Normalized defining polynomial
\( x^{20} + 35 x^{18} + 560 x^{16} - 119 x^{15} + 5075 x^{14} - 4515 x^{13} + 30625 x^{12} + \cdots + 29901760 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(77992289980729574777937412261962890625\)
\(\medspace = 3^{15}\cdot 5^{21}\cdot 7^{19}\)
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| Root discriminant: | \(78.45\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{105}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7}a^{10}$, $\frac{1}{7}a^{11}$, $\frac{1}{14}a^{12}-\frac{1}{14}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{14}a^{13}-\frac{1}{14}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{14}a^{14}-\frac{1}{14}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{140}a^{15}-\frac{1}{28}a^{14}-\frac{1}{28}a^{12}+\frac{1}{28}a^{11}-\frac{3}{140}a^{10}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{9}{20}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{140}a^{16}-\frac{1}{28}a^{14}-\frac{1}{28}a^{13}+\frac{1}{70}a^{11}-\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{3}{10}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{140}a^{17}-\frac{3}{140}a^{12}-\frac{1}{28}a^{11}-\frac{1}{14}a^{10}+\frac{1}{20}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a$, $\frac{1}{786328200}a^{18}+\frac{2250527}{786328200}a^{17}-\frac{1241381}{393164100}a^{16}+\frac{1324927}{393164100}a^{15}-\frac{220}{561663}a^{14}+\frac{1786021}{112332600}a^{13}-\frac{994913}{65527350}a^{12}+\frac{52375901}{786328200}a^{11}+\frac{11684417}{196582050}a^{10}-\frac{94099}{330390}a^{9}+\frac{3235159}{12481400}a^{8}+\frac{702637}{18722100}a^{7}-\frac{26145907}{112332600}a^{6}-\frac{381067}{1651950}a^{5}+\frac{2469889}{11233260}a^{4}-\frac{85259}{660780}a^{3}-\frac{1064303}{4493304}a^{2}+\frac{6631793}{22466520}a+\frac{659347}{2808315}$, $\frac{1}{73\cdots 00}a^{19}+\frac{19\cdots 11}{81\cdots 00}a^{18}-\frac{61\cdots 91}{91\cdots 75}a^{17}-\frac{92\cdots 49}{18\cdots 15}a^{16}-\frac{56\cdots 89}{17\cdots 25}a^{15}-\frac{25\cdots 03}{73\cdots 00}a^{14}-\frac{75\cdots 11}{36\cdots 00}a^{13}+\frac{29\cdots 99}{73\cdots 00}a^{12}+\frac{49\cdots 09}{73\cdots 60}a^{11}-\frac{44\cdots 28}{91\cdots 75}a^{10}+\frac{44\cdots 61}{10\cdots 00}a^{9}-\frac{17\cdots 79}{43\cdots 75}a^{8}-\frac{13\cdots 33}{10\cdots 00}a^{7}+\frac{57\cdots 91}{11\cdots 20}a^{6}+\frac{10\cdots 29}{52\cdots 00}a^{5}+\frac{70\cdots 65}{34\cdots 26}a^{4}+\frac{10\cdots 21}{69\cdots 20}a^{3}-\frac{48\cdots 87}{20\cdots 60}a^{2}-\frac{20\cdots 87}{52\cdots 90}a+\frac{56\cdots 54}{26\cdots 45}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{270}$, which has order $1080$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{270}$, which has order $1080$ (assuming GRH) |
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| Relative class number: | data not computed (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{76\cdots 09}{24\cdots 20}a^{19}+\frac{45\cdots 57}{24\cdots 20}a^{18}+\frac{20\cdots 13}{20\cdots 35}a^{17}-\frac{47\cdots 69}{81\cdots 40}a^{16}+\frac{35\cdots 19}{24\cdots 20}a^{15}-\frac{28\cdots 77}{47\cdots 20}a^{14}+\frac{28\cdots 29}{24\cdots 20}a^{13}-\frac{96\cdots 52}{60\cdots 05}a^{12}+\frac{51\cdots 37}{81\cdots 40}a^{11}-\frac{18\cdots 77}{11\cdots 20}a^{10}+\frac{80\cdots 59}{34\cdots 60}a^{9}-\frac{37\cdots 51}{11\cdots 20}a^{8}+\frac{22\cdots 83}{17\cdots 30}a^{7}+\frac{58\cdots 13}{34\cdots 60}a^{6}+\frac{13\cdots 33}{11\cdots 20}a^{5}+\frac{15\cdots 87}{17\cdots 63}a^{4}+\frac{63\cdots 35}{17\cdots 63}a^{3}+\frac{11\cdots 57}{11\cdots 42}a^{2}+\frac{14\cdots 43}{57\cdots 21}a+\frac{12\cdots 71}{17\cdots 63}$, $\frac{15\cdots 75}{19\cdots 48}a^{19}-\frac{23\cdots 45}{33\cdots 34}a^{18}+\frac{95\cdots 75}{45\cdots 12}a^{17}-\frac{14\cdots 10}{56\cdots 39}a^{16}+\frac{45\cdots 70}{16\cdots 17}a^{15}-\frac{29\cdots 75}{64\cdots 16}a^{14}+\frac{39\cdots 25}{19\cdots 48}a^{13}-\frac{11\cdots 75}{19\cdots 48}a^{12}+\frac{92\cdots 35}{64\cdots 16}a^{11}-\frac{13\cdots 83}{32\cdots 08}a^{10}+\frac{13\cdots 25}{19\cdots 48}a^{9}-\frac{60\cdots 25}{64\cdots 16}a^{8}+\frac{72\cdots 25}{19\cdots 48}a^{7}-\frac{73\cdots 05}{19\cdots 48}a^{6}+\frac{37\cdots 73}{16\cdots 54}a^{5}-\frac{13\cdots 25}{97\cdots 24}a^{4}+\frac{43\cdots 15}{19\cdots 48}a^{3}-\frac{52\cdots 50}{80\cdots 77}a^{2}+\frac{13\cdots 65}{64\cdots 16}a-\frac{90\cdots 28}{24\cdots 31}$, $\frac{23\cdots 43}{48\cdots 40}a^{19}+\frac{57\cdots 53}{24\cdots 20}a^{18}+\frac{12\cdots 37}{46\cdots 68}a^{17}+\frac{31\cdots 69}{40\cdots 70}a^{16}+\frac{30\cdots 19}{60\cdots 05}a^{15}+\frac{14\cdots 57}{16\cdots 80}a^{14}+\frac{20\cdots 47}{48\cdots 40}a^{13}+\frac{14\cdots 17}{57\cdots 84}a^{12}+\frac{22\cdots 77}{16\cdots 80}a^{11}-\frac{34\cdots 63}{11\cdots 20}a^{10}+\frac{68\cdots 43}{69\cdots 20}a^{9}-\frac{27\cdots 53}{23\cdots 40}a^{8}+\frac{46\cdots 39}{13\cdots 04}a^{7}+\frac{12\cdots 63}{69\cdots 20}a^{6}+\frac{15\cdots 28}{28\cdots 05}a^{5}+\frac{78\cdots 49}{69\cdots 52}a^{4}+\frac{34\cdots 29}{13\cdots 04}a^{3}+\frac{51\cdots 49}{23\cdots 84}a^{2}+\frac{75\cdots 13}{46\cdots 68}a+\frac{32\cdots 95}{17\cdots 63}$, $\frac{43\cdots 43}{97\cdots 28}a^{19}+\frac{57\cdots 05}{12\cdots 41}a^{18}+\frac{21\cdots 81}{16\cdots 80}a^{17}+\frac{24\cdots 49}{20\cdots 35}a^{16}+\frac{15\cdots 49}{86\cdots 15}a^{15}+\frac{25\cdots 81}{32\cdots 76}a^{14}+\frac{11\cdots 85}{97\cdots 28}a^{13}-\frac{50\cdots 59}{48\cdots 40}a^{12}+\frac{75\cdots 99}{16\cdots 80}a^{11}-\frac{15\cdots 61}{11\cdots 20}a^{10}+\frac{90\cdots 59}{13\cdots 04}a^{9}-\frac{33\cdots 27}{27\cdots 04}a^{8}+\frac{11\cdots 63}{69\cdots 20}a^{7}+\frac{21\cdots 01}{69\cdots 20}a^{6}+\frac{96\cdots 37}{57\cdots 10}a^{5}+\frac{13\cdots 25}{69\cdots 52}a^{4}+\frac{39\cdots 41}{13\cdots 04}a^{3}+\frac{60\cdots 51}{57\cdots 21}a^{2}+\frac{96\cdots 79}{46\cdots 68}a-\frac{54\cdots 82}{17\cdots 63}$, $\frac{15\cdots 21}{97\cdots 28}a^{19}+\frac{52\cdots 15}{24\cdots 82}a^{18}-\frac{18\cdots 19}{32\cdots 76}a^{17}+\frac{13\cdots 13}{20\cdots 35}a^{16}-\frac{45\cdots 03}{48\cdots 64}a^{15}+\frac{37\cdots 43}{32\cdots 76}a^{14}-\frac{12\cdots 45}{13\cdots 04}a^{13}+\frac{14\cdots 39}{97\cdots 28}a^{12}-\frac{96\cdots 27}{16\cdots 80}a^{11}+\frac{81\cdots 88}{57\cdots 21}a^{10}-\frac{39\cdots 07}{13\cdots 04}a^{9}+\frac{24\cdots 45}{46\cdots 68}a^{8}-\frac{12\cdots 79}{13\cdots 04}a^{7}-\frac{29\cdots 33}{69\cdots 20}a^{6}-\frac{17\cdots 01}{23\cdots 84}a^{5}-\frac{21\cdots 91}{17\cdots 63}a^{4}-\frac{49\cdots 55}{13\cdots 04}a^{3}-\frac{16\cdots 85}{23\cdots 84}a^{2}-\frac{11\cdots 55}{46\cdots 68}a+\frac{41\cdots 18}{17\cdots 63}$, $\frac{52\cdots 69}{12\cdots 10}a^{19}-\frac{36\cdots 79}{97\cdots 28}a^{18}+\frac{24\cdots 77}{16\cdots 80}a^{17}-\frac{11\cdots 33}{81\cdots 40}a^{16}+\frac{29\cdots 93}{12\cdots 10}a^{15}-\frac{32\cdots 39}{11\cdots 20}a^{14}+\frac{30\cdots 97}{13\cdots 04}a^{13}-\frac{97\cdots 59}{24\cdots 20}a^{12}+\frac{24\cdots 33}{16\cdots 80}a^{11}-\frac{42\cdots 43}{11\cdots 20}a^{10}+\frac{27\cdots 93}{34\cdots 60}a^{9}-\frac{63\cdots 99}{46\cdots 68}a^{8}+\frac{57\cdots 59}{17\cdots 30}a^{7}-\frac{61\cdots 93}{69\cdots 20}a^{6}+\frac{24\cdots 97}{11\cdots 20}a^{5}-\frac{13\cdots 09}{34\cdots 26}a^{4}+\frac{57\cdots 65}{69\cdots 52}a^{3}+\frac{35\cdots 05}{46\cdots 68}a^{2}+\frac{26\cdots 05}{46\cdots 68}a+\frac{14\cdots 23}{17\cdots 63}$, $\frac{10\cdots 77}{48\cdots 40}a^{19}+\frac{74\cdots 43}{24\cdots 20}a^{18}+\frac{11\cdots 97}{16\cdots 80}a^{17}+\frac{67\cdots 13}{81\cdots 40}a^{16}+\frac{60\cdots 17}{60\cdots 05}a^{15}+\frac{11\cdots 33}{16\cdots 80}a^{14}+\frac{35\cdots 67}{48\cdots 40}a^{13}-\frac{18\cdots 13}{48\cdots 40}a^{12}+\frac{45\cdots 27}{16\cdots 80}a^{11}-\frac{88\cdots 49}{11\cdots 20}a^{10}+\frac{33\cdots 87}{69\cdots 20}a^{9}-\frac{27\cdots 93}{23\cdots 40}a^{8}+\frac{66\cdots 01}{69\cdots 20}a^{7}+\frac{12\cdots 33}{69\cdots 20}a^{6}+\frac{29\cdots 64}{28\cdots 05}a^{5}+\frac{45\cdots 83}{34\cdots 26}a^{4}+\frac{34\cdots 21}{13\cdots 04}a^{3}+\frac{32\cdots 75}{23\cdots 84}a^{2}+\frac{81\cdots 45}{46\cdots 68}a-\frac{64\cdots 90}{17\cdots 63}$, $\frac{27\cdots 35}{12\cdots 41}a^{19}+\frac{98\cdots 11}{48\cdots 40}a^{18}+\frac{37\cdots 87}{32\cdots 76}a^{17}+\frac{14\cdots 23}{20\cdots 35}a^{16}+\frac{50\cdots 25}{24\cdots 82}a^{15}+\frac{42\cdots 71}{81\cdots 94}a^{14}+\frac{97\cdots 87}{48\cdots 40}a^{13}-\frac{69\cdots 41}{69\cdots 52}a^{12}+\frac{16\cdots 83}{16\cdots 80}a^{11}-\frac{52\cdots 17}{23\cdots 84}a^{10}+\frac{13\cdots 69}{34\cdots 26}a^{9}-\frac{25\cdots 43}{23\cdots 40}a^{8}+\frac{52\cdots 95}{34\cdots 26}a^{7}+\frac{23\cdots 77}{69\cdots 20}a^{6}+\frac{48\cdots 95}{23\cdots 84}a^{5}+\frac{19\cdots 33}{69\cdots 52}a^{4}+\frac{19\cdots 27}{17\cdots 63}a^{3}+\frac{32\cdots 07}{46\cdots 68}a^{2}+\frac{19\cdots 25}{27\cdots 04}a-\frac{32\cdots 69}{17\cdots 63}$, $\frac{12\cdots 57}{97\cdots 28}a^{19}+\frac{75\cdots 41}{28\cdots 20}a^{18}+\frac{37\cdots 81}{81\cdots 40}a^{17}+\frac{30\cdots 41}{40\cdots 70}a^{16}+\frac{17\cdots 89}{24\cdots 20}a^{15}+\frac{21\cdots 09}{32\cdots 76}a^{14}+\frac{67\cdots 31}{12\cdots 10}a^{13}-\frac{12\cdots 69}{69\cdots 20}a^{12}+\frac{16\cdots 79}{81\cdots 40}a^{11}-\frac{70\cdots 77}{11\cdots 20}a^{10}+\frac{54\cdots 83}{13\cdots 04}a^{9}-\frac{19\cdots 03}{11\cdots 20}a^{8}+\frac{27\cdots 53}{40\cdots 60}a^{7}+\frac{55\cdots 81}{34\cdots 60}a^{6}+\frac{43\cdots 29}{57\cdots 10}a^{5}+\frac{74\cdots 91}{69\cdots 52}a^{4}+\frac{35\cdots 87}{13\cdots 04}a^{3}+\frac{77\cdots 79}{46\cdots 68}a^{2}+\frac{19\cdots 29}{11\cdots 42}a-\frac{28\cdots 07}{17\cdots 63}$
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| Regulator: | \( 224767289.30957106 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 224767289.30957106 \cdot 1080}{2\cdot\sqrt{77992289980729574777937412261962890625}}\cr\approx \mathstrut & 1.31795236990188 \end{aligned}\] (assuming GRH)
Galois group
$C_{20}:C_4$ (as 20T18):
| A solvable group of order 80 |
| The 14 conjugacy class representatives for $C_{20}:C_4$ |
| Character table for $C_{20}:C_4$ |
Intermediate fields
| \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-42 +2 \sqrt{21}})\), 5.5.67528125.1, deg 10 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 sibling: | data not computed |
| Degree 40 siblings: | 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $20$ | $20$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(5\)
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.5.5.1 | $x^{5} + 20 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
| 5.10.11.1 | $x^{10} + 20 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
|
\(7\)
| 7.20.19.1 | $x^{20} + 7$ | $20$ | $1$ | $19$ | 20T18 | $$[\ ]_{20}^{4}$$ |