Normalized defining polynomial
\( x^{20} - 4 x^{19} + 177 x^{18} - 4 x^{17} + 8449 x^{16} + 23822 x^{15} + 168540 x^{14} + \cdots + 15554125329 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(4141864704128763454804479176602577540516903124992\)
\(\medspace = 2^{20}\cdot 17^{15}\cdot 53^{14}\)
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| Root discriminant: | \(269.69\) |
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| Galois root discriminant: | $2\cdot 17^{3/4}53^{3/4}\approx 328.90735776316194$ | ||
| Ramified primes: |
\(2\), \(17\), \(53\)
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| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{15}-\frac{4}{9}a^{14}-\frac{2}{9}a^{13}-\frac{4}{9}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{4}{9}a^{6}+\frac{1}{9}a^{5}+\frac{4}{9}a^{4}-\frac{4}{9}a^{3}+\frac{1}{9}a^{2}-\frac{2}{9}a+\frac{1}{3}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{15}-\frac{4}{9}a^{14}+\frac{4}{9}a^{13}+\frac{4}{9}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{4}{9}a^{7}+\frac{2}{9}a^{6}+\frac{1}{3}a^{5}-\frac{2}{9}a^{4}-\frac{1}{9}a^{3}+\frac{1}{3}a^{2}-\frac{1}{9}a-\frac{1}{3}$, $\frac{1}{37449}a^{18}-\frac{1754}{37449}a^{17}-\frac{227}{37449}a^{16}+\frac{183}{1387}a^{15}-\frac{2605}{12483}a^{14}-\frac{9034}{37449}a^{13}-\frac{15317}{37449}a^{12}-\frac{5735}{12483}a^{11}+\frac{3104}{12483}a^{10}-\frac{4366}{12483}a^{9}-\frac{16831}{37449}a^{8}+\frac{8527}{37449}a^{7}-\frac{17149}{37449}a^{6}-\frac{18146}{37449}a^{5}+\frac{436}{4161}a^{4}+\frac{3329}{37449}a^{3}-\frac{850}{37449}a^{2}+\frac{3335}{37449}a+\frac{3926}{12483}$, $\frac{1}{36\cdots 33}a^{19}-\frac{11\cdots 45}{12\cdots 11}a^{18}+\frac{55\cdots 30}{12\cdots 11}a^{17}+\frac{74\cdots 45}{36\cdots 33}a^{16}+\frac{36\cdots 36}{12\cdots 11}a^{15}-\frac{68\cdots 90}{36\cdots 33}a^{14}-\frac{65\cdots 62}{36\cdots 33}a^{13}+\frac{13\cdots 99}{36\cdots 33}a^{12}-\frac{13\cdots 17}{12\cdots 11}a^{11}-\frac{11\cdots 40}{40\cdots 37}a^{10}-\frac{14\cdots 73}{36\cdots 33}a^{9}-\frac{14\cdots 31}{36\cdots 33}a^{8}-\frac{49\cdots 81}{36\cdots 33}a^{7}-\frac{17\cdots 39}{36\cdots 33}a^{6}+\frac{81\cdots 10}{36\cdots 33}a^{5}-\frac{12\cdots 36}{36\cdots 33}a^{4}+\frac{13\cdots 14}{40\cdots 37}a^{3}-\frac{29\cdots 85}{12\cdots 11}a^{2}-\frac{13\cdots 96}{36\cdots 33}a+\frac{49\cdots 25}{12\cdots 11}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{6}\times C_{12}\times C_{22248}$, which has order $6407424$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{6}\times C_{12}\times C_{22248}$, which has order $6407424$ (assuming GRH) |
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| Relative class number: | $1601856$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{21\cdots 09}{11\cdots 59}a^{19}-\frac{52\cdots 50}{50\cdots 61}a^{18}+\frac{43\cdots 15}{12\cdots 51}a^{17}-\frac{46\cdots 57}{11\cdots 59}a^{16}+\frac{58\cdots 14}{36\cdots 53}a^{15}+\frac{32\cdots 05}{11\cdots 59}a^{14}+\frac{25\cdots 11}{11\cdots 59}a^{13}+\frac{88\cdots 89}{11\cdots 59}a^{12}+\frac{11\cdots 26}{36\cdots 53}a^{11}+\frac{85\cdots 15}{12\cdots 51}a^{10}+\frac{57\cdots 13}{11\cdots 59}a^{9}+\frac{55\cdots 27}{11\cdots 59}a^{8}+\frac{37\cdots 50}{11\cdots 59}a^{7}+\frac{11\cdots 03}{11\cdots 59}a^{6}+\frac{55\cdots 00}{11\cdots 59}a^{5}-\frac{36\cdots 98}{11\cdots 59}a^{4}+\frac{57\cdots 90}{36\cdots 53}a^{3}-\frac{76\cdots 28}{36\cdots 53}a^{2}+\frac{12\cdots 95}{11\cdots 59}a-\frac{70\cdots 13}{36\cdots 53}$, $\frac{13\cdots 98}{11\cdots 59}a^{19}-\frac{30\cdots 82}{36\cdots 53}a^{18}+\frac{86\cdots 45}{36\cdots 53}a^{17}-\frac{65\cdots 54}{11\cdots 59}a^{16}+\frac{40\cdots 14}{36\cdots 53}a^{15}+\frac{12\cdots 59}{11\cdots 59}a^{14}+\frac{17\cdots 50}{11\cdots 59}a^{13}+\frac{22\cdots 89}{11\cdots 59}a^{12}+\frac{74\cdots 49}{36\cdots 53}a^{11}+\frac{24\cdots 87}{12\cdots 51}a^{10}+\frac{37\cdots 45}{11\cdots 59}a^{9}-\frac{71\cdots 13}{11\cdots 59}a^{8}+\frac{30\cdots 24}{11\cdots 59}a^{7}+\frac{18\cdots 22}{11\cdots 59}a^{6}+\frac{11\cdots 97}{11\cdots 59}a^{5}-\frac{43\cdots 84}{11\cdots 59}a^{4}+\frac{14\cdots 98}{12\cdots 51}a^{3}-\frac{62\cdots 69}{36\cdots 53}a^{2}+\frac{12\cdots 43}{11\cdots 59}a-\frac{33\cdots 61}{36\cdots 53}$, $\frac{18\cdots 41}{36\cdots 53}a^{19}-\frac{75\cdots 56}{36\cdots 53}a^{18}+\frac{34\cdots 95}{36\cdots 53}a^{17}-\frac{34\cdots 49}{12\cdots 51}a^{16}+\frac{62\cdots 10}{12\cdots 51}a^{15}+\frac{58\cdots 53}{50\cdots 61}a^{14}+\frac{42\cdots 53}{36\cdots 53}a^{13}+\frac{47\cdots 59}{12\cdots 51}a^{12}+\frac{20\cdots 30}{12\cdots 51}a^{11}+\frac{56\cdots 45}{12\cdots 51}a^{10}+\frac{93\cdots 81}{36\cdots 53}a^{9}+\frac{20\cdots 48}{36\cdots 53}a^{8}+\frac{10\cdots 44}{36\cdots 53}a^{7}+\frac{18\cdots 19}{36\cdots 53}a^{6}+\frac{60\cdots 46}{12\cdots 51}a^{5}-\frac{27\cdots 48}{36\cdots 53}a^{4}+\frac{21\cdots 36}{36\cdots 53}a^{3}-\frac{24\cdots 09}{36\cdots 53}a^{2}+\frac{20\cdots 29}{12\cdots 51}a-\frac{26\cdots 51}{40\cdots 17}$, $\frac{49\cdots 05}{11\cdots 59}a^{19}-\frac{32\cdots 38}{12\cdots 51}a^{18}+\frac{30\cdots 35}{36\cdots 53}a^{17}-\frac{17\cdots 65}{11\cdots 59}a^{16}+\frac{13\cdots 42}{36\cdots 53}a^{15}+\frac{34\cdots 23}{11\cdots 59}a^{14}+\frac{61\cdots 11}{11\cdots 59}a^{13}+\frac{13\cdots 67}{11\cdots 59}a^{12}+\frac{26\cdots 24}{36\cdots 53}a^{11}+\frac{44\cdots 63}{40\cdots 17}a^{10}+\frac{13\cdots 03}{11\cdots 59}a^{9}+\frac{41\cdots 01}{11\cdots 59}a^{8}+\frac{98\cdots 98}{11\cdots 59}a^{7}+\frac{15\cdots 47}{11\cdots 59}a^{6}+\frac{29\cdots 94}{11\cdots 59}a^{5}-\frac{12\cdots 66}{11\cdots 59}a^{4}+\frac{14\cdots 78}{36\cdots 53}a^{3}-\frac{22\cdots 74}{40\cdots 17}a^{2}+\frac{37\cdots 81}{11\cdots 59}a-\frac{80\cdots 73}{50\cdots 61}$, $\frac{12\cdots 63}{70\cdots 07}a^{19}-\frac{31\cdots 28}{70\cdots 07}a^{18}+\frac{21\cdots 50}{70\cdots 07}a^{17}+\frac{32\cdots 01}{70\cdots 07}a^{16}+\frac{34\cdots 66}{23\cdots 69}a^{15}+\frac{44\cdots 21}{70\cdots 07}a^{14}+\frac{81\cdots 63}{23\cdots 69}a^{13}+\frac{97\cdots 32}{70\cdots 07}a^{12}+\frac{14\cdots 01}{25\cdots 41}a^{11}+\frac{38\cdots 74}{23\cdots 69}a^{10}+\frac{72\cdots 37}{95\cdots 59}a^{9}+\frac{45\cdots 89}{23\cdots 69}a^{8}+\frac{14\cdots 01}{23\cdots 69}a^{7}+\frac{43\cdots 23}{23\cdots 69}a^{6}+\frac{27\cdots 98}{70\cdots 07}a^{5}+\frac{22\cdots 22}{70\cdots 07}a^{4}+\frac{32\cdots 86}{70\cdots 07}a^{3}-\frac{65\cdots 45}{70\cdots 07}a^{2}+\frac{77\cdots 02}{70\cdots 07}a-\frac{41\cdots 01}{23\cdots 69}$, $\frac{67\cdots 57}{11\cdots 59}a^{19}-\frac{15\cdots 28}{12\cdots 51}a^{18}+\frac{36\cdots 71}{36\cdots 53}a^{17}+\frac{27\cdots 37}{11\cdots 59}a^{16}+\frac{15\cdots 00}{36\cdots 53}a^{15}+\frac{32\cdots 89}{11\cdots 59}a^{14}+\frac{82\cdots 41}{11\cdots 59}a^{13}+\frac{81\cdots 98}{11\cdots 59}a^{12}+\frac{34\cdots 44}{36\cdots 53}a^{11}+\frac{39\cdots 90}{40\cdots 17}a^{10}+\frac{16\cdots 63}{11\cdots 59}a^{9}+\frac{11\cdots 98}{11\cdots 59}a^{8}+\frac{12\cdots 54}{11\cdots 59}a^{7}+\frac{16\cdots 36}{15\cdots 83}a^{6}-\frac{13\cdots 64}{11\cdots 59}a^{5}+\frac{57\cdots 67}{11\cdots 59}a^{4}-\frac{38\cdots 02}{36\cdots 53}a^{3}+\frac{19\cdots 98}{13\cdots 39}a^{2}-\frac{12\cdots 97}{11\cdots 59}a+\frac{11\cdots 16}{36\cdots 53}$, $\frac{82\cdots 55}{21\cdots 21}a^{19}+\frac{11\cdots 23}{23\cdots 69}a^{18}-\frac{69\cdots 42}{70\cdots 07}a^{17}+\frac{17\cdots 83}{21\cdots 21}a^{16}+\frac{75\cdots 62}{70\cdots 07}a^{15}+\frac{72\cdots 27}{21\cdots 21}a^{14}+\frac{35\cdots 51}{21\cdots 21}a^{13}+\frac{10\cdots 85}{21\cdots 21}a^{12}+\frac{24\cdots 91}{70\cdots 07}a^{11}+\frac{56\cdots 55}{77\cdots 23}a^{10}+\frac{73\cdots 41}{21\cdots 21}a^{9}+\frac{25\cdots 76}{21\cdots 21}a^{8}+\frac{75\cdots 27}{21\cdots 21}a^{7}+\frac{12\cdots 91}{21\cdots 21}a^{6}+\frac{89\cdots 26}{21\cdots 21}a^{5}-\frac{96\cdots 82}{21\cdots 21}a^{4}-\frac{37\cdots 92}{70\cdots 07}a^{3}-\frac{93\cdots 44}{23\cdots 69}a^{2}+\frac{25\cdots 94}{21\cdots 21}a+\frac{37\cdots 07}{70\cdots 07}$, $\frac{69\cdots 85}{23\cdots 69}a^{19}-\frac{95\cdots 71}{70\cdots 07}a^{18}+\frac{37\cdots 85}{70\cdots 07}a^{17}-\frac{23\cdots 81}{70\cdots 07}a^{16}+\frac{20\cdots 26}{77\cdots 23}a^{15}+\frac{13\cdots 68}{23\cdots 69}a^{14}+\frac{33\cdots 47}{70\cdots 07}a^{13}+\frac{11\cdots 92}{70\cdots 07}a^{12}+\frac{14\cdots 18}{23\cdots 69}a^{11}+\frac{52\cdots 54}{23\cdots 69}a^{10}+\frac{28\cdots 08}{31\cdots 53}a^{9}+\frac{14\cdots 80}{70\cdots 07}a^{8}+\frac{59\cdots 73}{70\cdots 07}a^{7}+\frac{14\cdots 67}{70\cdots 07}a^{6}+\frac{24\cdots 31}{70\cdots 07}a^{5}+\frac{62\cdots 16}{23\cdots 69}a^{4}-\frac{17\cdots 00}{70\cdots 07}a^{3}-\frac{10\cdots 56}{70\cdots 07}a^{2}+\frac{17\cdots 11}{70\cdots 07}a+\frac{18\cdots 51}{23\cdots 69}$, $\frac{10\cdots 78}{21\cdots 21}a^{19}-\frac{92\cdots 04}{70\cdots 07}a^{18}+\frac{19\cdots 22}{23\cdots 69}a^{17}+\frac{24\cdots 71}{21\cdots 21}a^{16}+\frac{28\cdots 90}{70\cdots 07}a^{15}+\frac{35\cdots 77}{21\cdots 21}a^{14}+\frac{20\cdots 20}{21\cdots 21}a^{13}+\frac{78\cdots 79}{21\cdots 21}a^{12}+\frac{10\cdots 62}{70\cdots 07}a^{11}+\frac{10\cdots 95}{23\cdots 69}a^{10}+\frac{43\cdots 21}{21\cdots 21}a^{9}+\frac{10\cdots 87}{21\cdots 21}a^{8}+\frac{35\cdots 69}{21\cdots 21}a^{7}+\frac{10\cdots 95}{21\cdots 21}a^{6}+\frac{22\cdots 11}{21\cdots 21}a^{5}+\frac{64\cdots 86}{21\cdots 21}a^{4}+\frac{15\cdots 40}{70\cdots 07}a^{3}-\frac{19\cdots 15}{70\cdots 07}a^{2}+\frac{64\cdots 46}{21\cdots 21}a-\frac{34\cdots 46}{70\cdots 07}$
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| Regulator: | \( 8009866748.83 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 8009866748.83 \cdot 6407424}{2\cdot\sqrt{4141864704128763454804479176602577540516903124992}}\cr\approx \mathstrut & 1.20914807451 \end{aligned}\] (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
| \(\Q(\sqrt{901}) \), \(\Q(\sqrt{-102 +2 \sqrt{901}})\), 5.5.2382032.1, 10.10.426987989728139087104.1 |
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: | 20.0.78148390643938933109518475030237312085224587264.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $20$ | $20$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.4a1.1 | $x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$ | $2$ | $2$ | $4$ | $C_2^2$ | $$[2]^{2}$$ |
| 2.4.2.8a1.1 | $x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $$[2]^{4}$$ | |
| 2.4.2.8a1.1 | $x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $$[2]^{4}$$ | |
|
\(17\)
| 17.1.4.3a1.3 | $x^{4} + 153$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 17.2.4.6a1.2 | $x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
| 17.2.4.6a1.2 | $x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
|
\(53\)
| 53.1.2.1a1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 53.1.2.1a1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |