Normalized defining polynomial
\( x^{20} + 6 x^{18} + 141 x^{16} - 200 x^{15} + 1128 x^{14} + 12 x^{12} + 13488 x^{11} + 5688 x^{10} + \cdots + 26644 \)
Invariants
| Degree: | $20$ |
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| Signature: | $[0, 10]$ |
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| Discriminant: |
\(38975682627981862469140631179493376\)
\(\medspace = 2^{63}\cdot 3^{15}\cdot 131^{4}\)
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| Root discriminant: | \(53.65\) |
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| Galois root discriminant: | $2^{27/8}3^{3/4}131^{1/2}\approx 270.67791607719977$ | ||
| Ramified primes: |
\(2\), \(3\), \(131\)
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| Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 4.0.13824.1 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{9}-\frac{1}{8}a^{8}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{16}a^{14}+\frac{1}{16}a^{10}-\frac{3}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}-\frac{1}{8}a^{2}+\frac{1}{4}$, $\frac{1}{16}a^{15}+\frac{1}{16}a^{11}-\frac{3}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{16}-\frac{1}{16}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{16}a^{17}-\frac{1}{16}a^{13}-\frac{1}{8}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{6288}a^{18}+\frac{157}{6288}a^{17}+\frac{5}{262}a^{16}+\frac{167}{6288}a^{15}+\frac{185}{6288}a^{14}+\frac{229}{6288}a^{13}+\frac{157}{3144}a^{12}-\frac{407}{6288}a^{11}+\frac{277}{3144}a^{10}-\frac{623}{1572}a^{9}+\frac{659}{1572}a^{8}+\frac{1241}{3144}a^{7}-\frac{821}{3144}a^{6}+\frac{1259}{3144}a^{5}-\frac{271}{1572}a^{4}+\frac{523}{3144}a^{3}+\frac{103}{262}a^{2}-\frac{761}{1572}a+\frac{179}{786}$, $\frac{1}{42\cdots 96}a^{19}+\frac{95\cdots 20}{26\cdots 81}a^{18}-\frac{15\cdots 13}{70\cdots 16}a^{17}-\frac{54\cdots 19}{42\cdots 96}a^{16}-\frac{24\cdots 49}{26\cdots 81}a^{15}+\frac{43\cdots 31}{42\cdots 96}a^{14}+\frac{11\cdots 63}{21\cdots 48}a^{13}-\frac{23\cdots 97}{42\cdots 96}a^{12}+\frac{40\cdots 61}{42\cdots 96}a^{11}-\frac{15\cdots 79}{42\cdots 96}a^{10}-\frac{44\cdots 43}{10\cdots 24}a^{9}-\frac{10\cdots 89}{21\cdots 48}a^{8}+\frac{69\cdots 75}{26\cdots 81}a^{7}-\frac{99\cdots 59}{21\cdots 48}a^{6}+\frac{54\cdots 69}{10\cdots 24}a^{5}-\frac{21\cdots 79}{21\cdots 48}a^{4}-\frac{29\cdots 81}{70\cdots 16}a^{3}+\frac{26\cdots 23}{21\cdots 48}a^{2}+\frac{18\cdots 85}{52\cdots 62}a+\frac{49\cdots 73}{17\cdots 54}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{2}$, which has order $2$ (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: |
$\frac{33\cdots 19}{53\cdots 32}a^{19}-\frac{23\cdots 53}{53\cdots 32}a^{18}+\frac{11\cdots 59}{26\cdots 16}a^{17}-\frac{16\cdots 58}{66\cdots 29}a^{16}+\frac{48\cdots 61}{53\cdots 32}a^{15}-\frac{16\cdots 29}{88\cdots 72}a^{14}+\frac{15\cdots 51}{17\cdots 44}a^{13}-\frac{12\cdots 60}{22\cdots 43}a^{12}+\frac{13\cdots 05}{44\cdots 86}a^{11}+\frac{15\cdots 13}{17\cdots 44}a^{10}-\frac{48\cdots 83}{17\cdots 44}a^{9}+\frac{79\cdots 31}{88\cdots 72}a^{8}+\frac{15\cdots 27}{88\cdots 72}a^{7}+\frac{92\cdots 15}{44\cdots 86}a^{6}+\frac{12\cdots 33}{88\cdots 72}a^{5}-\frac{19\cdots 55}{66\cdots 29}a^{4}-\frac{15\cdots 77}{13\cdots 58}a^{3}+\frac{79\cdots 99}{26\cdots 16}a^{2}+\frac{22\cdots 69}{26\cdots 16}a+\frac{60\cdots 87}{13\cdots 58}$, $\frac{24\cdots 81}{26\cdots 81}a^{19}+\frac{24\cdots 45}{42\cdots 96}a^{18}+\frac{43\cdots 95}{26\cdots 81}a^{17}+\frac{20\cdots 59}{42\cdots 96}a^{16}+\frac{34\cdots 65}{42\cdots 96}a^{15}+\frac{78\cdots 79}{10\cdots 72}a^{14}-\frac{63\cdots 83}{70\cdots 16}a^{13}+\frac{13\cdots 25}{14\cdots 32}a^{12}-\frac{16\cdots 97}{14\cdots 32}a^{11}+\frac{26\cdots 21}{70\cdots 16}a^{10}+\frac{12\cdots 23}{35\cdots 08}a^{9}+\frac{65\cdots 25}{17\cdots 54}a^{8}+\frac{11\cdots 29}{70\cdots 16}a^{7}-\frac{50\cdots 09}{70\cdots 16}a^{6}+\frac{22\cdots 11}{35\cdots 08}a^{5}+\frac{19\cdots 25}{21\cdots 48}a^{4}-\frac{22\cdots 17}{21\cdots 48}a^{3}+\frac{23\cdots 05}{52\cdots 62}a^{2}-\frac{11\cdots 90}{26\cdots 81}a-\frac{10\cdots 15}{10\cdots 24}$, $\frac{67\cdots 05}{10\cdots 24}a^{19}-\frac{90\cdots 59}{42\cdots 96}a^{18}+\frac{94\cdots 51}{10\cdots 24}a^{17}-\frac{95\cdots 29}{42\cdots 96}a^{16}+\frac{57\cdots 35}{42\cdots 96}a^{15}-\frac{72\cdots 91}{14\cdots 32}a^{14}+\frac{17\cdots 01}{87\cdots 27}a^{13}-\frac{70\cdots 83}{14\cdots 32}a^{12}+\frac{14\cdots 17}{14\cdots 32}a^{11}-\frac{67\cdots 87}{70\cdots 16}a^{10}+\frac{24\cdots 65}{70\cdots 16}a^{9}+\frac{39\cdots 91}{17\cdots 54}a^{8}-\frac{35\cdots 33}{70\cdots 16}a^{7}+\frac{81\cdots 99}{70\cdots 16}a^{6}-\frac{44\cdots 51}{87\cdots 27}a^{5}-\frac{16\cdots 91}{21\cdots 48}a^{4}+\frac{62\cdots 13}{21\cdots 48}a^{3}-\frac{16\cdots 83}{52\cdots 62}a^{2}+\frac{25\cdots 03}{10\cdots 24}a-\frac{22\cdots 07}{10\cdots 24}$, $\frac{45\cdots 99}{10\cdots 24}a^{19}+\frac{43\cdots 03}{26\cdots 81}a^{18}+\frac{94\cdots 17}{42\cdots 96}a^{17}+\frac{45\cdots 79}{42\cdots 96}a^{16}+\frac{23\cdots 55}{42\cdots 96}a^{15}+\frac{53\cdots 25}{35\cdots 08}a^{14}+\frac{10\cdots 67}{14\cdots 32}a^{13}+\frac{29\cdots 31}{14\cdots 32}a^{12}-\frac{17\cdots 29}{14\cdots 32}a^{11}+\frac{55\cdots 19}{70\cdots 16}a^{10}+\frac{65\cdots 83}{35\cdots 08}a^{9}+\frac{48\cdots 03}{35\cdots 08}a^{8}+\frac{23\cdots 15}{70\cdots 16}a^{7}+\frac{51\cdots 91}{17\cdots 54}a^{6}+\frac{12\cdots 61}{70\cdots 16}a^{5}+\frac{78\cdots 43}{21\cdots 48}a^{4}+\frac{25\cdots 55}{21\cdots 48}a^{3}-\frac{34\cdots 15}{10\cdots 24}a^{2}-\frac{13\cdots 97}{10\cdots 24}a+\frac{31\cdots 15}{10\cdots 24}$, $\frac{11\cdots 47}{21\cdots 48}a^{19}-\frac{12\cdots 43}{52\cdots 62}a^{18}+\frac{33\cdots 39}{42\cdots 96}a^{17}-\frac{91\cdots 53}{42\cdots 96}a^{16}+\frac{49\cdots 85}{42\cdots 96}a^{15}-\frac{35\cdots 43}{70\cdots 16}a^{14}+\frac{26\cdots 35}{14\cdots 32}a^{13}-\frac{68\cdots 49}{14\cdots 32}a^{12}+\frac{12\cdots 83}{14\cdots 32}a^{11}-\frac{12\cdots 71}{17\cdots 54}a^{10}-\frac{40\cdots 49}{70\cdots 16}a^{9}+\frac{19\cdots 79}{87\cdots 27}a^{8}-\frac{30\cdots 03}{70\cdots 16}a^{7}+\frac{27\cdots 79}{35\cdots 08}a^{6}-\frac{38\cdots 87}{70\cdots 16}a^{5}-\frac{41\cdots 81}{21\cdots 48}a^{4}+\frac{58\cdots 35}{21\cdots 48}a^{3}-\frac{23\cdots 13}{52\cdots 62}a^{2}-\frac{11\cdots 15}{52\cdots 62}a+\frac{66\cdots 09}{10\cdots 24}$, $\frac{32\cdots 57}{42\cdots 96}a^{19}-\frac{10\cdots 47}{42\cdots 96}a^{18}+\frac{22\cdots 67}{42\cdots 96}a^{17}-\frac{36\cdots 69}{21\cdots 48}a^{16}+\frac{48\cdots 23}{42\cdots 96}a^{15}-\frac{27\cdots 69}{14\cdots 32}a^{14}+\frac{14\cdots 93}{14\cdots 32}a^{13}-\frac{35\cdots 79}{70\cdots 16}a^{12}+\frac{56\cdots 97}{70\cdots 16}a^{11}+\frac{64\cdots 83}{70\cdots 16}a^{10}+\frac{16\cdots 50}{87\cdots 27}a^{9}+\frac{73\cdots 99}{70\cdots 16}a^{8}+\frac{12\cdots 51}{70\cdots 16}a^{7}+\frac{26\cdots 77}{70\cdots 16}a^{6}+\frac{11\cdots 67}{70\cdots 16}a^{5}+\frac{22\cdots 15}{10\cdots 24}a^{4}-\frac{81\cdots 85}{52\cdots 62}a^{3}+\frac{10\cdots 69}{52\cdots 62}a^{2}+\frac{25\cdots 89}{10\cdots 24}a-\frac{65\cdots 55}{10\cdots 24}$, $\frac{45\cdots 81}{42\cdots 96}a^{19}+\frac{26\cdots 31}{10\cdots 24}a^{18}+\frac{13\cdots 81}{21\cdots 48}a^{17}+\frac{86\cdots 17}{52\cdots 62}a^{16}+\frac{63\cdots 41}{42\cdots 96}a^{15}-\frac{12\cdots 97}{70\cdots 16}a^{14}+\frac{78\cdots 87}{70\cdots 16}a^{13}+\frac{27\cdots 67}{70\cdots 16}a^{12}-\frac{39\cdots 87}{70\cdots 16}a^{11}+\frac{10\cdots 27}{70\cdots 16}a^{10}+\frac{54\cdots 55}{70\cdots 16}a^{9}+\frac{94\cdots 21}{70\cdots 16}a^{8}+\frac{23\cdots 69}{70\cdots 16}a^{7}+\frac{18\cdots 39}{35\cdots 08}a^{6}+\frac{10\cdots 19}{35\cdots 08}a^{5}+\frac{17\cdots 35}{10\cdots 24}a^{4}-\frac{73\cdots 35}{26\cdots 81}a^{3}+\frac{31\cdots 31}{10\cdots 24}a^{2}+\frac{59\cdots 41}{10\cdots 24}a+\frac{30\cdots 73}{10\cdots 24}$, $\frac{26\cdots 41}{35\cdots 08}a^{19}-\frac{40\cdots 77}{10\cdots 24}a^{18}+\frac{12\cdots 47}{26\cdots 81}a^{17}-\frac{89\cdots 43}{35\cdots 08}a^{16}+\frac{22\cdots 45}{21\cdots 48}a^{15}-\frac{43\cdots 65}{21\cdots 48}a^{14}+\frac{25\cdots 46}{26\cdots 81}a^{13}-\frac{11\cdots 75}{21\cdots 48}a^{12}+\frac{88\cdots 39}{21\cdots 48}a^{11}+\frac{21\cdots 87}{21\cdots 48}a^{10}-\frac{18\cdots 51}{10\cdots 24}a^{9}+\frac{29\cdots 05}{21\cdots 48}a^{8}+\frac{14\cdots 47}{10\cdots 24}a^{7}+\frac{36\cdots 79}{10\cdots 24}a^{6}+\frac{48\cdots 67}{26\cdots 81}a^{5}-\frac{65\cdots 73}{10\cdots 24}a^{4}-\frac{77\cdots 03}{10\cdots 24}a^{3}+\frac{80\cdots 43}{35\cdots 08}a^{2}+\frac{80\cdots 97}{52\cdots 62}a+\frac{67\cdots 95}{10\cdots 24}$, $\frac{38\cdots 55}{21\cdots 48}a^{19}+\frac{27\cdots 53}{14\cdots 32}a^{18}+\frac{12\cdots 35}{42\cdots 96}a^{17}-\frac{12\cdots 07}{42\cdots 96}a^{16}+\frac{24\cdots 47}{17\cdots 54}a^{15}-\frac{10\cdots 31}{42\cdots 96}a^{14}+\frac{40\cdots 85}{42\cdots 96}a^{13}+\frac{24\cdots 13}{42\cdots 96}a^{12}-\frac{14\cdots 59}{26\cdots 81}a^{11}+\frac{33\cdots 27}{21\cdots 48}a^{10}+\frac{60\cdots 49}{10\cdots 24}a^{9}+\frac{67\cdots 39}{10\cdots 24}a^{8}-\frac{48\cdots 01}{26\cdots 81}a^{7}-\frac{28\cdots 97}{21\cdots 48}a^{6}-\frac{77\cdots 49}{21\cdots 48}a^{5}+\frac{96\cdots 97}{70\cdots 16}a^{4}-\frac{33\cdots 88}{26\cdots 81}a^{3}-\frac{59\cdots 13}{26\cdots 81}a^{2}-\frac{67\cdots 55}{35\cdots 08}a-\frac{14\cdots 63}{10\cdots 24}$
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| Regulator: | \( 3152563834.73 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 3152563834.73 \cdot 2}{2\cdot\sqrt{38975682627981862469140631179493376}}\cr\approx \mathstrut & 1.53131912870 \end{aligned}\] (assuming GRH)
Galois group
$D_5^2.D_4$ (as 20T157):
| A solvable group of order 800 |
| The 26 conjugacy class representatives for $D_5^2.D_4$ |
| Character table for $D_5^2.D_4$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 4.0.13824.1, 10.6.279852217270272.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Arithmetically equivalent siblings: | data not computed |
| Minimal sibling: | 20.0.38975682627981862469140631179493376.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $20$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.9a1.5 | $x^{4} + 2 x^{2} + 6$ | $4$ | $1$ | $9$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ |
| 2.1.16.54o1.296 | $x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$ | $16$ | $1$ | $54$ | 16T54 | $$[2, 3, \frac{7}{2}, 4]^{2}$$ | |
|
\(3\)
| 3.1.4.3a1.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.4.4.12a1.3 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(131\)
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 131.2.1.0a1.1 | $x^{2} + 127 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 131.4.1.0a1.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 131.2.2.2a1.1 | $x^{4} + 254 x^{3} + 16133 x^{2} + 639 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
| 131.2.2.2a1.1 | $x^{4} + 254 x^{3} + 16133 x^{2} + 639 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
| 131.4.1.0a1.1 | $x^{4} + 9 x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |