Normalized defining polynomial
\( x^{18} - 12 x^{16} - 20 x^{15} + 45 x^{14} + 240 x^{13} + 180 x^{12} - 900 x^{11} - 2385 x^{10} + \cdots + 3100 \)
Invariants
| Degree: | $18$ |
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| Signature: | $(6, 6)$ |
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| Discriminant: |
\(5077997833420800000000000000000\)
\(\medspace = 2^{34}\cdot 3^{18}\cdot 5^{17}\)
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| |
| Root discriminant: | \(50.80\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{25\cdots 70}a^{17}-\frac{17\cdots 73}{25\cdots 67}a^{16}-\frac{16\cdots 77}{25\cdots 70}a^{15}+\frac{69\cdots 58}{25\cdots 67}a^{14}-\frac{98\cdots 61}{51\cdots 34}a^{13}-\frac{12\cdots 18}{25\cdots 67}a^{12}+\frac{10\cdots 33}{51\cdots 34}a^{11}-\frac{45\cdots 92}{25\cdots 67}a^{10}+\frac{20\cdots 45}{51\cdots 34}a^{9}-\frac{32\cdots 49}{25\cdots 67}a^{8}+\frac{14\cdots 09}{51\cdots 34}a^{7}-\frac{22\cdots 81}{25\cdots 67}a^{6}-\frac{10\cdots 59}{51\cdots 34}a^{5}-\frac{15\cdots 87}{25\cdots 67}a^{4}+\frac{50\cdots 53}{51\cdots 34}a^{3}+\frac{57\cdots 02}{25\cdots 67}a^{2}+\frac{99\cdots 08}{25\cdots 67}a+\frac{21\cdots 85}{25\cdots 67}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
$\frac{48\cdots 23}{18\cdots 30}a^{17}-\frac{24\cdots 35}{18\cdots 53}a^{16}-\frac{58\cdots 01}{18\cdots 30}a^{15}-\frac{63\cdots 29}{18\cdots 53}a^{14}+\frac{52\cdots 07}{36\cdots 06}a^{13}+\frac{10\cdots 56}{18\cdots 53}a^{12}+\frac{53\cdots 11}{36\cdots 06}a^{11}-\frac{47\cdots 58}{18\cdots 53}a^{10}-\frac{17\cdots 71}{36\cdots 06}a^{9}+\frac{10\cdots 68}{18\cdots 53}a^{8}+\frac{60\cdots 53}{36\cdots 06}a^{7}+\frac{46\cdots 17}{18\cdots 53}a^{6}-\frac{10\cdots 07}{36\cdots 06}a^{5}-\frac{92\cdots 93}{18\cdots 53}a^{4}-\frac{18\cdots 81}{36\cdots 06}a^{3}+\frac{40\cdots 53}{18\cdots 53}a^{2}+\frac{80\cdots 95}{18\cdots 53}a+\frac{20\cdots 62}{18\cdots 53}$, $\frac{51\cdots 79}{36\cdots 06}a^{17}-\frac{13\cdots 22}{18\cdots 53}a^{16}-\frac{31\cdots 99}{18\cdots 53}a^{15}-\frac{34\cdots 08}{18\cdots 53}a^{14}+\frac{28\cdots 81}{36\cdots 06}a^{13}+\frac{55\cdots 76}{18\cdots 53}a^{12}+\frac{14\cdots 04}{18\cdots 53}a^{11}-\frac{25\cdots 29}{18\cdots 53}a^{10}-\frac{97\cdots 41}{36\cdots 06}a^{9}+\frac{59\cdots 62}{18\cdots 53}a^{8}+\frac{16\cdots 00}{18\cdots 53}a^{7}+\frac{25\cdots 67}{18\cdots 53}a^{6}-\frac{92\cdots 79}{36\cdots 06}a^{5}-\frac{53\cdots 12}{18\cdots 53}a^{4}-\frac{54\cdots 57}{18\cdots 53}a^{3}+\frac{19\cdots 50}{18\cdots 53}a^{2}+\frac{51\cdots 65}{18\cdots 53}a+\frac{18\cdots 01}{18\cdots 53}$, $\frac{43\cdots 51}{12\cdots 35}a^{17}-\frac{19\cdots 89}{51\cdots 34}a^{16}-\frac{47\cdots 32}{12\cdots 35}a^{15}-\frac{67\cdots 24}{25\cdots 67}a^{14}+\frac{47\cdots 35}{25\cdots 67}a^{13}+\frac{31\cdots 95}{51\cdots 34}a^{12}-\frac{21\cdots 24}{25\cdots 67}a^{11}-\frac{76\cdots 40}{25\cdots 67}a^{10}-\frac{12\cdots 52}{25\cdots 67}a^{9}+\frac{11\cdots 45}{51\cdots 34}a^{8}+\frac{48\cdots 36}{25\cdots 67}a^{7}+\frac{61\cdots 57}{25\cdots 67}a^{6}-\frac{17\cdots 58}{25\cdots 67}a^{5}-\frac{28\cdots 89}{51\cdots 34}a^{4}-\frac{10\cdots 35}{25\cdots 67}a^{3}+\frac{76\cdots 74}{25\cdots 67}a^{2}+\frac{10\cdots 78}{25\cdots 67}a+\frac{26\cdots 89}{25\cdots 67}$, $\frac{27\cdots 99}{25\cdots 70}a^{17}-\frac{22\cdots 95}{25\cdots 67}a^{16}-\frac{31\cdots 53}{25\cdots 70}a^{15}-\frac{29\cdots 77}{25\cdots 67}a^{14}+\frac{29\cdots 73}{51\cdots 34}a^{13}+\frac{53\cdots 93}{25\cdots 67}a^{12}+\frac{98\cdots 41}{51\cdots 34}a^{11}-\frac{25\cdots 59}{25\cdots 67}a^{10}-\frac{89\cdots 21}{51\cdots 34}a^{9}+\frac{11\cdots 04}{25\cdots 67}a^{8}+\frac{32\cdots 51}{51\cdots 34}a^{7}+\frac{23\cdots 25}{25\cdots 67}a^{6}-\frac{55\cdots 61}{51\cdots 34}a^{5}-\frac{49\cdots 13}{25\cdots 67}a^{4}-\frac{87\cdots 69}{51\cdots 34}a^{3}+\frac{22\cdots 34}{25\cdots 67}a^{2}+\frac{41\cdots 11}{25\cdots 67}a+\frac{10\cdots 44}{25\cdots 67}$, $\frac{16\cdots 05}{18\cdots 53}a^{17}-\frac{29\cdots 57}{36\cdots 06}a^{16}-\frac{37\cdots 13}{36\cdots 06}a^{15}-\frac{16\cdots 29}{18\cdots 53}a^{14}+\frac{89\cdots 53}{18\cdots 53}a^{13}+\frac{63\cdots 73}{36\cdots 06}a^{12}+\frac{16\cdots 15}{36\cdots 06}a^{11}-\frac{15\cdots 16}{18\cdots 53}a^{10}-\frac{25\cdots 59}{18\cdots 53}a^{9}+\frac{17\cdots 49}{36\cdots 06}a^{8}+\frac{19\cdots 31}{36\cdots 06}a^{7}+\frac{13\cdots 74}{18\cdots 53}a^{6}-\frac{25\cdots 80}{18\cdots 53}a^{5}-\frac{59\cdots 55}{36\cdots 06}a^{4}-\frac{49\cdots 75}{36\cdots 06}a^{3}+\frac{14\cdots 40}{18\cdots 53}a^{2}+\frac{24\cdots 55}{18\cdots 53}a+\frac{58\cdots 18}{18\cdots 53}$, $\frac{29\cdots 87}{25\cdots 70}a^{17}-\frac{72\cdots 27}{51\cdots 34}a^{16}-\frac{16\cdots 82}{12\cdots 35}a^{15}-\frac{11\cdots 34}{25\cdots 67}a^{14}+\frac{32\cdots 77}{51\cdots 34}a^{13}+\frac{99\cdots 95}{51\cdots 34}a^{12}-\frac{20\cdots 10}{25\cdots 67}a^{11}-\frac{25\cdots 25}{25\cdots 67}a^{10}-\frac{65\cdots 49}{51\cdots 34}a^{9}+\frac{55\cdots 11}{51\cdots 34}a^{8}+\frac{15\cdots 16}{25\cdots 67}a^{7}+\frac{15\cdots 04}{25\cdots 67}a^{6}-\frac{18\cdots 45}{51\cdots 34}a^{5}-\frac{84\cdots 61}{51\cdots 34}a^{4}-\frac{16\cdots 08}{25\cdots 67}a^{3}+\frac{29\cdots 44}{25\cdots 67}a^{2}+\frac{57\cdots 31}{25\cdots 67}a-\frac{99\cdots 83}{25\cdots 67}$, $\frac{34\cdots 30}{25\cdots 67}a^{17}-\frac{64\cdots 69}{51\cdots 34}a^{16}-\frac{39\cdots 25}{25\cdots 67}a^{15}-\frac{33\cdots 30}{25\cdots 67}a^{14}+\frac{19\cdots 90}{25\cdots 67}a^{13}+\frac{13\cdots 75}{51\cdots 34}a^{12}-\frac{10\cdots 92}{25\cdots 67}a^{11}-\frac{32\cdots 67}{25\cdots 67}a^{10}-\frac{54\cdots 99}{25\cdots 67}a^{9}+\frac{42\cdots 09}{51\cdots 34}a^{8}+\frac{21\cdots 98}{25\cdots 67}a^{7}+\frac{28\cdots 53}{25\cdots 67}a^{6}-\frac{69\cdots 47}{25\cdots 67}a^{5}-\frac{13\cdots 29}{51\cdots 34}a^{4}-\frac{53\cdots 93}{25\cdots 67}a^{3}+\frac{33\cdots 87}{25\cdots 67}a^{2}+\frac{55\cdots 61}{25\cdots 67}a+\frac{13\cdots 87}{25\cdots 67}$, $\frac{44\cdots 68}{12\cdots 35}a^{17}-\frac{16\cdots 25}{51\cdots 34}a^{16}-\frac{49\cdots 66}{12\cdots 35}a^{15}-\frac{87\cdots 25}{25\cdots 67}a^{14}+\frac{47\cdots 76}{25\cdots 67}a^{13}+\frac{33\cdots 21}{51\cdots 34}a^{12}+\frac{50\cdots 39}{25\cdots 67}a^{11}-\frac{80\cdots 08}{25\cdots 67}a^{10}-\frac{13\cdots 65}{25\cdots 67}a^{9}+\frac{87\cdots 31}{51\cdots 34}a^{8}+\frac{51\cdots 49}{25\cdots 67}a^{7}+\frac{71\cdots 72}{25\cdots 67}a^{6}-\frac{11\cdots 18}{25\cdots 67}a^{5}-\frac{31\cdots 35}{51\cdots 34}a^{4}-\frac{13\cdots 49}{25\cdots 67}a^{3}+\frac{75\cdots 29}{25\cdots 67}a^{2}+\frac{12\cdots 93}{25\cdots 67}a+\frac{31\cdots 89}{25\cdots 67}$, $\frac{58\cdots 73}{25\cdots 70}a^{17}-\frac{12\cdots 77}{51\cdots 34}a^{16}-\frac{32\cdots 38}{12\cdots 35}a^{15}-\frac{48\cdots 88}{25\cdots 67}a^{14}+\frac{62\cdots 69}{51\cdots 34}a^{13}+\frac{21\cdots 57}{51\cdots 34}a^{12}-\frac{49\cdots 36}{25\cdots 67}a^{11}-\frac{52\cdots 29}{25\cdots 67}a^{10}-\frac{17\cdots 45}{51\cdots 34}a^{9}+\frac{58\cdots 41}{51\cdots 34}a^{8}+\frac{33\cdots 40}{25\cdots 67}a^{7}+\frac{45\cdots 36}{25\cdots 67}a^{6}-\frac{17\cdots 29}{51\cdots 34}a^{5}-\frac{19\cdots 95}{51\cdots 34}a^{4}-\frac{85\cdots 29}{25\cdots 67}a^{3}+\frac{43\cdots 04}{25\cdots 67}a^{2}+\frac{87\cdots 09}{25\cdots 67}a+\frac{26\cdots 17}{25\cdots 67}$, $\frac{28\cdots 09}{12\cdots 35}a^{17}+\frac{14\cdots 20}{25\cdots 67}a^{16}-\frac{65\cdots 43}{12\cdots 35}a^{15}-\frac{12\cdots 91}{25\cdots 67}a^{14}+\frac{33\cdots 32}{25\cdots 67}a^{13}+\frac{21\cdots 14}{25\cdots 67}a^{12}+\frac{14\cdots 72}{25\cdots 67}a^{11}-\frac{91\cdots 11}{25\cdots 67}a^{10}-\frac{18\cdots 68}{25\cdots 67}a^{9}-\frac{26\cdots 21}{25\cdots 67}a^{8}+\frac{61\cdots 56}{25\cdots 67}a^{7}+\frac{98\cdots 78}{25\cdots 67}a^{6}+\frac{12\cdots 49}{25\cdots 67}a^{5}-\frac{20\cdots 30}{25\cdots 67}a^{4}-\frac{20\cdots 44}{25\cdots 67}a^{3}+\frac{10\cdots 51}{25\cdots 67}a^{2}+\frac{17\cdots 41}{25\cdots 67}a+\frac{42\cdots 73}{25\cdots 67}$, $\frac{16\cdots 88}{25\cdots 67}a^{17}-\frac{15\cdots 88}{25\cdots 67}a^{16}-\frac{37\cdots 35}{51\cdots 34}a^{15}-\frac{16\cdots 56}{25\cdots 67}a^{14}+\frac{89\cdots 13}{25\cdots 67}a^{13}+\frac{31\cdots 99}{25\cdots 67}a^{12}+\frac{10\cdots 53}{51\cdots 34}a^{11}-\frac{15\cdots 76}{25\cdots 67}a^{10}-\frac{25\cdots 58}{25\cdots 67}a^{9}+\frac{82\cdots 80}{25\cdots 67}a^{8}+\frac{19\cdots 87}{51\cdots 34}a^{7}+\frac{13\cdots 88}{25\cdots 67}a^{6}-\frac{21\cdots 07}{25\cdots 67}a^{5}-\frac{28\cdots 99}{25\cdots 67}a^{4}-\frac{47\cdots 87}{51\cdots 34}a^{3}+\frac{13\cdots 13}{25\cdots 67}a^{2}+\frac{22\cdots 62}{25\cdots 67}a+\frac{56\cdots 84}{25\cdots 67}$
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| Regulator: | \( 1235939083.08 \) (assuming GRH) |
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| Unit signature rank: | \( 6 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 1235939083.08 \cdot 1}{2\cdot\sqrt{5077997833420800000000000000000}}\cr\approx \mathstrut & 1.07989119130 \end{aligned}\] (assuming GRH)
Galois group
$C_3\wr S_5:C_2$ (as 18T845):
| A non-solvable group of order 174960 |
| The 84 conjugacy class representatives for $C_3\wr S_5:C_2$ |
| Character table for $C_3\wr S_5:C_2$ |
Intermediate fields
| 6.2.3200000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Degree 45 siblings: | data not computed |
| Minimal sibling: | 18.6.5077997833420800000000000000000.2 |
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 | R | $18$ | ${\href{/padicField/11.5.0.1}{5} }^{3}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | $18$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{5}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.6.10a1.5 | $x^{6} + 2 x^{5} + 4 x + 2$ | $6$ | $1$ | $10$ | $S_4$ | $$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ |
| 2.1.12.24c1.14 | $x^{12} + 4 x^{8} + 2 x^{6} + 4 x^{5} + 4 x + 6$ | $12$ | $1$ | $24$ | $C_2 \times S_4$ | $$[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ | |
|
\(3\)
| 3.6.3.18a96.1 | $x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 26 x^{12} + 27 x^{11} + 39 x^{10} + 45 x^{9} + 57 x^{8} + 57 x^{7} + 73 x^{6} + 63 x^{5} + 51 x^{4} + 44 x^{3} + 42 x^{2} + 24 x + 11$ | $3$ | $6$ | $18$ | not computed | not computed |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.1.15.17a1.4 | $x^{15} + 20 x^{3} + 5$ | $15$ | $1$ | $17$ | $F_5 \times S_3$ | $$[\frac{5}{4}]_{12}^{2}$$ |