Normalized defining polynomial
\( x^{18} + 12 x^{16} - 2 x^{15} + 87 x^{14} - 2 x^{13} + 1065 x^{12} + 28 x^{11} - 16181 x^{10} + \cdots + 1609843 \)
Invariants
| Degree: | $18$ |
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| Signature: | $(0, 9)$ |
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| Discriminant: |
\(-72525903267959963648000000000000\)
\(\medspace = -\,2^{27}\cdot 5^{12}\cdot 19^{12}\)
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| Root discriminant: | \(58.89\) |
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| Galois root discriminant: | $2^{3/2}5^{2/3}19^{2/3}\approx 58.888080312522455$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
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| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_3\times S_3$ |
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| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 6.0.66724352.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}-\frac{1}{3}a^{4}+\frac{1}{6}a^{2}-\frac{1}{6}$, $\frac{1}{6}a^{7}-\frac{1}{3}a^{5}+\frac{1}{6}a^{3}-\frac{1}{6}a$, $\frac{1}{6}a^{8}-\frac{1}{2}a^{4}+\frac{1}{6}a^{2}-\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{5}+\frac{1}{6}a^{3}-\frac{1}{3}a$, $\frac{1}{6}a^{10}+\frac{1}{6}a^{4}+\frac{1}{6}a^{2}-\frac{1}{2}$, $\frac{1}{6}a^{11}+\frac{1}{6}a^{5}+\frac{1}{6}a^{3}-\frac{1}{2}a$, $\frac{1}{1008}a^{12}+\frac{1}{126}a^{10}-\frac{1}{28}a^{9}-\frac{1}{24}a^{8}-\frac{5}{84}a^{7}+\frac{1}{14}a^{6}+\frac{31}{84}a^{5}-\frac{211}{1008}a^{4}+\frac{37}{84}a^{3}+\frac{1}{252}a^{2}+\frac{1}{6}a-\frac{53}{1008}$, $\frac{1}{1008}a^{13}+\frac{1}{126}a^{11}-\frac{1}{28}a^{10}-\frac{1}{24}a^{9}-\frac{5}{84}a^{8}+\frac{1}{14}a^{7}+\frac{1}{28}a^{6}-\frac{211}{1008}a^{5}+\frac{3}{28}a^{4}+\frac{1}{252}a^{3}-\frac{1}{6}a^{2}-\frac{53}{1008}a+\frac{1}{3}$, $\frac{1}{1008}a^{14}-\frac{1}{28}a^{11}+\frac{31}{504}a^{10}+\frac{5}{84}a^{9}+\frac{1}{14}a^{8}+\frac{1}{84}a^{7}+\frac{53}{1008}a^{6}-\frac{29}{84}a^{5}+\frac{5}{28}a^{4}-\frac{5}{14}a^{3}-\frac{421}{1008}a^{2}-\frac{1}{6}a-\frac{31}{126}$, $\frac{1}{57456}a^{15}+\frac{1}{19152}a^{14}-\frac{5}{14364}a^{13}-\frac{13}{57456}a^{12}-\frac{1531}{28728}a^{11}-\frac{2029}{28728}a^{10}+\frac{179}{2394}a^{9}+\frac{35}{1368}a^{8}+\frac{3317}{57456}a^{7}+\frac{1313}{19152}a^{6}+\frac{818}{3591}a^{5}+\frac{18871}{57456}a^{4}-\frac{1213}{2736}a^{3}+\frac{23189}{57456}a^{2}+\frac{527}{2052}a+\frac{5765}{57456}$, $\frac{1}{9214306110576}a^{16}-\frac{10660903}{1535717685096}a^{15}-\frac{598772771}{9214306110576}a^{14}-\frac{184351687}{1316329444368}a^{13}-\frac{4275986837}{9214306110576}a^{12}+\frac{1321617761}{27753934068}a^{11}-\frac{90785091277}{1535717685096}a^{10}+\frac{17716264559}{219388240728}a^{9}+\frac{104281237607}{9214306110576}a^{8}-\frac{115699101989}{1535717685096}a^{7}-\frac{48121046479}{9214306110576}a^{6}-\frac{691615841753}{9214306110576}a^{5}-\frac{2416768523}{85317649172}a^{4}-\frac{2001421719875}{4607153055288}a^{3}-\frac{2143655550181}{9214306110576}a^{2}-\frac{220637290777}{1316329444368}a+\frac{184460271511}{1023811790064}$, $\frac{1}{70\cdots 12}a^{17}-\frac{656245228979}{25\cdots 56}a^{16}-\frac{33\cdots 19}{58\cdots 76}a^{15}+\frac{27\cdots 19}{70\cdots 12}a^{14}+\frac{42\cdots 71}{58\cdots 76}a^{13}-\frac{29\cdots 65}{17\cdots 28}a^{12}-\frac{15\cdots 41}{87\cdots 14}a^{11}+\frac{14\cdots 61}{50\cdots 08}a^{10}+\frac{49\cdots 97}{70\cdots 12}a^{9}+\frac{32\cdots 97}{23\cdots 04}a^{8}-\frac{55\cdots 50}{14\cdots 69}a^{7}+\frac{25\cdots 27}{70\cdots 12}a^{6}-\frac{66\cdots 35}{70\cdots 12}a^{5}+\frac{14\cdots 27}{70\cdots 12}a^{4}+\frac{40\cdots 47}{17\cdots 28}a^{3}+\frac{17\cdots 83}{23\cdots 04}a^{2}-\frac{24\cdots 73}{35\cdots 56}a-\frac{15\cdots 13}{50\cdots 08}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{28\cdots 71}{38\cdots 84}a^{17}+\frac{12\cdots 65}{23\cdots 04}a^{16}+\frac{61\cdots 35}{70\cdots 12}a^{15}+\frac{36\cdots 27}{11\cdots 52}a^{14}+\frac{10\cdots 27}{17\cdots 28}a^{13}+\frac{14\cdots 83}{70\cdots 12}a^{12}+\frac{25\cdots 35}{35\cdots 56}a^{11}+\frac{98\cdots 58}{23\cdots 53}a^{10}-\frac{20\cdots 49}{16\cdots 36}a^{9}-\frac{83\cdots 03}{77\cdots 68}a^{8}-\frac{11\cdots 97}{70\cdots 12}a^{7}+\frac{25\cdots 87}{11\cdots 52}a^{6}+\frac{44\cdots 51}{35\cdots 56}a^{5}+\frac{45\cdots 01}{35\cdots 56}a^{4}+\frac{13\cdots 11}{23\cdots 04}a^{3}-\frac{95\cdots 83}{35\cdots 56}a^{2}+\frac{11\cdots 97}{17\cdots 28}a-\frac{70\cdots 17}{70\cdots 12}$, $\frac{94\cdots 95}{35\cdots 56}a^{17}-\frac{42\cdots 75}{97\cdots 46}a^{16}+\frac{95\cdots 27}{35\cdots 56}a^{15}-\frac{34\cdots 71}{70\cdots 12}a^{14}+\frac{13\cdots 03}{70\cdots 12}a^{13}-\frac{20\cdots 15}{70\cdots 12}a^{12}+\frac{73\cdots 27}{29\cdots 38}a^{11}-\frac{67\cdots 55}{16\cdots 36}a^{10}-\frac{42\cdots 77}{87\cdots 14}a^{9}+\frac{29\cdots 57}{38\cdots 84}a^{8}+\frac{25\cdots 93}{50\cdots 08}a^{7}-\frac{48\cdots 03}{70\cdots 12}a^{6}+\frac{10\cdots 99}{23\cdots 04}a^{5}-\frac{52\cdots 35}{70\cdots 12}a^{4}+\frac{88\cdots 15}{50\cdots 08}a^{3}-\frac{28\cdots 31}{10\cdots 16}a^{2}+\frac{49\cdots 47}{23\cdots 04}a-\frac{28\cdots 41}{77\cdots 68}$, $\frac{98\cdots 63}{11\cdots 52}a^{17}+\frac{29\cdots 39}{77\cdots 68}a^{16}+\frac{64\cdots 99}{70\cdots 12}a^{15}+\frac{11\cdots 49}{33\cdots 72}a^{14}+\frac{39\cdots 77}{70\cdots 12}a^{13}+\frac{19\cdots 47}{35\cdots 56}a^{12}+\frac{25\cdots 85}{35\cdots 56}a^{11}+\frac{23\cdots 75}{35\cdots 56}a^{10}-\frac{49\cdots 33}{32\cdots 82}a^{9}-\frac{94\cdots 03}{23\cdots 04}a^{8}-\frac{25\cdots 47}{36\cdots 48}a^{7}+\frac{13\cdots 11}{23\cdots 04}a^{6}+\frac{90\cdots 67}{70\cdots 12}a^{5}-\frac{90\cdots 19}{70\cdots 12}a^{4}+\frac{14\cdots 63}{23\cdots 04}a^{3}+\frac{52\cdots 65}{10\cdots 16}a^{2}+\frac{52\cdots 31}{70\cdots 12}a+\frac{72\cdots 89}{43\cdots 07}$, $\frac{73\cdots 91}{10\cdots 39}a^{17}+\frac{96\cdots 23}{18\cdots 24}a^{16}+\frac{74\cdots 11}{95\cdots 51}a^{15}+\frac{16\cdots 33}{31\cdots 17}a^{14}+\frac{61\cdots 38}{13\cdots 93}a^{13}+\frac{35\cdots 30}{95\cdots 51}a^{12}+\frac{98\cdots 65}{13\cdots 93}a^{11}+\frac{96\cdots 93}{19\cdots 02}a^{10}-\frac{36\cdots 76}{31\cdots 17}a^{9}-\frac{17\cdots 53}{18\cdots 24}a^{8}-\frac{88\cdots 81}{95\cdots 51}a^{7}+\frac{84\cdots 13}{91\cdots 62}a^{6}+\frac{32\cdots 71}{95\cdots 51}a^{5}+\frac{54\cdots 07}{54\cdots 72}a^{4}+\frac{56\cdots 41}{10\cdots 39}a^{3}+\frac{32\cdots 48}{95\cdots 51}a^{2}+\frac{10\cdots 25}{13\cdots 93}a+\frac{82\cdots 67}{19\cdots 02}$, $\frac{38\cdots 83}{19\cdots 02}a^{17}+\frac{82\cdots 11}{18\cdots 24}a^{16}+\frac{25\cdots 27}{19\cdots 02}a^{15}+\frac{75\cdots 37}{38\cdots 04}a^{14}+\frac{10\cdots 98}{13\cdots 93}a^{13}+\frac{13\cdots 12}{95\cdots 51}a^{12}+\frac{82\cdots 36}{45\cdots 31}a^{11}+\frac{98\cdots 13}{31\cdots 17}a^{10}-\frac{77\cdots 95}{19\cdots 02}a^{9}-\frac{18\cdots 95}{18\cdots 24}a^{8}+\frac{35\cdots 91}{19\cdots 02}a^{7}+\frac{34\cdots 79}{66\cdots 84}a^{6}-\frac{16\cdots 27}{63\cdots 34}a^{5}-\frac{52\cdots 03}{54\cdots 72}a^{4}-\frac{30\cdots 63}{19\cdots 02}a^{3}+\frac{12\cdots 01}{38\cdots 04}a^{2}-\frac{35\cdots 74}{50\cdots 59}a+\frac{24\cdots 87}{63\cdots 34}$, $\frac{48\cdots 81}{62\cdots 01}a^{17}+\frac{38\cdots 35}{23\cdots 04}a^{16}+\frac{52\cdots 97}{70\cdots 12}a^{15}+\frac{18\cdots 81}{70\cdots 12}a^{14}+\frac{29\cdots 75}{70\cdots 12}a^{13}+\frac{11\cdots 99}{35\cdots 56}a^{12}+\frac{10\cdots 19}{16\cdots 36}a^{11}+\frac{26\cdots 97}{11\cdots 52}a^{10}-\frac{52\cdots 87}{35\cdots 56}a^{9}-\frac{51\cdots 93}{23\cdots 04}a^{8}+\frac{57\cdots 95}{36\cdots 48}a^{7}-\frac{37\cdots 87}{70\cdots 12}a^{6}+\frac{22\cdots 73}{23\cdots 04}a^{5}+\frac{77\cdots 79}{70\cdots 12}a^{4}+\frac{19\cdots 39}{70\cdots 12}a^{3}+\frac{44\cdots 27}{70\cdots 12}a^{2}+\frac{14\cdots 81}{33\cdots 72}a+\frac{35\cdots 17}{58\cdots 76}$, $\frac{16\cdots 21}{35\cdots 56}a^{17}+\frac{66\cdots 69}{17\cdots 28}a^{16}+\frac{18\cdots 61}{35\cdots 56}a^{15}+\frac{64\cdots 05}{16\cdots 36}a^{14}+\frac{59\cdots 57}{19\cdots 92}a^{13}+\frac{12\cdots 41}{46\cdots 06}a^{12}+\frac{83\cdots 17}{17\cdots 28}a^{11}+\frac{21\cdots 35}{58\cdots 76}a^{10}-\frac{26\cdots 63}{35\cdots 56}a^{9}-\frac{11\cdots 63}{17\cdots 28}a^{8}-\frac{91\cdots 89}{18\cdots 24}a^{7}+\frac{87\cdots 25}{16\cdots 36}a^{6}+\frac{13\cdots 39}{35\cdots 56}a^{5}+\frac{25\cdots 17}{25\cdots 04}a^{4}+\frac{16\cdots 35}{35\cdots 56}a^{3}+\frac{10\cdots 97}{44\cdots 16}a^{2}+\frac{30\cdots 01}{46\cdots 06}a+\frac{26\cdots 82}{48\cdots 23}$, $\frac{27\cdots 19}{25\cdots 36}a^{17}+\frac{32\cdots 73}{38\cdots 04}a^{16}+\frac{13\cdots 75}{76\cdots 08}a^{15}-\frac{98\cdots 93}{10\cdots 44}a^{14}+\frac{50\cdots 04}{35\cdots 13}a^{13}-\frac{26\cdots 87}{42\cdots 56}a^{12}+\frac{57\cdots 79}{38\cdots 04}a^{11}+\frac{10\cdots 01}{38\cdots 04}a^{10}-\frac{32\cdots 89}{25\cdots 36}a^{9}-\frac{10\cdots 21}{38\cdots 04}a^{8}-\frac{11\cdots 67}{10\cdots 44}a^{7}+\frac{25\cdots 17}{76\cdots 08}a^{6}+\frac{35\cdots 49}{36\cdots 48}a^{5}-\frac{29\cdots 09}{27\cdots 86}a^{4}+\frac{16\cdots 55}{10\cdots 44}a^{3}+\frac{35\cdots 93}{10\cdots 44}a^{2}+\frac{66\cdots 67}{38\cdots 04}a+\frac{87\cdots 05}{38\cdots 04}$
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| Regulator: | \( 188104111.949 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 188104111.949 \cdot 3}{2\cdot\sqrt{72525903267959963648000000000000}}\cr\approx \mathstrut & 0.505663714292 \end{aligned}\] (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
| \(\Q(\sqrt{-2}) \), 3.1.72200.1 x3, 3.3.361.1, 6.0.41702720000.1, 6.0.66724352.1, 6.0.115520000.2 x2, 9.3.376367048000000.3 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 6 sibling: | 6.0.115520000.2 |
| Degree 9 sibling: | 9.3.376367048000000.3 |
| Minimal sibling: | 6.0.115520000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | ${\href{/padicField/11.3.0.1}{3} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{6}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ |
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ | |
| 2.3.2.9a1.1 | $x^{6} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 3$ | $2$ | $3$ | $9$ | $C_6$ | $$[3]^{3}$$ | |
|
\(5\)
| 5.6.3.12a1.2 | $x^{18} + 3 x^{16} + 12 x^{15} + 6 x^{14} + 24 x^{13} + 61 x^{12} + 36 x^{11} + 66 x^{10} + 136 x^{9} + 69 x^{8} + 60 x^{7} + 121 x^{6} + 48 x^{5} + 18 x^{4} + 48 x^{3} + 12 x^{2} + 13$ | $3$ | $6$ | $12$ | $S_3 \times C_3$ | $$[\ ]_{3}^{6}$$ |
|
\(19\)
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 19.1.3.2a1.1 | $x^{3} + 19$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *18 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| *18 | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| *18 | 1.152.6t1.c.a | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *18 | 1.152.6t1.c.b | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| *18 | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *18 | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| *36 | 2.72200.3t2.b.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19^{2}$ | 3.1.72200.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| *36 | 2.3800.6t5.f.a | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19 $ | 18.0.72525903267959963648000000000000.4 | $S_3 \times C_3$ (as 18T3) | $0$ | $0$ |
| *36 | 2.3800.6t5.f.b | $2$ | $ 2^{3} \cdot 5^{2} \cdot 19 $ | 18.0.72525903267959963648000000000000.4 | $S_3 \times C_3$ (as 18T3) | $0$ | $0$ |