Normalized defining polynomial
\( x^{18} - 7 x^{17} + 17 x^{16} - 3 x^{15} - 51 x^{14} + 44 x^{13} + 219 x^{12} - 2249 x^{11} + \cdots - 78971 \)
Invariants
| Degree: | $18$ |
| |
| Signature: | $[2, 8]$ |
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| Discriminant: |
\(68187784456396954697778125000000\)
\(\medspace = 2^{6}\cdot 5^{11}\cdot 139^{4}\cdot 197^{6}\)
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| |
| Root discriminant: | \(58.69\) |
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| Galois root discriminant: | $2^{3/2}5^{3/4}139^{2/3}197^{1/2}\approx 3561.9034512990697$ | ||
| Ramified primes: |
\(2\), \(5\), \(139\), \(197\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{2}+\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{145}a^{16}+\frac{8}{145}a^{15}+\frac{1}{145}a^{14}-\frac{61}{145}a^{13}-\frac{2}{5}a^{12}+\frac{46}{145}a^{11}-\frac{48}{145}a^{10}+\frac{26}{145}a^{9}-\frac{4}{145}a^{8}-\frac{9}{29}a^{7}+\frac{54}{145}a^{6}+\frac{1}{145}a^{5}+\frac{13}{145}a^{4}-\frac{71}{145}a^{3}+\frac{1}{145}a^{2}-\frac{44}{145}a-\frac{7}{29}$, $\frac{1}{17\cdots 85}a^{17}-\frac{53\cdots 52}{17\cdots 85}a^{16}-\frac{54\cdots 12}{17\cdots 85}a^{15}+\frac{18\cdots 02}{17\cdots 85}a^{14}-\frac{12\cdots 68}{17\cdots 85}a^{13}-\frac{10\cdots 91}{17\cdots 85}a^{12}+\frac{48\cdots 44}{17\cdots 85}a^{11}-\frac{54\cdots 79}{34\cdots 97}a^{10}-\frac{44\cdots 66}{17\cdots 85}a^{9}-\frac{69\cdots 10}{34\cdots 97}a^{8}+\frac{12\cdots 06}{20\cdots 85}a^{7}+\frac{61\cdots 88}{17\cdots 85}a^{6}+\frac{76\cdots 78}{17\cdots 85}a^{5}+\frac{71\cdots 71}{17\cdots 85}a^{4}+\frac{85\cdots 84}{17\cdots 85}a^{3}-\frac{20\cdots 48}{17\cdots 85}a^{2}-\frac{83\cdots 47}{17\cdots 85}a+\frac{12\cdots 07}{34\cdots 97}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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: |
$\frac{10\cdots 59}{34\cdots 97}a^{17}-\frac{23\cdots 92}{17\cdots 85}a^{16}+\frac{52\cdots 38}{17\cdots 85}a^{15}+\frac{14\cdots 12}{17\cdots 85}a^{14}-\frac{14\cdots 26}{17\cdots 85}a^{13}-\frac{34\cdots 83}{17\cdots 85}a^{12}+\frac{74\cdots 94}{17\cdots 85}a^{11}-\frac{88\cdots 48}{17\cdots 85}a^{10}-\frac{58\cdots 73}{34\cdots 97}a^{9}+\frac{11\cdots 79}{34\cdots 97}a^{8}-\frac{52\cdots 89}{58\cdots 65}a^{7}-\frac{53\cdots 36}{17\cdots 85}a^{6}+\frac{23\cdots 01}{17\cdots 85}a^{5}-\frac{31\cdots 72}{17\cdots 85}a^{4}-\frac{35\cdots 89}{17\cdots 85}a^{3}+\frac{50\cdots 82}{17\cdots 85}a^{2}-\frac{44\cdots 77}{17\cdots 85}a-\frac{41\cdots 78}{17\cdots 85}$, $\frac{83\cdots 21}{45\cdots 65}a^{17}-\frac{58\cdots 01}{45\cdots 65}a^{16}+\frac{13\cdots 47}{45\cdots 65}a^{15}+\frac{36\cdots 92}{45\cdots 65}a^{14}-\frac{56\cdots 64}{45\cdots 65}a^{13}+\frac{71\cdots 45}{91\cdots 33}a^{12}+\frac{22\cdots 16}{45\cdots 65}a^{11}-\frac{18\cdots 31}{45\cdots 65}a^{10}+\frac{35\cdots 79}{45\cdots 65}a^{9}+\frac{48\cdots 51}{45\cdots 65}a^{8}-\frac{18\cdots 38}{45\cdots 65}a^{7}+\frac{19\cdots 43}{45\cdots 65}a^{6}+\frac{42\cdots 99}{45\cdots 65}a^{5}-\frac{27\cdots 32}{91\cdots 33}a^{4}-\frac{20\cdots 08}{45\cdots 65}a^{3}+\frac{18\cdots 81}{45\cdots 65}a^{2}-\frac{79\cdots 97}{45\cdots 65}a+\frac{25\cdots 70}{91\cdots 33}$, $\frac{46\cdots 65}{34\cdots 97}a^{17}-\frac{17\cdots 46}{17\cdots 85}a^{16}+\frac{48\cdots 94}{17\cdots 85}a^{15}-\frac{24\cdots 84}{17\cdots 85}a^{14}-\frac{11\cdots 88}{17\cdots 85}a^{13}+\frac{11\cdots 86}{17\cdots 85}a^{12}+\frac{55\cdots 62}{17\cdots 85}a^{11}-\frac{55\cdots 39}{17\cdots 85}a^{10}+\frac{27\cdots 64}{34\cdots 97}a^{9}+\frac{80\cdots 59}{34\cdots 97}a^{8}-\frac{16\cdots 92}{58\cdots 65}a^{7}+\frac{71\cdots 37}{17\cdots 85}a^{6}+\frac{62\cdots 18}{17\cdots 85}a^{5}-\frac{39\cdots 81}{17\cdots 85}a^{4}+\frac{11\cdots 03}{17\cdots 85}a^{3}+\frac{24\cdots 66}{17\cdots 85}a^{2}-\frac{41\cdots 06}{17\cdots 85}a+\frac{56\cdots 11}{17\cdots 85}$, $\frac{28\cdots 53}{17\cdots 85}a^{17}-\frac{39\cdots 31}{34\cdots 97}a^{16}+\frac{43\cdots 79}{17\cdots 85}a^{15}+\frac{70\cdots 93}{17\cdots 85}a^{14}-\frac{14\cdots 13}{17\cdots 85}a^{13}+\frac{21\cdots 97}{17\cdots 85}a^{12}+\frac{85\cdots 97}{17\cdots 85}a^{11}-\frac{65\cdots 16}{17\cdots 85}a^{10}+\frac{11\cdots 67}{17\cdots 85}a^{9}+\frac{14\cdots 93}{17\cdots 85}a^{8}-\frac{35\cdots 84}{11\cdots 93}a^{7}+\frac{43\cdots 13}{17\cdots 85}a^{6}+\frac{16\cdots 43}{17\cdots 85}a^{5}-\frac{42\cdots 02}{17\cdots 85}a^{4}-\frac{12\cdots 83}{17\cdots 85}a^{3}+\frac{13\cdots 70}{34\cdots 97}a^{2}-\frac{31\cdots 03}{17\cdots 85}a-\frac{41\cdots 28}{17\cdots 85}$, $\frac{63\cdots 79}{17\cdots 85}a^{17}-\frac{45\cdots 91}{17\cdots 85}a^{16}+\frac{11\cdots 56}{17\cdots 85}a^{15}-\frac{12\cdots 59}{34\cdots 97}a^{14}-\frac{16\cdots 92}{17\cdots 85}a^{13}-\frac{23\cdots 63}{17\cdots 85}a^{12}+\frac{14\cdots 63}{17\cdots 85}a^{11}-\frac{14\cdots 77}{17\cdots 85}a^{10}+\frac{32\cdots 31}{17\cdots 85}a^{9}+\frac{14\cdots 59}{17\cdots 85}a^{8}-\frac{25\cdots 32}{58\cdots 65}a^{7}+\frac{92\cdots 36}{17\cdots 85}a^{6}+\frac{11\cdots 02}{17\cdots 85}a^{5}-\frac{84\cdots 32}{17\cdots 85}a^{4}+\frac{25\cdots 49}{17\cdots 85}a^{3}+\frac{55\cdots 96}{17\cdots 85}a^{2}-\frac{11\cdots 71}{34\cdots 97}a+\frac{64\cdots 57}{17\cdots 85}$, $\frac{54\cdots 14}{17\cdots 85}a^{17}-\frac{79\cdots 71}{34\cdots 97}a^{16}+\frac{98\cdots 17}{17\cdots 85}a^{15}-\frac{11\cdots 56}{17\cdots 85}a^{14}-\frac{32\cdots 64}{17\cdots 85}a^{13}+\frac{25\cdots 11}{17\cdots 85}a^{12}+\frac{13\cdots 06}{17\cdots 85}a^{11}-\frac{12\cdots 18}{17\cdots 85}a^{10}+\frac{26\cdots 31}{17\cdots 85}a^{9}+\frac{22\cdots 54}{17\cdots 85}a^{8}-\frac{80\cdots 38}{11\cdots 93}a^{7}+\frac{14\cdots 24}{17\cdots 85}a^{6}+\frac{21\cdots 09}{17\cdots 85}a^{5}-\frac{91\cdots 21}{17\cdots 85}a^{4}+\frac{40\cdots 31}{17\cdots 85}a^{3}+\frac{18\cdots 79}{34\cdots 97}a^{2}-\frac{33\cdots 79}{17\cdots 85}a+\frac{12\cdots 46}{17\cdots 85}$, $\frac{12\cdots 03}{17\cdots 85}a^{17}-\frac{89\cdots 83}{17\cdots 85}a^{16}+\frac{21\cdots 26}{17\cdots 85}a^{15}-\frac{40\cdots 74}{17\cdots 85}a^{14}-\frac{75\cdots 37}{17\cdots 85}a^{13}+\frac{11\cdots 53}{34\cdots 97}a^{12}+\frac{31\cdots 58}{17\cdots 85}a^{11}-\frac{28\cdots 38}{17\cdots 85}a^{10}+\frac{58\cdots 72}{17\cdots 85}a^{9}+\frac{55\cdots 73}{17\cdots 85}a^{8}-\frac{92\cdots 16}{58\cdots 65}a^{7}+\frac{31\cdots 49}{17\cdots 85}a^{6}+\frac{54\cdots 77}{17\cdots 85}a^{5}-\frac{40\cdots 52}{34\cdots 97}a^{4}-\frac{27\cdots 69}{17\cdots 85}a^{3}+\frac{22\cdots 38}{17\cdots 85}a^{2}-\frac{88\cdots 01}{17\cdots 85}a+\frac{39\cdots 98}{34\cdots 97}$, $\frac{25\cdots 05}{34\cdots 97}a^{17}-\frac{93\cdots 18}{17\cdots 85}a^{16}+\frac{23\cdots 97}{17\cdots 85}a^{15}-\frac{31\cdots 02}{17\cdots 85}a^{14}-\frac{77\cdots 64}{17\cdots 85}a^{13}+\frac{65\cdots 38}{17\cdots 85}a^{12}+\frac{30\cdots 56}{17\cdots 85}a^{11}-\frac{29\cdots 87}{17\cdots 85}a^{10}+\frac{12\cdots 69}{34\cdots 97}a^{9}+\frac{10\cdots 99}{34\cdots 97}a^{8}-\frac{96\cdots 11}{58\cdots 65}a^{7}+\frac{35\cdots 21}{17\cdots 85}a^{6}+\frac{47\cdots 99}{17\cdots 85}a^{5}-\frac{21\cdots 53}{17\cdots 85}a^{4}+\frac{17\cdots 39}{17\cdots 85}a^{3}+\frac{21\cdots 33}{17\cdots 85}a^{2}-\frac{70\cdots 88}{17\cdots 85}a+\frac{30\cdots 18}{17\cdots 85}$, $\frac{49\cdots 62}{34\cdots 97}a^{17}-\frac{20\cdots 73}{17\cdots 85}a^{16}+\frac{72\cdots 57}{17\cdots 85}a^{15}-\frac{22\cdots 37}{34\cdots 97}a^{14}+\frac{41\cdots 39}{17\cdots 85}a^{13}+\frac{47\cdots 03}{17\cdots 85}a^{12}+\frac{46\cdots 29}{17\cdots 85}a^{11}-\frac{12\cdots 23}{34\cdots 97}a^{10}+\frac{20\cdots 09}{17\cdots 85}a^{9}-\frac{21\cdots 47}{17\cdots 85}a^{8}-\frac{49\cdots 36}{58\cdots 65}a^{7}+\frac{90\cdots 78}{17\cdots 85}a^{6}-\frac{42\cdots 19}{17\cdots 85}a^{5}-\frac{29\cdots 83}{17\cdots 85}a^{4}+\frac{48\cdots 16}{17\cdots 85}a^{3}-\frac{48\cdots 39}{17\cdots 85}a^{2}+\frac{37\cdots 63}{17\cdots 85}a-\frac{13\cdots 34}{17\cdots 85}$
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| Regulator: | \( 101821357.206 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 101821357.206 \cdot 8}{2\cdot\sqrt{68187784456396954697778125000000}}\cr\approx \mathstrut & 0.479230578232 \end{aligned}\] (assuming GRH)
Galois group
$A_4^3.(C_2\times S_4)$ (as 18T775):
| A solvable group of order 82944 |
| The 84 conjugacy class representatives for $A_4^3.(C_2\times S_4)$ |
| Character table for $A_4^3.(C_2\times S_4)$ |
Intermediate fields
| 3.3.985.1, 9.9.92322657333125.1 |
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 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.2.13637556891279390939555625000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a3.1 | $x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 6 x + 5$ | $2$ | $3$ | $6$ | $A_4$ | $$[2, 2]^{3}$$ |
| 2.12.1.0a1.1 | $x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $$[\ ]^{12}$$ | |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.2 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(139\)
| 139.2.3.4a1.2 | $x^{6} + 414 x^{5} + 57138 x^{4} + 2629728 x^{3} + 114276 x^{2} + 1656 x + 147$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ |
| 139.12.1.0a1.1 | $x^{12} + 120 x^{7} + 75 x^{6} + 41 x^{5} + 77 x^{4} + 106 x^{3} + 8 x^{2} + 10 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $$[\ ]^{12}$$ | |
|
\(197\)
| $\Q_{197}$ | $x + 195$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{197}$ | $x + 195$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 197.4.1.0a1.1 | $x^{4} + 16 x^{2} + 124 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 197.2.2.2a1.2 | $x^{4} + 384 x^{3} + 36868 x^{2} + 768 x + 201$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 197.4.2.4a1.2 | $x^{8} + 32 x^{6} + 248 x^{5} + 260 x^{4} + 3968 x^{3} + 15440 x^{2} + 496 x + 201$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ |