Normalized defining polynomial
\( x^{40} + 220 x^{38} + 21560 x^{36} + 1246300 x^{34} + 47425950 x^{32} + 1255302444 x^{30} + \cdots + 313993243201 \)
Invariants
| Degree: | $40$ |
| |
| Signature: | $(0, 20)$ |
| |
| Discriminant: |
\(287\!\cdots\!000\)
\(\medspace = 2^{40}\cdot 5^{70}\cdot 11^{36}\)
|
| |
| Root discriminant: | \(289.39\) |
| |
| 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$$\chi_{1100}(1,·)$, $\chi_{1100}(1003,·)$, $\chi_{1100}(513,·)$, $\chi_{1100}(307,·)$, $\chi_{1100}(779,·)$, $\chi_{1100}(269,·)$, $\chi_{1100}(877,·)$, $\chi_{1100}(17,·)$, $\chi_{1100}(1047,·)$, $\chi_{1100}(289,·)$, $\chi_{1100}(551,·)$, $\chi_{1100}(43,·)$, $\chi_{1100}(173,·)$, $\chi_{1100}(819,·)$, $\chi_{1100}(181,·)$, $\chi_{1100}(311,·)$, $\chi_{1100}(59,·)$, $\chi_{1100}(191,·)$, $\chi_{1100}(453,·)$, $\chi_{1100}(327,·)$, $\chi_{1100}(567,·)$, $\chi_{1100}(593,·)$, $\chi_{1100}(83,·)$, $\chi_{1100}(471,·)$, $\chi_{1100}(857,·)$, $\chi_{1100}(731,·)$, $\chi_{1100}(861,·)$, $\chi_{1100}(199,·)$, $\chi_{1100}(229,·)$, $\chi_{1100}(609,·)$, $\chi_{1100}(741,·)$, $\chi_{1100}(337,·)$, $\chi_{1100}(1063,·)$, $\chi_{1100}(749,·)$, $\chi_{1100}(497,·)$, $\chi_{1100}(839,·)$, $\chi_{1100}(723,·)$, $\chi_{1100}(887,·)$, $\chi_{1100}(633,·)$, $\chi_{1100}(1021,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}$, $\frac{1}{11}a^{11}$, $\frac{1}{11}a^{12}$, $\frac{1}{11}a^{13}$, $\frac{1}{11}a^{14}$, $\frac{1}{11}a^{15}$, $\frac{1}{11}a^{16}$, $\frac{1}{121}a^{17}+\frac{2}{11}a^{7}$, $\frac{1}{121}a^{18}+\frac{2}{11}a^{8}$, $\frac{1}{121}a^{19}+\frac{2}{11}a^{9}$, $\frac{1}{121}a^{20}$, $\frac{1}{121}a^{21}$, $\frac{1}{121}a^{22}$, $\frac{1}{121}a^{23}$, $\frac{1}{121}a^{24}$, $\frac{1}{1331}a^{25}+\frac{4}{121}a^{15}+\frac{4}{11}a^{5}$, $\frac{1}{1331}a^{26}+\frac{4}{121}a^{16}+\frac{4}{11}a^{6}$, $\frac{1}{560351}a^{27}+\frac{120}{560351}a^{25}+\frac{16}{50941}a^{23}-\frac{166}{50941}a^{21}-\frac{157}{50941}a^{19}+\frac{76}{50941}a^{17}+\frac{7}{50941}a^{15}-\frac{13}{4631}a^{13}-\frac{148}{4631}a^{11}+\frac{2139}{4631}a^{9}+\frac{1688}{4631}a^{7}+\frac{1437}{4631}a^{5}-\frac{209}{421}a^{3}-\frac{135}{421}a$, $\frac{1}{1681053}a^{28}-\frac{301}{1681053}a^{26}+\frac{16}{152823}a^{24}-\frac{166}{152823}a^{22}-\frac{578}{152823}a^{20}+\frac{497}{152823}a^{18}-\frac{559}{50941}a^{16}-\frac{13}{13893}a^{14}+\frac{91}{4631}a^{12}+\frac{455}{13893}a^{10}+\frac{230}{1263}a^{8}-\frac{1626}{4631}a^{6}+\frac{212}{1263}a^{4}-\frac{556}{1263}a^{2}-\frac{1}{3}$, $\frac{1}{1681053}a^{29}-\frac{1}{1681053}a^{27}-\frac{41}{152823}a^{25}-\frac{38}{13893}a^{23}+\frac{142}{152823}a^{21}+\frac{128}{152823}a^{19}-\frac{116}{50941}a^{17}-\frac{511}{13893}a^{15}+\frac{54}{4631}a^{13}+\frac{260}{13893}a^{11}+\frac{1363}{13893}a^{9}+\frac{37}{4631}a^{7}-\frac{439}{1263}a^{5}-\frac{106}{1263}a^{3}-\frac{505}{1263}a$, $\frac{1}{18491583}a^{30}-\frac{298}{1681053}a^{26}-\frac{127}{50941}a^{24}-\frac{39}{50941}a^{22}+\frac{1534}{560351}a^{20}+\frac{358}{152823}a^{18}+\frac{3470}{152823}a^{16}+\frac{358}{13893}a^{14}-\frac{296}{13893}a^{12}-\frac{1920}{50941}a^{10}+\frac{2192}{13893}a^{8}-\frac{538}{13893}a^{6}-\frac{220}{1263}a^{4}+\frac{248}{1263}a^{2}+\frac{1}{3}$, $\frac{1}{18491583}a^{31}-\frac{1}{1681053}a^{27}-\frac{42}{560351}a^{25}-\frac{139}{50941}a^{23}+\frac{1369}{560351}a^{21}+\frac{460}{152823}a^{19}-\frac{481}{152823}a^{17}+\frac{4754}{152823}a^{15}-\frac{368}{13893}a^{13}-\frac{1007}{50941}a^{11}-\frac{5392}{13893}a^{9}+\frac{3176}{13893}a^{7}+\frac{6316}{13893}a^{5}+\frac{62}{1263}a^{3}-\frac{521}{1263}a$, $\frac{1}{18491583}a^{32}-\frac{427}{1681053}a^{26}-\frac{401}{152823}a^{24}+\frac{2281}{1681053}a^{22}-\frac{118}{152823}a^{20}+\frac{16}{152823}a^{18}+\frac{3077}{152823}a^{16}-\frac{127}{4631}a^{14}-\frac{6}{50941}a^{12}+\frac{115}{13893}a^{10}+\frac{1902}{4631}a^{8}+\frac{1438}{13893}a^{6}+\frac{274}{1263}a^{4}+\frac{62}{421}a^{2}-\frac{1}{3}$, $\frac{1}{18491583}a^{33}-\frac{1}{1681053}a^{27}-\frac{2}{152823}a^{25}-\frac{6101}{1681053}a^{23}-\frac{106}{152823}a^{21}+\frac{73}{152823}a^{19}+\frac{89}{152823}a^{17}+\frac{193}{4631}a^{15}-\frac{1788}{50941}a^{13}+\frac{217}{13893}a^{11}-\frac{1690}{4631}a^{9}-\frac{3173}{13893}a^{7}-\frac{221}{1263}a^{5}-\frac{146}{421}a^{3}+\frac{167}{1263}a$, $\frac{1}{18491583}a^{34}-\frac{323}{1681053}a^{26}-\frac{1975}{560351}a^{24}-\frac{272}{152823}a^{22}-\frac{505}{152823}a^{20}+\frac{586}{152823}a^{18}+\frac{1564}{50941}a^{16}-\frac{5507}{152823}a^{14}+\frac{490}{13893}a^{12}+\frac{437}{13893}a^{10}-\frac{643}{13893}a^{8}+\frac{6584}{13893}a^{6}-\frac{226}{1263}a^{4}-\frac{389}{1263}a^{2}-\frac{1}{3}$, $\frac{1}{203407413}a^{35}-\frac{1}{1681053}a^{27}+\frac{401}{6163861}a^{25}+\frac{4720}{1681053}a^{23}-\frac{4303}{1681053}a^{21}-\frac{496}{152823}a^{19}+\frac{57}{50941}a^{17}-\frac{13427}{1681053}a^{15}+\frac{6538}{152823}a^{13}+\frac{6044}{152823}a^{11}+\frac{4006}{13893}a^{9}+\frac{4501}{13893}a^{7}-\frac{3328}{13893}a^{5}+\frac{122}{1263}a^{3}-\frac{308}{1263}a$, $\frac{1}{203407413}a^{36}-\frac{2108}{18491583}a^{26}+\frac{1632}{560351}a^{24}-\frac{2043}{560351}a^{22}+\frac{63}{50941}a^{20}-\frac{595}{152823}a^{18}-\frac{31874}{1681053}a^{16}+\frac{6395}{152823}a^{14}-\frac{4846}{152823}a^{12}-\frac{197}{4631}a^{10}+\frac{4505}{13893}a^{8}+\frac{517}{1263}a^{6}+\frac{334}{1263}a^{4}+\frac{133}{421}a^{2}-\frac{1}{3}$, $\frac{1}{203407413}a^{37}+\frac{4}{18491583}a^{27}+\frac{50}{560351}a^{25}-\frac{41}{560351}a^{23}-\frac{36}{50941}a^{21}-\frac{427}{152823}a^{19}+\frac{3601}{1681053}a^{17}-\frac{6154}{152823}a^{15}-\frac{4516}{152823}a^{13}+\frac{14}{4631}a^{11}+\frac{3455}{13893}a^{9}+\frac{1403}{13893}a^{7}+\frac{1718}{13893}a^{5}-\frac{192}{421}a^{3}+\frac{182}{1263}a$, $\frac{1}{42\cdots 27}a^{38}+\frac{89\cdots 44}{14\cdots 09}a^{36}+\frac{22\cdots 54}{12\cdots 19}a^{34}-\frac{18\cdots 30}{12\cdots 19}a^{32}+\frac{52\cdots 52}{38\cdots 57}a^{30}-\frac{16\cdots 39}{12\cdots 19}a^{28}+\frac{11\cdots 25}{38\cdots 57}a^{26}-\frac{88\cdots 54}{35\cdots 87}a^{24}+\frac{17\cdots 94}{35\cdots 87}a^{22}-\frac{30\cdots 84}{11\cdots 29}a^{20}-\frac{72\cdots 38}{35\cdots 87}a^{18}+\frac{82\cdots 77}{35\cdots 87}a^{16}+\frac{19\cdots 74}{31\cdots 17}a^{14}-\frac{64\cdots 32}{31\cdots 17}a^{12}+\frac{51\cdots 23}{31\cdots 17}a^{10}+\frac{72\cdots 65}{29\cdots 47}a^{8}+\frac{12\cdots 10}{29\cdots 47}a^{6}+\frac{30\cdots 84}{26\cdots 77}a^{4}-\frac{13\cdots 14}{88\cdots 59}a^{2}+\frac{49\cdots 68}{49\cdots 99}$, $\frac{1}{42\cdots 27}a^{39}+\frac{89\cdots 44}{14\cdots 09}a^{37}+\frac{34\cdots 15}{14\cdots 09}a^{35}-\frac{18\cdots 30}{12\cdots 19}a^{33}+\frac{52\cdots 52}{38\cdots 57}a^{31}-\frac{16\cdots 39}{12\cdots 19}a^{29}+\frac{15\cdots 91}{38\cdots 57}a^{27}-\frac{17\cdots 75}{38\cdots 57}a^{25}-\frac{44\cdots 73}{35\cdots 87}a^{23}-\frac{11\cdots 30}{11\cdots 29}a^{21}-\frac{83\cdots 43}{35\cdots 87}a^{19}-\frac{32\cdots 67}{35\cdots 87}a^{17}+\frac{85\cdots 20}{35\cdots 87}a^{15}+\frac{10\cdots 14}{29\cdots 47}a^{13}+\frac{50\cdots 31}{31\cdots 17}a^{11}-\frac{50\cdots 25}{29\cdots 47}a^{9}-\frac{18\cdots 12}{29\cdots 47}a^{7}+\frac{12\cdots 17}{29\cdots 47}a^{5}+\frac{41\cdots 21}{88\cdots 59}a^{3}+\frac{49\cdots 68}{49\cdots 99}a$
| 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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| Relative class number: | data not computed |
Unit group
| Rank: | $19$ |
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| Torsion generator: |
\( \frac{18715453815770535446224232999160305107250}{910117316266742442051858770431021195939787761530183} a^{39} + \frac{4102145866190137224371566345385642549485250}{910117316266742442051858770431021195939787761530183} a^{37} + \frac{400161706585494048375017780757732965812975450}{910117316266742442051858770431021195939787761530183} a^{35} + \frac{2090810175503125112278142542802122621663054250}{82737937842431131095623524584638290539980705593653} a^{33} + \frac{78986571596642215174992620122410202381949122525}{82737937842431131095623524584638290539980705593653} a^{31} + \frac{2071396598536900706194123285150766175249436348750}{82737937842431131095623524584638290539980705593653} a^{29} + \frac{38812859493405170709609141501461513630956035647500}{82737937842431131095623524584638290539980705593653} a^{27} + \frac{175598096409040278509465599733034547156757207645571}{27579312614143710365207841528212763513326901864551} a^{25} + \frac{42934976271855693257930603729988740215996192290950}{683784610268025876823334913922630500330419054493} a^{23} + \frac{3365643611165042524098971437967127134815608179354300}{7521630712948284645056684053148935503634609599423} a^{21} + \frac{5702417055554204117576169849253936977092656915743125}{2507210237649428215018894684382978501211536533141} a^{19} + \frac{60809726404382941679963203210723316749890881285903500}{7521630712948284645056684053148935503634609599423} a^{17} + \frac{147465994187748950138255523719147615845583547141568745}{7521630712948284645056684053148935503634609599423} a^{15} + \frac{21599245198826132240274378785107818371386868682531025}{683784610268025876823334913922630500330419054493} a^{13} + \frac{7520915759373652796338851798314383673751111882125475}{227928203422675292274444971307543500110139684831} a^{11} + \frac{451293783360727274843091249505503436727583896970000}{20720745765697753843131361027958500010012698621} a^{9} + \frac{545398033247519873503759199148524919261287599166500}{62162237297093261529394083083875500030038095863} a^{7} + \frac{42170499844318363183132737110726025057382011463445}{20720745765697753843131361027958500010012698621} a^{5} + \frac{1378470362979883096263172033432958690012969992300}{5651112481553932866308553007625045457276190533} a^{3} + \frac{59664826323921443855308774134941880178520900}{4474356675814673686705109269695206221121291} a \)
(order $4$)
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| Fundamental units: | not computed |
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| Regulator: | not computed |
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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 = \mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot R \cdot h}{4\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 | ${\href{/padicField/13.4.0.1}{4} }^{10}$ | $20^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{8}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | ${\href{/padicField/47.4.0.1}{4} }^{10}$ | $20^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 | $$[\ ]_{10}^{2}$$ |