Normalized defining polynomial
\( x^{40} - 220 x^{38} + 21560 x^{36} - 1246300 x^{34} + 47425950 x^{32} - 1255559844 x^{30} + \cdots + 313993243201 \)
Invariants
| Degree: | $40$ |
| |
| Signature: | $(40, 0)$ |
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| Discriminant: |
\(287\!\cdots\!000\)
\(\medspace = 2^{40}\cdot 5^{70}\cdot 11^{36}\)
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| |
| Root discriminant: | \(289.39\) |
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| Galois root discriminant: | $2\cdot 5^{7/4}11^{9/10}\approx 289.3882675684651$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_2\times C_{20}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1100=2^{2}\cdot 5^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1100}(1,·)$, $\chi_{1100}(513,·)$, $\chi_{1100}(269,·)$, $\chi_{1100}(877,·)$, $\chi_{1100}(17,·)$, $\chi_{1100}(919,·)$, $\chi_{1100}(927,·)$, $\chi_{1100}(289,·)$, $\chi_{1100}(359,·)$, $\chi_{1100}(811,·)$, $\chi_{1100}(173,·)$, $\chi_{1100}(351,·)$, $\chi_{1100}(871,·)$, $\chi_{1100}(181,·)$, $\chi_{1100}(1083,·)$, $\chi_{1100}(831,·)$, $\chi_{1100}(1099,·)$, $\chi_{1100}(453,·)$, $\chi_{1100}(587,·)$, $\chi_{1100}(79,·)$, $\chi_{1100}(337,·)$, $\chi_{1100}(467,·)$, $\chi_{1100}(857,·)$, $\chi_{1100}(603,·)$, $\chi_{1100}(861,·)$, $\chi_{1100}(223,·)$, $\chi_{1100}(609,·)$, $\chi_{1100}(229,·)$, $\chi_{1100}(741,·)$, $\chi_{1100}(593,·)$, $\chi_{1100}(491,·)$, $\chi_{1100}(647,·)$, $\chi_{1100}(749,·)$, $\chi_{1100}(239,·)$, $\chi_{1100}(497,·)$, $\chi_{1100}(243,·)$, $\chi_{1100}(633,·)$, $\chi_{1100}(763,·)$, $\chi_{1100}(1021,·)$, $\chi_{1100}(507,·)$$\rbrace$ | ||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}$, $\frac{1}{11}a^{11}$, $\frac{1}{11}a^{12}$, $\frac{1}{11}a^{13}$, $\frac{1}{11}a^{14}$, $\frac{1}{11}a^{15}$, $\frac{1}{11}a^{16}$, $\frac{1}{121}a^{17}+\frac{2}{11}a^{7}$, $\frac{1}{121}a^{18}+\frac{2}{11}a^{8}$, $\frac{1}{121}a^{19}+\frac{2}{11}a^{9}$, $\frac{1}{121}a^{20}$, $\frac{1}{121}a^{21}$, $\frac{1}{121}a^{22}$, $\frac{1}{121}a^{23}$, $\frac{1}{121}a^{24}$, $\frac{1}{1331}a^{25}+\frac{4}{121}a^{15}+\frac{4}{11}a^{5}$, $\frac{1}{1331}a^{26}+\frac{4}{121}a^{16}+\frac{4}{11}a^{6}$, $\frac{1}{1331}a^{27}-\frac{4}{11}a^{7}$, $\frac{1}{3993}a^{28}+\frac{1}{3993}a^{26}+\frac{1}{363}a^{24}+\frac{1}{363}a^{22}+\frac{1}{363}a^{20}+\frac{1}{363}a^{18}+\frac{5}{121}a^{16}+\frac{1}{33}a^{14}+\frac{1}{33}a^{10}-\frac{2}{33}a^{8}+\frac{5}{11}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{3993}a^{29}+\frac{1}{3993}a^{27}-\frac{1}{3993}a^{25}+\frac{1}{363}a^{23}+\frac{1}{363}a^{21}+\frac{1}{363}a^{19}-\frac{4}{363}a^{15}+\frac{1}{33}a^{11}-\frac{2}{33}a^{9}-\frac{5}{11}a^{7}+\frac{7}{33}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{43923}a^{30}-\frac{1}{3993}a^{26}+\frac{5}{1331}a^{20}+\frac{1}{363}a^{18}+\frac{16}{363}a^{16}+\frac{1}{33}a^{14}-\frac{1}{33}a^{12}+\frac{5}{121}a^{10}-\frac{1}{3}a^{8}-\frac{10}{33}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}$, $\frac{1}{18491583}a^{31}+\frac{4}{152823}a^{29}+\frac{568}{1681053}a^{27}+\frac{64}{1681053}a^{25}-\frac{307}{152823}a^{23}-\frac{6167}{1681053}a^{21}-\frac{2}{4631}a^{19}-\frac{203}{152823}a^{17}+\frac{2080}{50941}a^{15}+\frac{272}{13893}a^{13}+\frac{2050}{152823}a^{11}-\frac{2208}{4631}a^{9}+\frac{4085}{13893}a^{7}-\frac{5222}{13893}a^{5}-\frac{506}{1263}a^{3}+\frac{163}{421}a$, $\frac{1}{18491583}a^{32}+\frac{21}{6163861}a^{30}+\frac{49}{560351}a^{28}+\frac{64}{1681053}a^{26}+\frac{535}{152823}a^{24}+\frac{3095}{1681053}a^{22}+\frac{2221}{1681053}a^{20}+\frac{218}{152823}a^{18}-\frac{6811}{152823}a^{16}-\frac{190}{4631}a^{14}+\frac{2227}{50941}a^{12}-\frac{452}{152823}a^{10}-\frac{603}{4631}a^{8}+\frac{6566}{13893}a^{6}-\frac{506}{1263}a^{4}+\frac{163}{421}a^{2}-\frac{1}{3}$, $\frac{1}{18491583}a^{33}-\frac{3}{50941}a^{29}-\frac{356}{1681053}a^{27}+\frac{590}{1681053}a^{25}-\frac{6442}{1681053}a^{23}+\frac{158}{152823}a^{21}+\frac{587}{152823}a^{19}-\frac{337}{152823}a^{17}-\frac{515}{50941}a^{15}-\frac{402}{50941}a^{13}-\frac{415}{13893}a^{11}-\frac{1692}{4631}a^{9}-\frac{1978}{13893}a^{7}-\frac{1171}{13893}a^{5}-\frac{157}{421}a^{3}+\frac{347}{1263}a$, $\frac{1}{18491583}a^{34}+\frac{58}{6163861}a^{30}+\frac{65}{1681053}a^{28}-\frac{84}{560351}a^{26}-\frac{1811}{1681053}a^{24}+\frac{193}{50941}a^{22}+\frac{749}{560351}a^{20}+\frac{28}{50941}a^{18}-\frac{936}{50941}a^{16}+\frac{3425}{152823}a^{14}-\frac{415}{13893}a^{12}-\frac{4474}{152823}a^{10}-\frac{162}{421}a^{8}+\frac{6407}{13893}a^{6}+\frac{371}{1263}a^{4}-\frac{165}{421}a^{2}-\frac{1}{3}$, $\frac{1}{203407413}a^{35}-\frac{116}{1681053}a^{29}+\frac{21}{560351}a^{27}-\frac{1580}{18491583}a^{25}+\frac{1580}{560351}a^{23}+\frac{168}{560351}a^{21}-\frac{147}{50941}a^{19}+\frac{105}{50941}a^{17}+\frac{19001}{1681053}a^{15}+\frac{1514}{152823}a^{13}+\frac{6319}{152823}a^{11}-\frac{676}{4631}a^{9}+\frac{5545}{13893}a^{7}+\frac{6275}{13893}a^{5}+\frac{127}{421}a^{3}-\frac{425}{1263}a$, $\frac{1}{140554522383}a^{36}-\frac{334}{12777683853}a^{34}-\frac{233}{12777683853}a^{32}+\frac{138113}{12777683853}a^{30}-\frac{26351}{387202541}a^{28}+\frac{1436362}{12777683853}a^{26}-\frac{1405300}{1161607623}a^{24}-\frac{1464877}{387202541}a^{22}+\frac{779954}{1161607623}a^{20}+\frac{152173}{105600693}a^{18}-\frac{729636}{387202541}a^{16}+\frac{702847}{105600693}a^{14}+\frac{130382}{35200231}a^{12}+\frac{589486}{35200231}a^{10}-\frac{139943}{290911}a^{8}-\frac{4000762}{9600063}a^{6}+\frac{131298}{290911}a^{4}+\frac{18112}{290911}a^{2}+\frac{886}{2073}$, $\frac{1}{140554522383}a^{37}-\frac{73}{46851507461}a^{35}-\frac{233}{12777683853}a^{33}-\frac{29}{4259227951}a^{31}+\frac{43440}{387202541}a^{29}+\frac{1162726}{12777683853}a^{27}+\frac{190777}{12777683853}a^{25}-\frac{410882}{105600693}a^{23}-\frac{2804954}{1161607623}a^{21}-\frac{131930}{35200231}a^{19}+\frac{1581188}{1161607623}a^{17}+\frac{36993785}{1161607623}a^{15}-\frac{4669738}{105600693}a^{13}-\frac{3707717}{105600693}a^{11}-\frac{3248557}{9600063}a^{9}+\frac{1598556}{3200021}a^{7}+\frac{343000}{9600063}a^{5}+\frac{105869}{290911}a^{3}-\frac{311084}{872733}a$, $\frac{1}{22\cdots 77}a^{38}-\frac{38\cdots 86}{20\cdots 07}a^{36}-\frac{50\cdots 67}{20\cdots 07}a^{34}+\frac{33\cdots 08}{60\cdots 79}a^{32}-\frac{12\cdots 73}{16\cdots 67}a^{30}-\frac{63\cdots 00}{67\cdots 69}a^{28}-\frac{57\cdots 64}{18\cdots 37}a^{26}+\frac{23\cdots 28}{18\cdots 37}a^{24}+\frac{14\cdots 82}{55\cdots 89}a^{22}+\frac{31\cdots 23}{16\cdots 67}a^{20}+\frac{61\cdots 10}{18\cdots 37}a^{18}+\frac{16\cdots 05}{55\cdots 89}a^{16}+\frac{42\cdots 47}{16\cdots 67}a^{14}-\frac{44\cdots 60}{50\cdots 99}a^{12}+\frac{67\cdots 95}{50\cdots 99}a^{10}+\frac{36\cdots 99}{50\cdots 99}a^{8}-\frac{59\cdots 46}{15\cdots 97}a^{6}+\frac{52\cdots 82}{13\cdots 27}a^{4}+\frac{44\cdots 48}{13\cdots 27}a^{2}-\frac{82\cdots 77}{32\cdots 87}$, $\frac{1}{22\cdots 77}a^{39}-\frac{38\cdots 86}{20\cdots 07}a^{37}+\frac{17\cdots 64}{73\cdots 59}a^{35}+\frac{33\cdots 08}{60\cdots 79}a^{33}+\frac{12\cdots 98}{20\cdots 07}a^{31}-\frac{64\cdots 86}{20\cdots 07}a^{29}+\frac{21\cdots 02}{60\cdots 79}a^{27}+\frac{14\cdots 79}{67\cdots 69}a^{25}-\frac{41\cdots 07}{60\cdots 79}a^{23}-\frac{46\cdots 92}{18\cdots 37}a^{21}+\frac{65\cdots 01}{60\cdots 79}a^{19}+\frac{58\cdots 70}{16\cdots 67}a^{17}-\frac{32\cdots 34}{18\cdots 37}a^{15}-\frac{26\cdots 44}{55\cdots 89}a^{13}+\frac{61\cdots 18}{15\cdots 97}a^{11}-\frac{47\cdots 48}{15\cdots 97}a^{9}+\frac{64\cdots 79}{15\cdots 97}a^{7}-\frac{68\cdots 68}{15\cdots 97}a^{5}+\frac{12\cdots 35}{45\cdots 09}a^{3}-\frac{14\cdots 23}{13\cdots 27}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
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| Narrow class group: | not computed |
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Unit group
| Rank: | $39$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: | not computed |
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| Regulator: | not computed |
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| Unit signature rank: | not computed |
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 = \mathstrut &\frac{2^{40}\cdot(2\pi)^{0}\cdot R \cdot h}{2\cdot\sqrt{287896772221388971765277468942920243004336953163146972656250000000000000000000000000000000000000000}}\cr\mathstrut & \text{
Galois group
$C_2\times C_{20}$ (as 40T2):
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2\times C_{20}$ |
| Character table for $C_2\times C_{20}$ |
Intermediate fields
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 | R | ${\href{/padicField/3.4.0.1}{4} }^{10}$ | R | $20^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{10}$ | $20^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | ${\href{/padicField/47.4.0.1}{4} }^{10}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $2$ | $20$ | $40$ | |||
|
\(5\)
| 5.1.20.35a1.4 | $x^{20} + 20 x^{16} + 80$ | $20$ | $1$ | $35$ | 20T1 | $$[2]_{4}$$ |
| 5.1.20.35a1.4 | $x^{20} + 20 x^{16} + 80$ | $20$ | $1$ | $35$ | 20T1 | $$[2]_{4}$$ | |
|
\(11\)
| 11.2.10.18a1.10 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35884 x + 1123$ | $10$ | $2$ | $18$ | 20T3 | $$[\ ]_{10}^{2}$$ |
| 11.2.10.18a1.10 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241653 x^{10} + 2355135020 x^{9} + 1953240660 x^{8} + 1157466240 x^{7} + 496075680 x^{6} + 154293888 x^{5} + 34538880 x^{4} + 5429760 x^{3} + 569600 x^{2} + 35884 x + 1123$ | $10$ | $2$ | $18$ | 20T3 | $$[\ ]_{10}^{2}$$ |