Normalized defining polynomial
\( x^{20} - 5 x^{19} + 76 x^{18} - 247 x^{17} + 1197 x^{16} - 8474 x^{15} + 15561 x^{14} - 112347 x^{13} + \cdots + 31141888 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(2, 9)$ |
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| Discriminant: |
\(-206007596521214410095208558252435839890349094339\)
\(\medspace = -\,19^{37}\)
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| Root discriminant: | \(232.11\) |
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| Galois root discriminant: | $19^{683/342}\approx 357.90532196302917$ | ||
| Ramified primes: |
\(19\)
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| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{1}{16}a^{4}$, $\frac{1}{32}a^{12}-\frac{1}{16}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{1}{16}a^{6}+\frac{3}{16}a^{5}-\frac{1}{32}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{32}a^{13}+\frac{1}{32}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{3}{32}a^{5}+\frac{1}{16}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{10}-\frac{1}{8}a^{7}+\frac{1}{32}a^{6}-\frac{1}{8}a^{5}+\frac{1}{16}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{64}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{11}+\frac{1}{64}a^{10}-\frac{3}{64}a^{9}-\frac{1}{8}a^{8}+\frac{7}{64}a^{7}-\frac{1}{64}a^{6}-\frac{11}{64}a^{5}-\frac{1}{32}a^{4}-\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{64}a^{16}-\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{11}+\frac{1}{64}a^{9}-\frac{3}{64}a^{8}+\frac{1}{32}a^{7}+\frac{3}{32}a^{6}+\frac{3}{64}a^{5}+\frac{1}{16}a^{4}+\frac{1}{8}a^{2}$, $\frac{1}{128}a^{17}-\frac{1}{128}a^{15}-\frac{1}{64}a^{13}+\frac{3}{128}a^{11}+\frac{3}{64}a^{9}-\frac{1}{128}a^{7}+\frac{1}{16}a^{6}+\frac{25}{128}a^{5}+\frac{1}{16}a^{4}+\frac{3}{16}a^{3}+\frac{1}{16}a^{2}+\frac{1}{4}a$, $\frac{1}{512}a^{18}-\frac{1}{256}a^{17}+\frac{1}{512}a^{16}-\frac{1}{256}a^{15}+\frac{1}{256}a^{14}-\frac{1}{256}a^{13}-\frac{7}{512}a^{12}+\frac{5}{256}a^{11}-\frac{7}{256}a^{10}-\frac{7}{256}a^{9}+\frac{9}{512}a^{8}+\frac{29}{256}a^{7}+\frac{49}{512}a^{6}-\frac{5}{32}a^{5}+\frac{7}{64}a^{4}+\frac{9}{64}a^{3}+\frac{5}{16}a^{2}$, $\frac{1}{10\cdots 32}a^{19}-\frac{32\cdots 85}{10\cdots 32}a^{18}-\frac{96\cdots 71}{35\cdots 44}a^{17}-\frac{45\cdots 89}{10\cdots 32}a^{16}-\frac{18\cdots 37}{26\cdots 08}a^{15}-\frac{91\cdots 93}{44\cdots 68}a^{14}-\frac{72\cdots 67}{35\cdots 44}a^{13}-\frac{32\cdots 35}{35\cdots 44}a^{12}+\frac{33\cdots 11}{13\cdots 04}a^{11}+\frac{93\cdots 33}{88\cdots 36}a^{10}-\frac{20\cdots 51}{35\cdots 44}a^{9}+\frac{61\cdots 63}{10\cdots 32}a^{8}-\frac{42\cdots 79}{35\cdots 44}a^{7}+\frac{24\cdots 27}{35\cdots 44}a^{6}-\frac{32\cdots 15}{26\cdots 08}a^{5}-\frac{22\cdots 99}{11\cdots 92}a^{4}+\frac{16\cdots 19}{44\cdots 68}a^{3}+\frac{95\cdots 05}{41\cdots 22}a^{2}+\frac{17\cdots 95}{83\cdots 44}a-\frac{18\cdots 77}{20\cdots 61}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $10$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{52\cdots 05}{17\cdots 72}a^{19}-\frac{22\cdots 89}{17\cdots 72}a^{18}+\frac{37\cdots 45}{17\cdots 72}a^{17}-\frac{99\cdots 01}{17\cdots 72}a^{16}+\frac{24\cdots 45}{88\cdots 36}a^{15}-\frac{98\cdots 95}{44\cdots 68}a^{14}+\frac{47\cdots 35}{17\cdots 72}a^{13}-\frac{48\cdots 21}{17\cdots 72}a^{12}+\frac{65\cdots 71}{88\cdots 36}a^{11}-\frac{15\cdots 51}{11\cdots 92}a^{10}+\frac{16\cdots 71}{17\cdots 72}a^{9}-\frac{16\cdots 89}{17\cdots 72}a^{8}+\frac{74\cdots 93}{17\cdots 72}a^{7}-\frac{18\cdots 75}{17\cdots 72}a^{6}+\frac{20\cdots 17}{88\cdots 36}a^{5}-\frac{12\cdots 85}{22\cdots 84}a^{4}-\frac{67\cdots 13}{22\cdots 84}a^{3}-\frac{13\cdots 65}{11\cdots 92}a^{2}-\frac{72\cdots 81}{13\cdots 74}a-\frac{59\cdots 93}{69\cdots 87}$, $\frac{13\cdots 99}{17\cdots 72}a^{19}-\frac{39\cdots 07}{17\cdots 72}a^{18}+\frac{77\cdots 41}{17\cdots 72}a^{17}-\frac{91\cdots 87}{17\cdots 72}a^{16}+\frac{38\cdots 63}{22\cdots 84}a^{15}-\frac{22\cdots 59}{55\cdots 96}a^{14}-\frac{49\cdots 31}{17\cdots 72}a^{13}-\frac{49\cdots 63}{17\cdots 72}a^{12}+\frac{37\cdots 05}{44\cdots 68}a^{11}+\frac{61\cdots 99}{44\cdots 68}a^{10}+\frac{16\cdots 69}{17\cdots 72}a^{9}+\frac{15\cdots 49}{17\cdots 72}a^{8}-\frac{16\cdots 35}{17\cdots 72}a^{7}-\frac{12\cdots 93}{17\cdots 72}a^{6}-\frac{16\cdots 69}{44\cdots 68}a^{5}-\frac{99\cdots 47}{22\cdots 84}a^{4}-\frac{26\cdots 29}{22\cdots 84}a^{3}-\frac{17\cdots 41}{27\cdots 48}a^{2}-\frac{32\cdots 11}{27\cdots 48}a+\frac{48\cdots 25}{69\cdots 87}$, $\frac{31\cdots 47}{35\cdots 44}a^{19}-\frac{14\cdots 03}{35\cdots 44}a^{18}+\frac{20\cdots 57}{35\cdots 44}a^{17}-\frac{53\cdots 83}{35\cdots 44}a^{16}+\frac{85\cdots 49}{22\cdots 84}a^{15}-\frac{52\cdots 67}{11\cdots 92}a^{14}+\frac{13\cdots 41}{35\cdots 44}a^{13}-\frac{13\cdots 71}{35\cdots 44}a^{12}+\frac{10\cdots 05}{88\cdots 36}a^{11}-\frac{90\cdots 53}{88\cdots 36}a^{10}+\frac{33\cdots 09}{35\cdots 44}a^{9}+\frac{48\cdots 33}{35\cdots 44}a^{8}-\frac{20\cdots 95}{35\cdots 44}a^{7}-\frac{18\cdots 49}{35\cdots 44}a^{6}-\frac{14\cdots 87}{44\cdots 68}a^{5}-\frac{63\cdots 69}{22\cdots 84}a^{4}-\frac{46\cdots 53}{44\cdots 68}a^{3}-\frac{44\cdots 97}{11\cdots 92}a^{2}-\frac{29\cdots 61}{27\cdots 48}a+\frac{43\cdots 83}{69\cdots 87}$, $\frac{11\cdots 35}{88\cdots 36}a^{19}-\frac{31\cdots 53}{35\cdots 44}a^{18}+\frac{18\cdots 61}{17\cdots 72}a^{17}-\frac{15\cdots 73}{35\cdots 44}a^{16}+\frac{26\cdots 67}{17\cdots 72}a^{15}-\frac{17\cdots 09}{17\cdots 72}a^{14}+\frac{49\cdots 11}{17\cdots 72}a^{13}-\frac{39\cdots 81}{35\cdots 44}a^{12}+\frac{73\cdots 29}{17\cdots 72}a^{11}-\frac{15\cdots 33}{17\cdots 72}a^{10}+\frac{56\cdots 77}{17\cdots 72}a^{9}-\frac{23\cdots 89}{35\cdots 44}a^{8}+\frac{22\cdots 95}{17\cdots 72}a^{7}-\frac{11\cdots 05}{35\cdots 44}a^{6}+\frac{22\cdots 29}{88\cdots 36}a^{5}-\frac{35\cdots 87}{44\cdots 68}a^{4}+\frac{15\cdots 55}{44\cdots 68}a^{3}-\frac{19\cdots 25}{27\cdots 48}a^{2}+\frac{94\cdots 25}{27\cdots 48}a+\frac{92\cdots 59}{69\cdots 87}$, $\frac{12\cdots 57}{10\cdots 32}a^{19}-\frac{31\cdots 27}{53\cdots 16}a^{18}+\frac{31\cdots 19}{35\cdots 44}a^{17}-\frac{14\cdots 19}{53\cdots 16}a^{16}+\frac{69\cdots 85}{53\cdots 16}a^{15}-\frac{17\cdots 27}{17\cdots 72}a^{14}+\frac{58\cdots 23}{35\cdots 44}a^{13}-\frac{22\cdots 75}{17\cdots 72}a^{12}+\frac{19\cdots 37}{53\cdots 16}a^{11}-\frac{14\cdots 33}{17\cdots 72}a^{10}+\frac{15\cdots 15}{35\cdots 44}a^{9}-\frac{32\cdots 21}{53\cdots 16}a^{8}+\frac{77\cdots 07}{35\cdots 44}a^{7}-\frac{46\cdots 13}{88\cdots 36}a^{6}+\frac{17\cdots 19}{41\cdots 22}a^{5}-\frac{11\cdots 05}{44\cdots 68}a^{4}-\frac{91\cdots 35}{55\cdots 96}a^{3}-\frac{97\cdots 69}{16\cdots 88}a^{2}-\frac{22\cdots 77}{20\cdots 61}a-\frac{96\cdots 53}{20\cdots 61}$, $\frac{20\cdots 17}{17\cdots 72}a^{19}+\frac{68\cdots 09}{88\cdots 36}a^{18}-\frac{21\cdots 93}{17\cdots 72}a^{17}+\frac{75\cdots 63}{88\cdots 36}a^{16}-\frac{84\cdots 85}{22\cdots 84}a^{15}+\frac{72\cdots 83}{88\cdots 36}a^{14}-\frac{16\cdots 79}{17\cdots 72}a^{13}+\frac{18\cdots 21}{88\cdots 36}a^{12}-\frac{20\cdots 29}{22\cdots 84}a^{11}+\frac{35\cdots 61}{88\cdots 36}a^{10}-\frac{12\cdots 39}{17\cdots 72}a^{9}+\frac{27\cdots 21}{88\cdots 36}a^{8}-\frac{10\cdots 97}{17\cdots 72}a^{7}+\frac{48\cdots 19}{44\cdots 68}a^{6}-\frac{32\cdots 13}{88\cdots 36}a^{5}+\frac{15\cdots 01}{11\cdots 92}a^{4}-\frac{10\cdots 73}{11\cdots 92}a^{3}+\frac{35\cdots 87}{11\cdots 92}a^{2}-\frac{70\cdots 88}{69\cdots 87}a+\frac{40\cdots 07}{69\cdots 87}$, $\frac{19\cdots 59}{10\cdots 32}a^{19}-\frac{13\cdots 09}{26\cdots 08}a^{18}+\frac{33\cdots 41}{35\cdots 44}a^{17}-\frac{10\cdots 91}{26\cdots 08}a^{16}-\frac{30\cdots 83}{53\cdots 16}a^{15}-\frac{69\cdots 59}{17\cdots 72}a^{14}-\frac{99\cdots 27}{35\cdots 44}a^{13}+\frac{10\cdots 87}{44\cdots 68}a^{12}-\frac{99\cdots 15}{53\cdots 16}a^{11}+\frac{27\cdots 35}{17\cdots 72}a^{10}-\frac{66\cdots 71}{35\cdots 44}a^{9}+\frac{19\cdots 17}{13\cdots 04}a^{8}-\frac{10\cdots 11}{35\cdots 44}a^{7}+\frac{91\cdots 67}{17\cdots 72}a^{6}-\frac{69\cdots 73}{33\cdots 76}a^{5}+\frac{25\cdots 05}{44\cdots 68}a^{4}-\frac{12\cdots 63}{22\cdots 84}a^{3}-\frac{14\cdots 63}{83\cdots 44}a^{2}-\frac{50\cdots 97}{83\cdots 44}a+\frac{71\cdots 79}{20\cdots 61}$, $\frac{10\cdots 75}{26\cdots 08}a^{19}-\frac{88\cdots 59}{53\cdots 16}a^{18}+\frac{12\cdots 21}{44\cdots 68}a^{17}-\frac{39\cdots 79}{53\cdots 16}a^{16}+\frac{10\cdots 53}{26\cdots 08}a^{15}-\frac{25\cdots 75}{88\cdots 36}a^{14}+\frac{83\cdots 11}{22\cdots 84}a^{13}-\frac{66\cdots 25}{17\cdots 72}a^{12}+\frac{26\cdots 51}{26\cdots 08}a^{11}-\frac{18\cdots 47}{88\cdots 36}a^{10}+\frac{55\cdots 71}{44\cdots 68}a^{9}-\frac{69\cdots 59}{53\cdots 16}a^{8}+\frac{67\cdots 63}{11\cdots 92}a^{7}-\frac{25\cdots 33}{17\cdots 72}a^{6}+\frac{66\cdots 27}{13\cdots 04}a^{5}-\frac{89\cdots 87}{11\cdots 92}a^{4}-\frac{79\cdots 93}{22\cdots 84}a^{3}-\frac{15\cdots 15}{83\cdots 44}a^{2}-\frac{47\cdots 61}{83\cdots 44}a-\frac{27\cdots 79}{20\cdots 61}$, $\frac{13\cdots 71}{35\cdots 44}a^{19}-\frac{27\cdots 55}{44\cdots 68}a^{18}+\frac{40\cdots 63}{35\cdots 44}a^{17}+\frac{40\cdots 47}{44\cdots 68}a^{16}-\frac{15\cdots 11}{17\cdots 72}a^{15}+\frac{64\cdots 55}{17\cdots 72}a^{14}-\frac{93\cdots 49}{35\cdots 44}a^{13}+\frac{77\cdots 75}{88\cdots 36}a^{12}-\frac{55\cdots 55}{17\cdots 72}a^{11}+\frac{19\cdots 97}{17\cdots 72}a^{10}-\frac{87\cdots 53}{35\cdots 44}a^{9}+\frac{66\cdots 63}{88\cdots 36}a^{8}-\frac{53\cdots 97}{35\cdots 44}a^{7}+\frac{44\cdots 19}{17\cdots 72}a^{6}-\frac{17\cdots 13}{27\cdots 48}a^{5}+\frac{16\cdots 93}{44\cdots 68}a^{4}-\frac{31\cdots 95}{22\cdots 84}a^{3}+\frac{59\cdots 89}{27\cdots 48}a^{2}-\frac{92\cdots 61}{69\cdots 87}a+\frac{52\cdots 67}{69\cdots 87}$, $\frac{21\cdots 47}{53\cdots 16}a^{19}-\frac{47\cdots 83}{13\cdots 04}a^{18}+\frac{55\cdots 11}{17\cdots 72}a^{17}-\frac{21\cdots 05}{13\cdots 04}a^{16}+\frac{21\cdots 81}{83\cdots 44}a^{15}-\frac{21\cdots 83}{88\cdots 36}a^{14}+\frac{18\cdots 49}{17\cdots 72}a^{13}-\frac{16\cdots 07}{22\cdots 84}a^{12}+\frac{36\cdots 45}{33\cdots 76}a^{11}-\frac{15\cdots 89}{88\cdots 36}a^{10}+\frac{49\cdots 89}{17\cdots 72}a^{9}-\frac{12\cdots 39}{66\cdots 52}a^{8}-\frac{42\cdots 21}{17\cdots 72}a^{7}-\frac{18\cdots 57}{88\cdots 36}a^{6}+\frac{14\cdots 81}{26\cdots 08}a^{5}+\frac{43\cdots 40}{69\cdots 87}a^{4}+\frac{18\cdots 11}{27\cdots 48}a^{3}+\frac{73\cdots 97}{33\cdots 76}a^{2}+\frac{48\cdots 31}{41\cdots 22}a+\frac{47\cdots 83}{20\cdots 61}$
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| Regulator: | \( 496592090559000000 \) (assuming GRH) |
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| Unit signature rank: | \( 2 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 496592090559000000 \cdot 1}{2\cdot\sqrt{206007596521214410095208558252435839890349094339}}\cr\approx \mathstrut & 33.3969642233715 \end{aligned}\] (assuming GRH)
Galois group
$\PGL(2,19)$ (as 20T362):
| A non-solvable group of order 6840 |
| The 21 conjugacy class representatives for $\PGL(2,19)$ |
| Character table for $\PGL(2,19)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 40 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{9}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | $20$ | $18{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20$ | $18{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | $18{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 19.1.19.37a1.13 | $x^{19} + 4332 x + 19$ | $19$ | $1$ | $37$ | $F_{19}$ | $$[\frac{37}{18}]_{18}$$ |
Spectrum of ring of integers
Additional information
This field is associated with the 19-torsion points on any elliptic curve in the isogeny class 19.a.