Normalized defining polynomial
\( x^{18} - 6 x^{17} - 24 x^{16} + 184 x^{15} + 279 x^{14} - 2850 x^{13} + 262 x^{12} + 16494 x^{11} + \cdots + 509627 \)
Invariants
| Degree: | $18$ |
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| Signature: | $[0, 9]$ |
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| Discriminant: |
\(-1261446981901370189638661902928499\)
\(\medspace = -\,3^{24}\cdot 7^{12}\cdot 19^{9}\)
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| Root discriminant: | \(69.01\) |
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| Galois root discriminant: | $3^{4/3}7^{2/3}19^{1/2}\approx 69.01399479708432$ | ||
| Ramified primes: |
\(3\), \(7\), \(19\)
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| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_3\times S_3$ |
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| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 6.0.16468459.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{10}a^{12}-\frac{1}{10}a^{10}-\frac{1}{10}a^{9}+\frac{3}{10}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{2}a^{5}+\frac{3}{10}a^{4}-\frac{3}{10}a^{3}-\frac{3}{10}$, $\frac{1}{10}a^{13}-\frac{1}{10}a^{11}-\frac{1}{10}a^{10}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{3}{10}a^{7}-\frac{1}{2}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{3}{10}a-\frac{1}{2}$, $\frac{1}{10}a^{14}-\frac{1}{10}a^{11}+\frac{1}{5}a^{10}+\frac{1}{10}a^{9}+\frac{1}{10}a^{8}-\frac{3}{10}a^{7}+\frac{1}{10}a^{6}+\frac{1}{5}a^{5}+\frac{3}{10}a^{4}-\frac{3}{10}a^{3}-\frac{3}{10}a^{2}-\frac{3}{10}$, $\frac{1}{100}a^{15}-\frac{3}{100}a^{14}+\frac{3}{100}a^{13}-\frac{1}{25}a^{12}-\frac{2}{25}a^{11}+\frac{3}{20}a^{10}-\frac{1}{4}a^{9}-\frac{29}{100}a^{8}+\frac{3}{100}a^{7}-\frac{1}{2}a^{6}-\frac{1}{25}a^{5}+\frac{1}{4}a^{4}+\frac{1}{20}a^{3}+\frac{19}{100}a^{2}+\frac{9}{50}a-\frac{7}{100}$, $\frac{1}{8300}a^{16}+\frac{3}{830}a^{15}+\frac{117}{4150}a^{14}-\frac{73}{1660}a^{13}+\frac{1}{166}a^{12}-\frac{69}{8300}a^{11}+\frac{10}{83}a^{10}-\frac{336}{2075}a^{9}+\frac{1813}{4150}a^{8}-\frac{2891}{8300}a^{7}+\frac{373}{4150}a^{6}-\frac{1277}{8300}a^{5}-\frac{183}{830}a^{4}-\frac{969}{2075}a^{3}-\frac{139}{1660}a^{2}-\frac{3333}{8300}a+\frac{2209}{8300}$, $\frac{1}{81\cdots 00}a^{17}-\frac{70\cdots 97}{16\cdots 60}a^{16}-\frac{30\cdots 09}{81\cdots 00}a^{15}+\frac{87\cdots 16}{20\cdots 75}a^{14}-\frac{14\cdots 27}{40\cdots 50}a^{13}-\frac{30\cdots 47}{81\cdots 00}a^{12}-\frac{15\cdots 81}{81\cdots 00}a^{11}-\frac{16\cdots 19}{81\cdots 00}a^{10}+\frac{12\cdots 21}{81\cdots 00}a^{9}-\frac{97\cdots 81}{20\cdots 75}a^{8}-\frac{13\cdots 49}{40\cdots 50}a^{7}+\frac{28\cdots 43}{81\cdots 00}a^{6}+\frac{66\cdots 67}{81\cdots 00}a^{5}-\frac{32\cdots 21}{81\cdots 00}a^{4}-\frac{16\cdots 78}{81\cdots 03}a^{3}+\frac{52\cdots 49}{16\cdots 60}a^{2}+\frac{29\cdots 63}{81\cdots 30}a+\frac{15\cdots 93}{40\cdots 50}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{3}$, which has order $3$ (assuming GRH) |
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| Narrow class group: | $C_{3}$, which has order $3$ (assuming GRH) |
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Unit group
| Rank: | $8$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{49\cdots 08}{81\cdots 03}a^{17}-\frac{34\cdots 37}{81\cdots 03}a^{16}-\frac{83\cdots 60}{81\cdots 03}a^{15}+\frac{99\cdots 52}{81\cdots 03}a^{14}+\frac{17\cdots 80}{81\cdots 03}a^{13}-\frac{14\cdots 28}{81\cdots 03}a^{12}+\frac{21\cdots 84}{81\cdots 03}a^{11}+\frac{73\cdots 42}{81\cdots 03}a^{10}-\frac{13\cdots 48}{81\cdots 03}a^{9}-\frac{46\cdots 98}{81\cdots 03}a^{8}+\frac{11\cdots 12}{81\cdots 03}a^{7}+\frac{80\cdots 08}{81\cdots 03}a^{6}+\frac{43\cdots 84}{81\cdots 03}a^{5}-\frac{62\cdots 41}{81\cdots 03}a^{4}+\frac{31\cdots 92}{81\cdots 03}a^{3}+\frac{65\cdots 14}{81\cdots 03}a^{2}-\frac{71\cdots 72}{81\cdots 03}a-\frac{22\cdots 39}{81\cdots 03}$, $\frac{49\cdots 08}{81\cdots 03}a^{17}-\frac{34\cdots 37}{81\cdots 03}a^{16}-\frac{83\cdots 60}{81\cdots 03}a^{15}+\frac{99\cdots 52}{81\cdots 03}a^{14}+\frac{17\cdots 80}{81\cdots 03}a^{13}-\frac{14\cdots 28}{81\cdots 03}a^{12}+\frac{21\cdots 84}{81\cdots 03}a^{11}+\frac{73\cdots 42}{81\cdots 03}a^{10}-\frac{13\cdots 48}{81\cdots 03}a^{9}-\frac{46\cdots 98}{81\cdots 03}a^{8}+\frac{11\cdots 12}{81\cdots 03}a^{7}+\frac{80\cdots 08}{81\cdots 03}a^{6}+\frac{43\cdots 84}{81\cdots 03}a^{5}-\frac{62\cdots 41}{81\cdots 03}a^{4}+\frac{31\cdots 92}{81\cdots 03}a^{3}+\frac{65\cdots 14}{81\cdots 03}a^{2}-\frac{71\cdots 72}{81\cdots 03}a+\frac{59\cdots 64}{81\cdots 03}$, $\frac{17\cdots 43}{81\cdots 00}a^{17}-\frac{16\cdots 13}{81\cdots 30}a^{16}-\frac{24\cdots 59}{40\cdots 50}a^{15}+\frac{90\cdots 71}{16\cdots 60}a^{14}-\frac{70\cdots 75}{81\cdots 03}a^{13}-\frac{60\cdots 57}{81\cdots 00}a^{12}+\frac{19\cdots 49}{81\cdots 30}a^{11}+\frac{39\cdots 02}{20\cdots 75}a^{10}-\frac{56\cdots 41}{40\cdots 50}a^{9}-\frac{42\cdots 73}{81\cdots 00}a^{8}+\frac{34\cdots 19}{40\cdots 50}a^{7}-\frac{10\cdots 21}{81\cdots 00}a^{6}+\frac{30\cdots 07}{81\cdots 30}a^{5}+\frac{35\cdots 01}{40\cdots 50}a^{4}-\frac{12\cdots 01}{16\cdots 60}a^{3}-\frac{44\cdots 59}{81\cdots 00}a^{2}+\frac{90\cdots 37}{81\cdots 00}a-\frac{21\cdots 13}{40\cdots 15}$, $\frac{22\cdots 07}{81\cdots 00}a^{17}-\frac{65\cdots 37}{81\cdots 30}a^{16}-\frac{43\cdots 91}{40\cdots 50}a^{15}+\frac{39\cdots 79}{16\cdots 60}a^{14}+\frac{16\cdots 55}{81\cdots 03}a^{13}-\frac{28\cdots 93}{81\cdots 00}a^{12}-\frac{16\cdots 99}{81\cdots 30}a^{11}+\frac{30\cdots 73}{20\cdots 75}a^{10}+\frac{54\cdots 41}{40\cdots 50}a^{9}-\frac{13\cdots 77}{81\cdots 00}a^{8}-\frac{17\cdots 69}{40\cdots 50}a^{7}-\frac{36\cdots 29}{81\cdots 00}a^{6}+\frac{71\cdots 93}{81\cdots 30}a^{5}+\frac{58\cdots 99}{40\cdots 50}a^{4}-\frac{11\cdots 49}{16\cdots 60}a^{3}-\frac{11\cdots 91}{81\cdots 00}a^{2}+\frac{32\cdots 13}{81\cdots 00}a+\frac{31\cdots 23}{40\cdots 15}$, $\frac{10\cdots 47}{81\cdots 00}a^{17}-\frac{22\cdots 47}{40\cdots 50}a^{16}-\frac{16\cdots 67}{40\cdots 50}a^{15}+\frac{28\cdots 89}{16\cdots 60}a^{14}+\frac{26\cdots 67}{40\cdots 50}a^{13}-\frac{44\cdots 53}{16\cdots 60}a^{12}-\frac{17\cdots 69}{40\cdots 50}a^{11}+\frac{65\cdots 31}{40\cdots 50}a^{10}+\frac{12\cdots 99}{40\cdots 50}a^{9}-\frac{66\cdots 23}{81\cdots 00}a^{8}-\frac{22\cdots 96}{40\cdots 15}a^{7}+\frac{24\cdots 87}{81\cdots 00}a^{6}+\frac{85\cdots 19}{20\cdots 75}a^{5}-\frac{13\cdots 21}{40\cdots 50}a^{4}-\frac{41\cdots 11}{81\cdots 00}a^{3}+\frac{37\cdots 51}{81\cdots 00}a^{2}+\frac{57\cdots 59}{81\cdots 00}a-\frac{24\cdots 21}{40\cdots 50}$, $\frac{20\cdots 89}{40\cdots 50}a^{17}-\frac{71\cdots 93}{40\cdots 50}a^{16}-\frac{36\cdots 33}{20\cdots 75}a^{15}+\frac{11\cdots 67}{20\cdots 75}a^{14}+\frac{13\cdots 79}{40\cdots 50}a^{13}-\frac{17\cdots 79}{20\cdots 75}a^{12}-\frac{55\cdots 91}{20\cdots 75}a^{11}+\frac{20\cdots 49}{40\cdots 50}a^{10}+\frac{73\cdots 11}{40\cdots 50}a^{9}-\frac{87\cdots 69}{40\cdots 50}a^{8}-\frac{20\cdots 49}{40\cdots 50}a^{7}-\frac{10\cdots 73}{20\cdots 75}a^{6}+\frac{62\cdots 19}{40\cdots 50}a^{5}+\frac{25\cdots 71}{40\cdots 50}a^{4}-\frac{12\cdots 17}{40\cdots 50}a^{3}-\frac{64\cdots 61}{81\cdots 30}a^{2}+\frac{83\cdots 87}{20\cdots 75}a+\frac{60\cdots 91}{20\cdots 75}$, $\frac{19\cdots 81}{81\cdots 00}a^{17}-\frac{27\cdots 29}{40\cdots 15}a^{16}-\frac{14\cdots 13}{20\cdots 75}a^{15}+\frac{14\cdots 03}{81\cdots 00}a^{14}+\frac{19\cdots 03}{20\cdots 75}a^{13}-\frac{46\cdots 29}{16\cdots 60}a^{12}-\frac{87\cdots 83}{20\cdots 75}a^{11}+\frac{50\cdots 13}{40\cdots 50}a^{10}+\frac{11\cdots 53}{40\cdots 50}a^{9}-\frac{65\cdots 97}{81\cdots 00}a^{8}+\frac{16\cdots 89}{40\cdots 50}a^{7}-\frac{83\cdots 97}{81\cdots 00}a^{6}+\frac{10\cdots 71}{20\cdots 75}a^{5}-\frac{29\cdots 33}{40\cdots 50}a^{4}+\frac{50\cdots 61}{16\cdots 60}a^{3}-\frac{42\cdots 57}{81\cdots 00}a^{2}+\frac{26\cdots 11}{81\cdots 00}a-\frac{11\cdots 89}{40\cdots 50}$, $\frac{15\cdots 26}{20\cdots 75}a^{17}-\frac{30\cdots 74}{40\cdots 15}a^{16}+\frac{17\cdots 16}{20\cdots 75}a^{15}+\frac{38\cdots 84}{20\cdots 75}a^{14}-\frac{72\cdots 74}{20\cdots 75}a^{13}-\frac{44\cdots 67}{20\cdots 75}a^{12}+\frac{15\cdots 44}{20\cdots 75}a^{11}+\frac{48\cdots 91}{20\cdots 75}a^{10}-\frac{25\cdots 44}{20\cdots 75}a^{9}-\frac{79\cdots 24}{20\cdots 75}a^{8}+\frac{47\cdots 32}{20\cdots 75}a^{7}+\frac{64\cdots 33}{20\cdots 75}a^{6}-\frac{76\cdots 28}{20\cdots 75}a^{5}-\frac{91\cdots 91}{20\cdots 75}a^{4}+\frac{30\cdots 88}{40\cdots 15}a^{3}+\frac{10\cdots 48}{40\cdots 15}a^{2}-\frac{33\cdots 76}{40\cdots 15}a+\frac{72\cdots 76}{20\cdots 75}$
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| Regulator: | \( 283548193.676 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 283548193.676 \cdot 3}{2\cdot\sqrt{1261446981901370189638661902928499}}\cr\approx \mathstrut & 0.182768967430 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times S_3$ (as 18T3):
| A solvable group of order 18 |
| The 9 conjugacy class representatives for $S_3 \times C_3$ |
| Character table for $S_3 \times C_3$ |
Intermediate fields
| \(\Q(\sqrt{-19}) \), 3.1.1539.1 x3, \(\Q(\zeta_{7})^+\), 6.0.45001899.2, 6.0.16468459.1, 6.0.108049559499.14 x2, 9.3.428848701651531.5 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 sibling: | 6.0.108049559499.14 |
| Degree 9 sibling: | 9.3.428848701651531.5 |
| Minimal sibling: | 6.0.108049559499.14 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.24a1.1 | $x^{18} + 6 x^{16} + 15 x^{14} + 6 x^{13} + 29 x^{12} + 24 x^{11} + 51 x^{10} + 36 x^{9} + 72 x^{8} + 60 x^{7} + 85 x^{6} + 78 x^{5} + 69 x^{4} + 44 x^{3} + 60 x^{2} + 48 x + 23$ | $3$ | $6$ | $24$ | $S_3 \times C_3$ | not computed |
|
\(7\)
| 7.3.3.6a1.3 | $x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ |
| 7.3.3.6a1.3 | $x^{9} + 18 x^{8} + 108 x^{7} + 228 x^{6} + 144 x^{5} + 432 x^{4} + 48 x^{3} + 288 x^{2} + 71$ | $3$ | $3$ | $6$ | $C_3^2$ | $$[\ ]_{3}^{3}$$ | |
|
\(19\)
| 19.3.2.3a1.2 | $x^{6} + 8 x^{4} + 34 x^{3} + 16 x^{2} + 136 x + 308$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
| 19.3.2.3a1.2 | $x^{6} + 8 x^{4} + 34 x^{3} + 16 x^{2} + 136 x + 308$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 19.3.2.3a1.2 | $x^{6} + 8 x^{4} + 34 x^{3} + 16 x^{2} + 136 x + 308$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |