Normalized defining polynomial
\( x^{18} - 8 x^{17} + 37 x^{16} - 144 x^{15} + 480 x^{14} - 960 x^{13} + 1800 x^{12} - 2304 x^{11} + \cdots - 299763 \)
Invariants
| Degree: | $18$ |
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| Signature: | $(2, 8)$ |
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| Discriminant: |
\(24334781169379526551941294720024576\)
\(\medspace = 2^{54}\cdot 3^{38}\)
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| Root discriminant: | \(81.35\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\)
|
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{11}+\frac{1}{12}a^{10}+\frac{1}{12}a^{9}-\frac{1}{12}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{12}a^{12}-\frac{1}{6}a^{9}-\frac{1}{4}a^{8}-\frac{1}{6}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{12}a^{13}-\frac{1}{6}a^{10}-\frac{1}{4}a^{9}-\frac{1}{6}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{12}a^{14}-\frac{1}{12}a^{10}+\frac{1}{6}a^{9}+\frac{1}{6}a^{8}-\frac{1}{6}a^{7}+\frac{1}{12}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{12}a^{15}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{72}a^{16}-\frac{1}{6}a^{10}-\frac{1}{4}a^{8}-\frac{1}{18}a^{7}-\frac{1}{2}a^{5}-\frac{1}{3}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{6}a+\frac{1}{8}$, $\frac{1}{61\cdots 68}a^{17}+\frac{25\cdots 17}{15\cdots 42}a^{16}-\frac{13\cdots 59}{51\cdots 14}a^{15}+\frac{32\cdots 31}{10\cdots 28}a^{14}-\frac{10\cdots 83}{10\cdots 28}a^{13}+\frac{61\cdots 09}{17\cdots 38}a^{12}+\frac{18\cdots 66}{85\cdots 19}a^{11}-\frac{30\cdots 89}{34\cdots 76}a^{10}+\frac{24\cdots 31}{85\cdots 19}a^{9}+\frac{17\cdots 97}{77\cdots 71}a^{8}-\frac{52\cdots 89}{77\cdots 71}a^{7}-\frac{17\cdots 67}{10\cdots 28}a^{6}+\frac{20\cdots 49}{10\cdots 28}a^{5}+\frac{23\cdots 47}{51\cdots 14}a^{4}-\frac{56\cdots 05}{17\cdots 38}a^{3}-\frac{11\cdots 81}{10\cdots 28}a^{2}-\frac{15\cdots 63}{20\cdots 56}a-\frac{46\cdots 17}{17\cdots 38}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{24\cdots 25}{30\cdots 84}a^{17}+\frac{27\cdots 71}{15\cdots 42}a^{16}-\frac{17\cdots 59}{10\cdots 28}a^{15}+\frac{14\cdots 39}{25\cdots 57}a^{14}-\frac{28\cdots 75}{10\cdots 28}a^{13}+\frac{97\cdots 48}{85\cdots 19}a^{12}-\frac{60\cdots 92}{85\cdots 19}a^{11}+\frac{14\cdots 93}{34\cdots 76}a^{10}-\frac{78\cdots 49}{17\cdots 38}a^{9}-\frac{23\cdots 31}{30\cdots 84}a^{8}+\frac{31\cdots 61}{30\cdots 84}a^{7}+\frac{43\cdots 49}{51\cdots 14}a^{6}+\frac{63\cdots 95}{10\cdots 28}a^{5}-\frac{49\cdots 89}{51\cdots 14}a^{4}+\frac{51\cdots 57}{85\cdots 19}a^{3}-\frac{94\cdots 05}{10\cdots 28}a^{2}+\frac{59\cdots 71}{10\cdots 28}a-\frac{91\cdots 41}{34\cdots 76}$, $\frac{77\cdots 79}{20\cdots 56}a^{17}+\frac{79\cdots 59}{22\cdots 84}a^{16}-\frac{29\cdots 45}{17\cdots 38}a^{15}+\frac{56\cdots 10}{85\cdots 19}a^{14}-\frac{25\cdots 79}{11\cdots 92}a^{13}+\frac{82\cdots 31}{17\cdots 38}a^{12}-\frac{87\cdots 63}{11\cdots 92}a^{11}+\frac{10\cdots 81}{11\cdots 92}a^{10}+\frac{33\cdots 37}{34\cdots 76}a^{9}-\frac{63\cdots 83}{51\cdots 14}a^{8}+\frac{14\cdots 29}{57\cdots 46}a^{7}-\frac{20\cdots 47}{28\cdots 73}a^{6}+\frac{37\cdots 29}{34\cdots 76}a^{5}-\frac{11\cdots 19}{57\cdots 46}a^{4}+\frac{28\cdots 95}{11\cdots 92}a^{3}-\frac{83\cdots 87}{34\cdots 76}a^{2}+\frac{87\cdots 09}{22\cdots 84}a-\frac{63\cdots 47}{22\cdots 84}$, $\frac{79\cdots 69}{61\cdots 68}a^{17}+\frac{78\cdots 39}{61\cdots 68}a^{16}-\frac{63\cdots 97}{10\cdots 28}a^{15}+\frac{23\cdots 17}{10\cdots 28}a^{14}-\frac{81\cdots 45}{10\cdots 28}a^{13}+\frac{61\cdots 03}{34\cdots 76}a^{12}-\frac{30\cdots 59}{11\cdots 92}a^{11}+\frac{13\cdots 95}{34\cdots 76}a^{10}+\frac{17\cdots 76}{85\cdots 19}a^{9}-\frac{73\cdots 77}{15\cdots 42}a^{8}+\frac{29\cdots 17}{30\cdots 84}a^{7}-\frac{22\cdots 51}{10\cdots 28}a^{6}+\frac{61\cdots 39}{10\cdots 28}a^{5}-\frac{59\cdots 91}{10\cdots 28}a^{4}+\frac{31\cdots 11}{34\cdots 76}a^{3}-\frac{11\cdots 35}{10\cdots 28}a^{2}+\frac{12\cdots 55}{20\cdots 56}a-\frac{18\cdots 07}{68\cdots 52}$, $\frac{87\cdots 73}{61\cdots 68}a^{17}-\frac{69\cdots 63}{61\cdots 68}a^{16}+\frac{51\cdots 69}{10\cdots 28}a^{15}-\frac{19\cdots 01}{10\cdots 28}a^{14}+\frac{16\cdots 58}{25\cdots 57}a^{13}-\frac{40\cdots 15}{34\cdots 76}a^{12}+\frac{17\cdots 48}{85\cdots 19}a^{11}-\frac{20\cdots 96}{85\cdots 19}a^{10}-\frac{39\cdots 17}{85\cdots 19}a^{9}+\frac{11\cdots 39}{30\cdots 84}a^{8}-\frac{14\cdots 63}{30\cdots 84}a^{7}+\frac{26\cdots 59}{10\cdots 28}a^{6}-\frac{44\cdots 34}{25\cdots 57}a^{5}+\frac{73\cdots 79}{10\cdots 28}a^{4}-\frac{46\cdots 64}{85\cdots 19}a^{3}+\frac{31\cdots 55}{51\cdots 14}a^{2}-\frac{25\cdots 43}{20\cdots 56}a+\frac{21\cdots 97}{68\cdots 52}$, $\frac{58\cdots 93}{15\cdots 42}a^{17}-\frac{46\cdots 75}{15\cdots 42}a^{16}+\frac{69\cdots 13}{51\cdots 14}a^{15}-\frac{26\cdots 67}{51\cdots 14}a^{14}+\frac{44\cdots 92}{25\cdots 57}a^{13}-\frac{11\cdots 89}{34\cdots 76}a^{12}+\frac{34\cdots 09}{57\cdots 46}a^{11}-\frac{12\cdots 75}{17\cdots 38}a^{10}-\frac{10\cdots 19}{85\cdots 19}a^{9}+\frac{30\cdots 11}{30\cdots 84}a^{8}-\frac{98\cdots 08}{77\cdots 71}a^{7}+\frac{18\cdots 51}{25\cdots 57}a^{6}-\frac{24\cdots 99}{51\cdots 14}a^{5}+\frac{20\cdots 29}{10\cdots 28}a^{4}-\frac{14\cdots 26}{85\cdots 19}a^{3}+\frac{44\cdots 34}{25\cdots 57}a^{2}-\frac{94\cdots 05}{25\cdots 57}a+\frac{42\cdots 69}{34\cdots 76}$, $\frac{86\cdots 21}{68\cdots 52}a^{17}+\frac{11\cdots 11}{68\cdots 52}a^{16}-\frac{32\cdots 79}{34\cdots 76}a^{15}+\frac{41\cdots 07}{11\cdots 92}a^{14}-\frac{10\cdots 06}{85\cdots 19}a^{13}+\frac{97\cdots 35}{34\cdots 76}a^{12}-\frac{51\cdots 81}{17\cdots 38}a^{11}+\frac{74\cdots 19}{85\cdots 19}a^{10}+\frac{43\cdots 58}{28\cdots 73}a^{9}-\frac{10\cdots 37}{11\cdots 92}a^{8}+\frac{76\cdots 47}{34\cdots 76}a^{7}-\frac{27\cdots 41}{11\cdots 92}a^{6}+\frac{24\cdots 77}{28\cdots 73}a^{5}+\frac{55\cdots 17}{11\cdots 92}a^{4}+\frac{60\cdots 75}{57\cdots 46}a^{3}+\frac{14\cdots 37}{57\cdots 46}a^{2}-\frac{26\cdots 93}{22\cdots 84}a+\frac{24\cdots 25}{22\cdots 84}$, $\frac{10\cdots 77}{20\cdots 56}a^{17}+\frac{75\cdots 09}{20\cdots 56}a^{16}-\frac{25\cdots 39}{17\cdots 38}a^{15}+\frac{14\cdots 28}{28\cdots 73}a^{14}-\frac{42\cdots 70}{28\cdots 73}a^{13}+\frac{28\cdots 05}{17\cdots 38}a^{12}-\frac{99\cdots 31}{85\cdots 19}a^{11}-\frac{40\cdots 23}{28\cdots 73}a^{10}+\frac{28\cdots 79}{11\cdots 92}a^{9}-\frac{11\cdots 95}{10\cdots 28}a^{8}+\frac{15\cdots 07}{51\cdots 14}a^{7}-\frac{31\cdots 57}{57\cdots 46}a^{6}-\frac{22\cdots 53}{85\cdots 19}a^{5}-\frac{10\cdots 90}{85\cdots 19}a^{4}+\frac{82\cdots 34}{28\cdots 73}a^{3}+\frac{51\cdots 07}{17\cdots 38}a^{2}+\frac{83\cdots 09}{68\cdots 52}a+\frac{11\cdots 05}{22\cdots 84}$, $\frac{11\cdots 33}{61\cdots 68}a^{17}-\frac{12\cdots 69}{61\cdots 68}a^{16}-\frac{13\cdots 97}{25\cdots 57}a^{15}+\frac{38\cdots 27}{10\cdots 28}a^{14}-\frac{16\cdots 23}{10\cdots 28}a^{13}+\frac{55\cdots 48}{85\cdots 19}a^{12}-\frac{12\cdots 92}{85\cdots 19}a^{11}+\frac{25\cdots 81}{11\cdots 92}a^{10}-\frac{46\cdots 45}{57\cdots 46}a^{9}-\frac{24\cdots 89}{30\cdots 84}a^{8}+\frac{49\cdots 76}{77\cdots 71}a^{7}-\frac{12\cdots 99}{10\cdots 28}a^{6}+\frac{37\cdots 39}{10\cdots 28}a^{5}-\frac{12\cdots 94}{25\cdots 57}a^{4}+\frac{64\cdots 58}{85\cdots 19}a^{3}-\frac{88\cdots 81}{10\cdots 28}a^{2}+\frac{91\cdots 09}{20\cdots 56}a-\frac{12\cdots 49}{68\cdots 52}$, $\frac{89\cdots 87}{15\cdots 42}a^{17}-\frac{36\cdots 98}{77\cdots 71}a^{16}+\frac{21\cdots 23}{10\cdots 28}a^{15}-\frac{78\cdots 51}{10\cdots 28}a^{14}+\frac{24\cdots 79}{10\cdots 28}a^{13}-\frac{41\cdots 09}{11\cdots 92}a^{12}+\frac{33\cdots 45}{11\cdots 92}a^{11}+\frac{60\cdots 21}{11\cdots 92}a^{10}-\frac{16\cdots 87}{34\cdots 76}a^{9}+\frac{60\cdots 59}{30\cdots 84}a^{8}-\frac{67\cdots 93}{30\cdots 84}a^{7}+\frac{78\cdots 89}{10\cdots 28}a^{6}+\frac{43\cdots 17}{10\cdots 28}a^{5}+\frac{37\cdots 51}{10\cdots 28}a^{4}+\frac{43\cdots 41}{34\cdots 76}a^{3}-\frac{35\cdots 91}{10\cdots 28}a^{2}+\frac{20\cdots 75}{10\cdots 28}a-\frac{33\cdots 85}{34\cdots 76}$
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| Regulator: | \( 208003889751.14267 \) (assuming GRH) |
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| Unit signature rank: | \( 2 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 208003889751.14267 \cdot 1}{2\cdot\sqrt{24334781169379526551941294720024576}}\cr\approx \mathstrut & 6.47778719130032 \end{aligned}\] (assuming GRH)
Galois group
| A non-solvable group of order 3201186852864000 |
| The 200 conjugacy class representatives for $A_{18}$ |
| Character table for $A_{18}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | $17{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/23.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.16.52e2.58 | $x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{5} + 16 x^{4} + 2$ | $16$ | $1$ | $52$ | 16T1658 | not computed | |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 3.1.6.11a2.5 | $x^{6} + 9 x^{2} + 9 x + 6$ | $6$ | $1$ | $11$ | $S_3^2$ | $$[2, \frac{5}{2}]_{2}^{2}$$ | |
| 3.1.9.26a1.15 | $x^{9} + 54 x^{4} + 27 x^{2} + 27 x + 3$ | $9$ | $1$ | $26$ | $(C_3^3:C_3):C_2$ | $$[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}, \frac{7}{2}]_{2}$$ |