$ (as 18T576):
| A solvable group of order 11664 |
| The 49 conjugacy class representatives for $C_3^4:(C_6^2:C_4)$ |
| Character table for $C_3^4:(C_6^2:C_4)$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 6.2.11943936.2 |
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 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 | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.4.10a1.1 | $x^{4} + 4 x^{3} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ | |
| 2.1.4.11a1.12 | $x^{4} + 8 x^{3} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ | |
| 2.1.8.26c1.6 | $x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 18$ | $8$ | $1$ | $26$ | $C_2^2:C_4$ | $$[2, 3, \frac{7}{2}, 4]$$ | |
|
\(3\)
| 3.2.9.30b27.5 | $x^{18} + 21 x^{17} + 213 x^{16} + 1392 x^{15} + 6582 x^{14} + 23940 x^{13} + 69468 x^{12} + 164544 x^{11} + 322728 x^{10} + 528470 x^{9} + 724902 x^{8} + 831987 x^{7} + 794322 x^{6} + 623622 x^{5} + 394896 x^{4} + 195372 x^{3} + 71496 x^{2} + 17400 x + 2147$ | $9$ | $2$ | $30$ | not computed | not computed |