Normalized defining polynomial
\( x^{20} - x^{19} + 61 x^{18} - 97 x^{17} + 1959 x^{16} - 3824 x^{15} + 42001 x^{14} - 86676 x^{13} + \cdots + 4751163121 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(193315443721344440894830801055908203125\)
\(\medspace = 5^{15}\cdot 11^{16}\cdot 13^{10}\)
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| Root discriminant: | \(82.09\) |
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| Galois root discriminant: | $5^{3/4}11^{4/5}13^{1/2}\approx 82.09436114174476$ | ||
| Ramified primes: |
\(5\), \(11\), \(13\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(715=5\cdot 11\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{715}(1,·)$, $\chi_{715}(196,·)$, $\chi_{715}(456,·)$, $\chi_{715}(521,·)$, $\chi_{715}(586,·)$, $\chi_{715}(12,·)$, $\chi_{715}(14,·)$, $\chi_{715}(207,·)$, $\chi_{715}(144,·)$, $\chi_{715}(339,·)$, $\chi_{715}(532,·)$, $\chi_{715}(597,·)$, $\chi_{715}(599,·)$, $\chi_{715}(664,·)$, $\chi_{715}(38,·)$, $\chi_{715}(103,·)$, $\chi_{715}(168,·)$, $\chi_{715}(298,·)$, $\chi_{715}(493,·)$, $\chi_{715}(467,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19}a^{15}+\frac{2}{19}a^{14}+\frac{7}{19}a^{13}-\frac{6}{19}a^{12}-\frac{9}{19}a^{11}-\frac{5}{19}a^{10}-\frac{8}{19}a^{9}+\frac{9}{19}a^{8}-\frac{1}{19}a^{7}-\frac{9}{19}a^{6}-\frac{8}{19}a^{5}-\frac{9}{19}a^{3}+\frac{5}{19}a^{2}-\frac{5}{19}a+\frac{4}{19}$, $\frac{1}{19}a^{16}+\frac{3}{19}a^{14}-\frac{1}{19}a^{13}+\frac{3}{19}a^{12}-\frac{6}{19}a^{11}+\frac{2}{19}a^{10}+\frac{6}{19}a^{9}-\frac{7}{19}a^{7}-\frac{9}{19}a^{6}-\frac{3}{19}a^{5}-\frac{9}{19}a^{4}+\frac{4}{19}a^{3}+\frac{4}{19}a^{2}-\frac{5}{19}a-\frac{8}{19}$, $\frac{1}{1691}a^{17}-\frac{42}{1691}a^{16}-\frac{43}{1691}a^{15}+\frac{370}{1691}a^{14}-\frac{524}{1691}a^{13}-\frac{464}{1691}a^{12}+\frac{421}{1691}a^{11}-\frac{8}{89}a^{10}+\frac{135}{1691}a^{9}-\frac{592}{1691}a^{8}+\frac{293}{1691}a^{7}-\frac{503}{1691}a^{6}-\frac{674}{1691}a^{5}+\frac{6}{19}a^{4}-\frac{320}{1691}a^{3}+\frac{34}{1691}a^{2}-\frac{841}{1691}a-\frac{30}{89}$, $\frac{1}{221521}a^{18}-\frac{65}{221521}a^{17}+\frac{2169}{221521}a^{16}-\frac{2913}{221521}a^{15}-\frac{312}{221521}a^{14}+\frac{51460}{221521}a^{13}+\frac{37081}{221521}a^{12}+\frac{14373}{221521}a^{11}-\frac{18174}{221521}a^{10}-\frac{51668}{221521}a^{9}+\frac{41410}{221521}a^{8}+\frac{22128}{221521}a^{7}-\frac{29511}{221521}a^{6}+\frac{4226}{11659}a^{5}+\frac{55661}{221521}a^{4}+\frac{81264}{221521}a^{3}-\frac{61965}{221521}a^{2}-\frac{91231}{221521}a+\frac{3983}{11659}$, $\frac{1}{21\cdots 19}a^{19}+\frac{15\cdots 26}{11\cdots 01}a^{18}-\frac{30\cdots 74}{21\cdots 19}a^{17}-\frac{37\cdots 37}{21\cdots 19}a^{16}+\frac{15\cdots 87}{21\cdots 19}a^{15}+\frac{90\cdots 05}{21\cdots 19}a^{14}+\frac{92\cdots 48}{21\cdots 19}a^{13}-\frac{80\cdots 79}{21\cdots 19}a^{12}+\frac{85\cdots 74}{21\cdots 19}a^{11}-\frac{34\cdots 67}{21\cdots 19}a^{10}-\frac{83\cdots 41}{21\cdots 19}a^{9}+\frac{45\cdots 88}{21\cdots 19}a^{8}+\frac{83\cdots 06}{21\cdots 19}a^{7}-\frac{74\cdots 26}{21\cdots 19}a^{6}+\frac{49\cdots 67}{21\cdots 19}a^{5}+\frac{64\cdots 95}{21\cdots 19}a^{4}-\frac{32\cdots 06}{11\cdots 01}a^{3}-\frac{67\cdots 75}{21\cdots 19}a^{2}+\frac{60\cdots 92}{21\cdots 19}a-\frac{84\cdots 34}{21\cdots 19}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{331922}$, which has order $331922$ (assuming GRH) |
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| Narrow class group: | $C_{331922}$, which has order $331922$ (assuming GRH) |
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| Relative class number: | $331922$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{13\cdots 22}{75\cdots 91}a^{19}-\frac{24\cdots 29}{75\cdots 91}a^{18}+\frac{32\cdots 53}{75\cdots 91}a^{17}-\frac{11\cdots 31}{75\cdots 91}a^{16}+\frac{22\cdots 16}{75\cdots 91}a^{15}-\frac{32\cdots 37}{75\cdots 91}a^{14}+\frac{70\cdots 04}{75\cdots 91}a^{13}-\frac{31\cdots 11}{39\cdots 89}a^{12}+\frac{12\cdots 96}{75\cdots 91}a^{11}-\frac{77\cdots 05}{75\cdots 91}a^{10}+\frac{14\cdots 49}{75\cdots 91}a^{9}-\frac{71\cdots 83}{75\cdots 91}a^{8}+\frac{10\cdots 84}{75\cdots 91}a^{7}-\frac{45\cdots 52}{75\cdots 91}a^{6}+\frac{51\cdots 88}{75\cdots 91}a^{5}-\frac{15\cdots 51}{57\cdots 61}a^{4}+\frac{14\cdots 00}{75\cdots 91}a^{3}-\frac{54\cdots 48}{75\cdots 91}a^{2}+\frac{17\cdots 41}{75\cdots 91}a-\frac{25\cdots 38}{75\cdots 91}$, $\frac{48\cdots 30}{11\cdots 01}a^{19}-\frac{42\cdots 00}{11\cdots 01}a^{18}+\frac{28\cdots 70}{11\cdots 01}a^{17}-\frac{44\cdots 25}{11\cdots 01}a^{16}+\frac{85\cdots 50}{11\cdots 01}a^{15}-\frac{17\cdots 90}{11\cdots 01}a^{14}+\frac{17\cdots 40}{11\cdots 01}a^{13}-\frac{37\cdots 60}{11\cdots 01}a^{12}+\frac{25\cdots 40}{11\cdots 01}a^{11}-\frac{52\cdots 24}{11\cdots 01}a^{10}+\frac{26\cdots 60}{11\cdots 01}a^{9}-\frac{47\cdots 70}{11\cdots 01}a^{8}+\frac{22\cdots 70}{12\cdots 09}a^{7}-\frac{28\cdots 10}{11\cdots 01}a^{6}+\frac{95\cdots 78}{11\cdots 01}a^{5}-\frac{10\cdots 60}{11\cdots 01}a^{4}+\frac{28\cdots 60}{11\cdots 01}a^{3}-\frac{21\cdots 70}{11\cdots 01}a^{2}+\frac{39\cdots 75}{11\cdots 01}a-\frac{18\cdots 29}{11\cdots 01}$, $\frac{17\cdots 66}{11\cdots 01}a^{19}+\frac{62\cdots 24}{11\cdots 01}a^{18}+\frac{93\cdots 77}{11\cdots 01}a^{17}-\frac{26\cdots 29}{11\cdots 01}a^{16}+\frac{19\cdots 73}{88\cdots 71}a^{15}-\frac{17\cdots 61}{11\cdots 01}a^{14}+\frac{45\cdots 11}{11\cdots 01}a^{13}-\frac{39\cdots 27}{11\cdots 01}a^{12}+\frac{60\cdots 83}{11\cdots 01}a^{11}-\frac{47\cdots 99}{11\cdots 01}a^{10}+\frac{59\cdots 28}{11\cdots 01}a^{9}-\frac{36\cdots 12}{11\cdots 01}a^{8}+\frac{44\cdots 29}{11\cdots 01}a^{7}-\frac{23\cdots 65}{11\cdots 01}a^{6}+\frac{25\cdots 44}{11\cdots 01}a^{5}-\frac{15\cdots 96}{11\cdots 01}a^{4}+\frac{10\cdots 43}{11\cdots 01}a^{3}-\frac{69\cdots 12}{11\cdots 01}a^{2}+\frac{22\cdots 61}{11\cdots 01}a-\frac{25\cdots 32}{11\cdots 01}$, $\frac{17\cdots 66}{11\cdots 01}a^{19}+\frac{62\cdots 24}{11\cdots 01}a^{18}+\frac{93\cdots 77}{11\cdots 01}a^{17}-\frac{26\cdots 29}{11\cdots 01}a^{16}+\frac{19\cdots 73}{88\cdots 71}a^{15}-\frac{17\cdots 61}{11\cdots 01}a^{14}+\frac{45\cdots 11}{11\cdots 01}a^{13}-\frac{39\cdots 27}{11\cdots 01}a^{12}+\frac{60\cdots 83}{11\cdots 01}a^{11}-\frac{47\cdots 99}{11\cdots 01}a^{10}+\frac{59\cdots 28}{11\cdots 01}a^{9}-\frac{36\cdots 12}{11\cdots 01}a^{8}+\frac{44\cdots 29}{11\cdots 01}a^{7}-\frac{23\cdots 65}{11\cdots 01}a^{6}+\frac{25\cdots 44}{11\cdots 01}a^{5}-\frac{15\cdots 96}{11\cdots 01}a^{4}+\frac{10\cdots 43}{11\cdots 01}a^{3}-\frac{69\cdots 12}{11\cdots 01}a^{2}+\frac{22\cdots 61}{11\cdots 01}a-\frac{14\cdots 31}{11\cdots 01}$, $\frac{87\cdots 26}{11\cdots 01}a^{19}+\frac{33\cdots 44}{11\cdots 01}a^{18}+\frac{43\cdots 37}{11\cdots 01}a^{17}+\frac{13\cdots 96}{11\cdots 01}a^{16}+\frac{98\cdots 83}{11\cdots 01}a^{15}+\frac{33\cdots 59}{11\cdots 01}a^{14}+\frac{13\cdots 11}{11\cdots 01}a^{13}+\frac{59\cdots 83}{11\cdots 01}a^{12}+\frac{11\cdots 03}{11\cdots 01}a^{11}+\frac{82\cdots 27}{11\cdots 01}a^{10}+\frac{74\cdots 58}{11\cdots 01}a^{9}+\frac{81\cdots 93}{11\cdots 01}a^{8}+\frac{40\cdots 89}{11\cdots 01}a^{7}+\frac{49\cdots 25}{11\cdots 01}a^{6}+\frac{31\cdots 66}{11\cdots 01}a^{5}+\frac{14\cdots 89}{11\cdots 01}a^{4}+\frac{23\cdots 33}{11\cdots 01}a^{3}+\frac{78\cdots 73}{11\cdots 01}a^{2}+\frac{63\cdots 01}{11\cdots 01}a-\frac{14\cdots 37}{11\cdots 01}$, $\frac{12\cdots 43}{21\cdots 19}a^{19}-\frac{28\cdots 61}{21\cdots 19}a^{18}+\frac{72\cdots 47}{21\cdots 19}a^{17}-\frac{20\cdots 18}{21\cdots 19}a^{16}+\frac{23\cdots 03}{21\cdots 19}a^{15}-\frac{70\cdots 08}{21\cdots 19}a^{14}+\frac{50\cdots 87}{21\cdots 19}a^{13}-\frac{14\cdots 74}{21\cdots 19}a^{12}+\frac{76\cdots 05}{21\cdots 19}a^{11}-\frac{20\cdots 71}{21\cdots 19}a^{10}+\frac{83\cdots 11}{21\cdots 19}a^{9}-\frac{18\cdots 60}{21\cdots 19}a^{8}+\frac{62\cdots 00}{21\cdots 19}a^{7}-\frac{11\cdots 66}{21\cdots 19}a^{6}+\frac{16\cdots 49}{11\cdots 01}a^{5}-\frac{40\cdots 07}{21\cdots 19}a^{4}+\frac{88\cdots 38}{21\cdots 19}a^{3}-\frac{76\cdots 16}{21\cdots 19}a^{2}+\frac{11\cdots 86}{21\cdots 19}a-\frac{52\cdots 51}{21\cdots 19}$, $\frac{46\cdots 98}{21\cdots 19}a^{19}-\frac{15\cdots 80}{21\cdots 19}a^{18}+\frac{28\cdots 75}{21\cdots 19}a^{17}-\frac{10\cdots 05}{21\cdots 19}a^{16}+\frac{95\cdots 55}{21\cdots 19}a^{15}-\frac{25\cdots 13}{16\cdots 49}a^{14}+\frac{21\cdots 65}{21\cdots 19}a^{13}-\frac{36\cdots 39}{11\cdots 01}a^{12}+\frac{33\cdots 39}{21\cdots 19}a^{11}-\frac{94\cdots 35}{21\cdots 19}a^{10}+\frac{37\cdots 82}{21\cdots 19}a^{9}-\frac{87\cdots 65}{21\cdots 19}a^{8}+\frac{28\cdots 83}{21\cdots 19}a^{7}-\frac{52\cdots 37}{21\cdots 19}a^{6}+\frac{14\cdots 50}{21\cdots 19}a^{5}-\frac{17\cdots 22}{21\cdots 19}a^{4}+\frac{42\cdots 43}{21\cdots 19}a^{3}-\frac{24\cdots 85}{21\cdots 19}a^{2}+\frac{58\cdots 17}{21\cdots 19}a+\frac{18\cdots 74}{21\cdots 19}$, $\frac{10\cdots 32}{21\cdots 19}a^{19}-\frac{60\cdots 19}{21\cdots 19}a^{18}+\frac{62\cdots 13}{21\cdots 19}a^{17}-\frac{82\cdots 71}{21\cdots 19}a^{16}+\frac{18\cdots 26}{21\cdots 19}a^{15}-\frac{33\cdots 57}{21\cdots 19}a^{14}+\frac{37\cdots 24}{21\cdots 19}a^{13}-\frac{43\cdots 39}{12\cdots 09}a^{12}+\frac{53\cdots 16}{21\cdots 19}a^{11}-\frac{10\cdots 05}{21\cdots 19}a^{10}+\frac{56\cdots 29}{21\cdots 19}a^{9}-\frac{92\cdots 78}{21\cdots 19}a^{8}+\frac{42\cdots 74}{21\cdots 19}a^{7}-\frac{55\cdots 32}{21\cdots 19}a^{6}+\frac{20\cdots 72}{21\cdots 19}a^{5}-\frac{20\cdots 26}{21\cdots 19}a^{4}+\frac{66\cdots 30}{21\cdots 19}a^{3}-\frac{40\cdots 13}{21\cdots 19}a^{2}+\frac{10\cdots 96}{21\cdots 19}a-\frac{48\cdots 14}{21\cdots 19}$, $\frac{25\cdots 76}{16\cdots 49}a^{19}+\frac{83\cdots 17}{21\cdots 19}a^{18}+\frac{16\cdots 86}{21\cdots 19}a^{17}+\frac{29\cdots 28}{21\cdots 19}a^{16}+\frac{41\cdots 53}{21\cdots 19}a^{15}+\frac{60\cdots 74}{21\cdots 19}a^{14}+\frac{66\cdots 73}{21\cdots 19}a^{13}+\frac{51\cdots 72}{11\cdots 01}a^{12}+\frac{78\cdots 13}{21\cdots 19}a^{11}+\frac{13\cdots 64}{21\cdots 19}a^{10}+\frac{72\cdots 91}{21\cdots 19}a^{9}+\frac{14\cdots 19}{21\cdots 19}a^{8}+\frac{52\cdots 95}{21\cdots 19}a^{7}+\frac{89\cdots 33}{21\cdots 19}a^{6}+\frac{32\cdots 44}{21\cdots 19}a^{5}+\frac{28\cdots 05}{21\cdots 19}a^{4}+\frac{16\cdots 17}{21\cdots 19}a^{3}+\frac{27\cdots 04}{21\cdots 19}a^{2}+\frac{37\cdots 90}{21\cdots 19}a-\frac{41\cdots 66}{21\cdots 19}$
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| Regulator: | \( 140644.599182 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 140644.599182 \cdot 331922}{2\cdot\sqrt{193315443721344440894830801055908203125}}\cr\approx \mathstrut & 0.160988372888 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-130 +26 \sqrt{5}})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | $20$ | R | $20$ | R | R | $20$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | $20$ | $20$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.5.4.15a1.2 | $x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 869 x^{2} + 432 x + 81$ | $4$ | $5$ | $15$ | 20T1 | not computed |
|
\(11\)
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |
| 11.2.5.8a1.2 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31367 x^{5} + 29970 x^{4} + 14840 x^{3} + 4000 x^{2} + 560 x + 43$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ | |
|
\(13\)
| 13.10.2.10a1.1 | $x^{20} + 14 x^{15} + 10 x^{14} + 16 x^{13} + 2 x^{12} + 2 x^{11} + 53 x^{10} + 70 x^{9} + 137 x^{8} + 94 x^{7} + 88 x^{6} + 54 x^{5} + 37 x^{4} + 34 x^{3} + 5 x^{2} + 17 x + 4$ | $2$ | $10$ | $10$ | 20T1 | $$[\ ]_{2}^{10}$$ |