Normalized defining polynomial
\( x^{24} - 12 x^{23} + 69 x^{22} - 253 x^{21} + 261 x^{20} + 2703 x^{19} - 5731 x^{18} - 47088 x^{17} + \cdots + 434205631 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
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| Discriminant: |
\(54372494083060640194576311797602218575775623321533203125\)
\(\medspace = 5^{33}\cdot 43^{20}\)
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| |
| Root discriminant: | \(210.04\) |
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| Galois root discriminant: | $5^{31/20}43^{9/10}\approx 357.7050872928245$ | ||
| Ramified primes: |
\(5\), \(43\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{43}a^{10}-\frac{5}{43}a^{9}-\frac{21}{43}a^{8}-\frac{15}{43}a^{7}-\frac{3}{43}a^{6}+\frac{19}{43}a^{5}+\frac{10}{43}a^{4}-\frac{9}{43}a^{3}-\frac{1}{43}a^{2}-\frac{19}{43}a+\frac{16}{43}$, $\frac{1}{43}a^{11}-\frac{3}{43}a^{9}+\frac{9}{43}a^{8}+\frac{8}{43}a^{7}+\frac{4}{43}a^{6}+\frac{19}{43}a^{5}-\frac{2}{43}a^{4}-\frac{3}{43}a^{3}+\frac{19}{43}a^{2}+\frac{7}{43}a-\frac{6}{43}$, $\frac{1}{43}a^{12}-\frac{6}{43}a^{9}-\frac{12}{43}a^{8}+\frac{2}{43}a^{7}+\frac{10}{43}a^{6}+\frac{12}{43}a^{5}-\frac{16}{43}a^{4}-\frac{8}{43}a^{3}+\frac{4}{43}a^{2}-\frac{20}{43}a+\frac{5}{43}$, $\frac{1}{43}a^{13}+\frac{1}{43}a^{9}+\frac{5}{43}a^{8}+\frac{6}{43}a^{7}-\frac{6}{43}a^{6}+\frac{12}{43}a^{5}+\frac{9}{43}a^{4}-\frac{7}{43}a^{3}+\frac{17}{43}a^{2}+\frac{20}{43}a+\frac{10}{43}$, $\frac{1}{1849}a^{14}-\frac{7}{1849}a^{13}+\frac{12}{1849}a^{12}+\frac{19}{1849}a^{11}-\frac{10}{1849}a^{10}-\frac{291}{1849}a^{9}-\frac{244}{1849}a^{8}-\frac{481}{1849}a^{7}-\frac{663}{1849}a^{6}+\frac{92}{1849}a^{5}+\frac{278}{1849}a^{4}+\frac{184}{1849}a^{3}-\frac{496}{1849}a^{2}-\frac{243}{1849}a-\frac{300}{1849}$, $\frac{1}{1849}a^{15}+\frac{6}{1849}a^{13}+\frac{17}{1849}a^{12}-\frac{6}{1849}a^{11}-\frac{17}{1849}a^{10}+\frac{643}{1849}a^{9}-\frac{82}{1849}a^{8}-\frac{891}{1849}a^{7}+\frac{181}{1849}a^{6}+\frac{793}{1849}a^{5}+\frac{195}{1849}a^{4}+\frac{319}{1849}a^{3}-\frac{576}{1849}a^{2}+\frac{536}{1849}a+\frac{480}{1849}$, $\frac{1}{1849}a^{16}+\frac{16}{1849}a^{13}+\frac{8}{1849}a^{12}-\frac{2}{1849}a^{11}+\frac{15}{1849}a^{10}+\frac{460}{1849}a^{9}+\frac{143}{1849}a^{8}-\frac{459}{1849}a^{7}-\frac{776}{1849}a^{6}+\frac{632}{1849}a^{5}+\frac{844}{1849}a^{4}+\frac{40}{1849}a^{3}+\frac{717}{1849}a^{2}+\frac{390}{1849}a-\frac{737}{1849}$, $\frac{1}{79507}a^{17}+\frac{13}{79507}a^{16}-\frac{9}{79507}a^{15}+\frac{1}{79507}a^{14}-\frac{77}{79507}a^{13}+\frac{242}{79507}a^{12}-\frac{629}{79507}a^{11}+\frac{356}{79507}a^{10}+\frac{29727}{79507}a^{9}+\frac{3863}{79507}a^{8}+\frac{18854}{79507}a^{7}-\frac{1484}{79507}a^{6}-\frac{27794}{79507}a^{5}+\frac{22459}{79507}a^{4}-\frac{10285}{79507}a^{3}-\frac{34597}{79507}a^{2}-\frac{8155}{79507}a+\frac{11798}{79507}$, $\frac{1}{79507}a^{18}-\frac{6}{79507}a^{16}-\frac{11}{79507}a^{15}-\frac{4}{79507}a^{14}+\frac{770}{79507}a^{13}+\frac{138}{79507}a^{12}-\frac{497}{79507}a^{11}-\frac{572}{79507}a^{10}-\frac{15755}{79507}a^{9}-\frac{26420}{79507}a^{8}+\frac{19842}{79507}a^{7}+\frac{29123}{79507}a^{6}-\frac{34566}{79507}a^{5}-\frac{30750}{79507}a^{4}-\frac{33977}{79507}a^{3}+\frac{1200}{79507}a^{2}-\frac{8693}{79507}a+\frac{20432}{79507}$, $\frac{1}{79507}a^{19}-\frac{19}{79507}a^{16}-\frac{15}{79507}a^{15}+\frac{2}{79507}a^{14}+\frac{278}{79507}a^{13}-\frac{894}{79507}a^{12}-\frac{648}{79507}a^{11}-\frac{504}{79507}a^{10}-\frac{13780}{79507}a^{9}-\frac{39110}{79507}a^{8}-\frac{11263}{79507}a^{7}-\frac{23432}{79507}a^{6}+\frac{8714}{79507}a^{5}+\frac{24796}{79507}a^{4}-\frac{29937}{79507}a^{3}-\frac{5704}{79507}a^{2}-\frac{4375}{79507}a-\frac{3129}{79507}$, $\frac{1}{21684617136755}a^{20}-\frac{2}{4336923427351}a^{19}-\frac{18245987}{4336923427351}a^{18}+\frac{570861}{4336923427351}a^{17}-\frac{441687111}{4336923427351}a^{16}+\frac{3420385388}{21684617136755}a^{15}-\frac{734529808}{4336923427351}a^{14}+\frac{25835169810}{4336923427351}a^{13}-\frac{41976234326}{4336923427351}a^{12}-\frac{46496790130}{4336923427351}a^{11}-\frac{202664567106}{21684617136755}a^{10}-\frac{754475572679}{4336923427351}a^{9}+\frac{1416048902287}{4336923427351}a^{8}-\frac{620899423602}{4336923427351}a^{7}+\frac{1521919425614}{4336923427351}a^{6}+\frac{10551317257417}{21684617136755}a^{5}-\frac{1052331411451}{4336923427351}a^{4}+\frac{1450367620557}{4336923427351}a^{3}+\frac{420501003211}{4336923427351}a^{2}+\frac{85797438239}{4336923427351}a-\frac{10625111840989}{21684617136755}$, $\frac{1}{412007725598345}a^{21}-\frac{1}{412007725598345}a^{20}+\frac{309040153}{82401545119669}a^{19}+\frac{3}{4336923427351}a^{18}-\frac{20105351}{4336923427351}a^{17}-\frac{3396258}{699503778605}a^{16}-\frac{8890657928}{412007725598345}a^{15}-\frac{4176558759}{82401545119669}a^{14}+\frac{43078596620}{4336923427351}a^{13}-\frac{685784539306}{82401545119669}a^{12}-\frac{2725674159191}{412007725598345}a^{11}+\frac{2648797568066}{412007725598345}a^{10}-\frac{6154481452496}{82401545119669}a^{9}+\frac{963788235149}{4336923427351}a^{8}-\frac{8200890516204}{82401545119669}a^{7}-\frac{91205385704488}{412007725598345}a^{6}-\frac{161157005170482}{412007725598345}a^{5}+\frac{26430507767052}{82401545119669}a^{4}+\frac{18929724389777}{82401545119669}a^{3}+\frac{600620934887}{4336923427351}a^{2}-\frac{18827449427184}{412007725598345}a+\frac{29734956006769}{412007725598345}$, $\frac{1}{11\cdots 25}a^{22}-\frac{11}{11\cdots 25}a^{21}-\frac{2363891886}{11\cdots 25}a^{20}+\frac{4727783849}{22\cdots 85}a^{19}-\frac{688121396220918}{11\cdots 15}a^{18}+\frac{30\cdots 27}{59\cdots 75}a^{17}-\frac{23\cdots 53}{11\cdots 25}a^{16}-\frac{10\cdots 13}{11\cdots 25}a^{15}-\frac{42\cdots 54}{22\cdots 85}a^{14}-\frac{25\cdots 11}{22\cdots 85}a^{13}+\frac{92\cdots 29}{36\cdots 75}a^{12}-\frac{11\cdots 64}{11\cdots 25}a^{11}+\frac{90\cdots 66}{11\cdots 25}a^{10}+\frac{10\cdots 16}{22\cdots 85}a^{9}-\frac{16\cdots 22}{44\cdots 17}a^{8}+\frac{31\cdots 97}{11\cdots 25}a^{7}-\frac{22\cdots 22}{11\cdots 25}a^{6}-\frac{83\cdots 78}{59\cdots 75}a^{5}+\frac{18\cdots 28}{44\cdots 17}a^{4}+\frac{89\cdots 93}{22\cdots 85}a^{3}-\frac{51\cdots 69}{11\cdots 25}a^{2}-\frac{80\cdots 24}{59\cdots 75}a-\frac{52\cdots 51}{11\cdots 25}$, $\frac{1}{56\cdots 25}a^{23}+\frac{1}{56\cdots 25}a^{22}+\frac{363492632}{56\cdots 25}a^{21}-\frac{7455168037}{56\cdots 25}a^{20}+\frac{54\cdots 41}{11\cdots 25}a^{19}+\frac{13\cdots 47}{29\cdots 75}a^{18}-\frac{10\cdots 22}{56\cdots 25}a^{17}+\frac{11\cdots 51}{56\cdots 25}a^{16}+\frac{11\cdots 24}{56\cdots 25}a^{15}+\frac{20\cdots 86}{11\cdots 25}a^{14}+\frac{20\cdots 39}{56\cdots 25}a^{13}-\frac{19\cdots 76}{56\cdots 25}a^{12}+\frac{21\cdots 63}{20\cdots 75}a^{11}-\frac{10\cdots 03}{56\cdots 25}a^{10}+\frac{35\cdots 07}{11\cdots 25}a^{9}-\frac{22\cdots 53}{56\cdots 25}a^{8}+\frac{18\cdots 17}{56\cdots 25}a^{7}-\frac{18\cdots 34}{29\cdots 75}a^{6}-\frac{16\cdots 09}{56\cdots 25}a^{5}-\frac{17\cdots 02}{11\cdots 25}a^{4}-\frac{91\cdots 14}{56\cdots 25}a^{3}+\frac{19\cdots 66}{56\cdots 25}a^{2}-\frac{35\cdots 73}{56\cdots 25}a+\frac{19\cdots 88}{56\cdots 25}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{10}$, which has order $40$ (assuming GRH) |
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| Narrow class group: | $C_{10}\times C_{2}\times C_{2}\times C_{2}$, which has order $80$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{826153934680396}{22\cdots 85}a^{22}-\frac{90\cdots 56}{22\cdots 85}a^{21}+\frac{47\cdots 29}{22\cdots 85}a^{20}-\frac{31\cdots 66}{44\cdots 17}a^{19}+\frac{455430581873349}{23\cdots 43}a^{18}+\frac{11\cdots 02}{11\cdots 15}a^{17}-\frac{21\cdots 43}{22\cdots 85}a^{16}-\frac{43\cdots 68}{22\cdots 85}a^{15}+\frac{24\cdots 61}{44\cdots 17}a^{14}+\frac{22\cdots 07}{44\cdots 17}a^{13}-\frac{26\cdots 31}{72\cdots 35}a^{12}+\frac{68\cdots 31}{22\cdots 85}a^{11}+\frac{12\cdots 21}{22\cdots 85}a^{10}-\frac{50\cdots 59}{44\cdots 17}a^{9}-\frac{41\cdots 47}{44\cdots 17}a^{8}+\frac{12\cdots 07}{22\cdots 85}a^{7}-\frac{68\cdots 62}{22\cdots 85}a^{6}-\frac{12\cdots 33}{11\cdots 15}a^{5}+\frac{52\cdots 33}{44\cdots 17}a^{4}+\frac{35\cdots 85}{44\cdots 17}a^{3}-\frac{50\cdots 04}{22\cdots 85}a^{2}+\frac{22\cdots 46}{11\cdots 15}a+\frac{87\cdots 99}{22\cdots 85}$, $\frac{1691295442620}{55\cdots 01}a^{22}+\frac{18604249868820}{55\cdots 01}a^{21}-\frac{477742674904188}{27\cdots 05}a^{20}+\frac{304336604399676}{55\cdots 01}a^{19}+\frac{1937658107152}{55\cdots 01}a^{18}-\frac{49\cdots 44}{55\cdots 01}a^{17}+\frac{44\cdots 56}{55\cdots 01}a^{16}+\frac{46\cdots 76}{27\cdots 05}a^{15}-\frac{59\cdots 63}{12\cdots 07}a^{14}-\frac{30\cdots 13}{55\cdots 01}a^{13}+\frac{59\cdots 10}{17\cdots 71}a^{12}-\frac{12\cdots 03}{55\cdots 01}a^{11}-\frac{17\cdots 47}{27\cdots 05}a^{10}+\frac{56\cdots 62}{55\cdots 01}a^{9}+\frac{43\cdots 01}{55\cdots 01}a^{8}-\frac{24\cdots 95}{55\cdots 01}a^{7}+\frac{78\cdots 51}{55\cdots 01}a^{6}+\frac{28\cdots 24}{27\cdots 05}a^{5}-\frac{70\cdots 16}{55\cdots 01}a^{4}+\frac{20\cdots 53}{55\cdots 01}a^{3}+\frac{10\cdots 57}{55\cdots 01}a^{2}-\frac{10\cdots 11}{55\cdots 01}a-\frac{12\cdots 98}{27\cdots 05}$, $\frac{22\cdots 53}{11\cdots 25}a^{22}-\frac{24\cdots 83}{11\cdots 25}a^{21}+\frac{12\cdots 42}{11\cdots 25}a^{20}-\frac{69\cdots 03}{22\cdots 85}a^{19}-\frac{26\cdots 14}{11\cdots 15}a^{18}+\frac{38\cdots 31}{59\cdots 75}a^{17}-\frac{51\cdots 34}{11\cdots 25}a^{16}-\frac{13\cdots 14}{11\cdots 25}a^{15}+\frac{67\cdots 08}{22\cdots 85}a^{14}+\frac{13\cdots 62}{22\cdots 85}a^{13}-\frac{95\cdots 38}{36\cdots 75}a^{12}+\frac{39\cdots 58}{11\cdots 25}a^{11}+\frac{90\cdots 23}{11\cdots 25}a^{10}-\frac{19\cdots 67}{22\cdots 85}a^{9}-\frac{59\cdots 58}{44\cdots 17}a^{8}+\frac{53\cdots 41}{11\cdots 25}a^{7}-\frac{11\cdots 16}{11\cdots 25}a^{6}-\frac{66\cdots 09}{59\cdots 75}a^{5}+\frac{54\cdots 76}{44\cdots 17}a^{4}-\frac{30\cdots 96}{22\cdots 85}a^{3}-\frac{25\cdots 07}{11\cdots 25}a^{2}+\frac{12\cdots 03}{59\cdots 75}a+\frac{49\cdots 97}{11\cdots 25}$, $\frac{33\cdots 82}{11\cdots 25}a^{22}+\frac{36\cdots 02}{11\cdots 25}a^{21}-\frac{21\cdots 18}{11\cdots 25}a^{20}+\frac{17\cdots 22}{22\cdots 85}a^{19}-\frac{12\cdots 89}{11\cdots 15}a^{18}-\frac{39\cdots 39}{59\cdots 75}a^{17}+\frac{20\cdots 71}{11\cdots 25}a^{16}+\frac{75\cdots 56}{11\cdots 25}a^{15}-\frac{10\cdots 22}{22\cdots 85}a^{14}+\frac{28\cdots 22}{22\cdots 85}a^{13}+\frac{38\cdots 47}{36\cdots 75}a^{12}-\frac{18\cdots 02}{11\cdots 25}a^{11}+\frac{27\cdots 08}{11\cdots 25}a^{10}+\frac{99\cdots 13}{22\cdots 85}a^{9}-\frac{53\cdots 25}{44\cdots 17}a^{8}+\frac{53\cdots 96}{11\cdots 25}a^{7}+\frac{24\cdots 04}{11\cdots 25}a^{6}-\frac{16\cdots 39}{59\cdots 75}a^{5}-\frac{27\cdots 42}{44\cdots 17}a^{4}+\frac{33\cdots 09}{22\cdots 85}a^{3}+\frac{54\cdots 58}{11\cdots 25}a^{2}-\frac{74\cdots 82}{59\cdots 75}a+\frac{47\cdots 87}{11\cdots 25}$, $\frac{15\cdots 93}{11\cdots 25}a^{22}-\frac{16\cdots 23}{11\cdots 25}a^{21}+\frac{82\cdots 92}{11\cdots 25}a^{20}-\frac{48\cdots 23}{22\cdots 85}a^{19}-\frac{19\cdots 44}{11\cdots 15}a^{18}+\frac{27\cdots 11}{59\cdots 75}a^{17}-\frac{48\cdots 04}{11\cdots 25}a^{16}-\frac{88\cdots 14}{11\cdots 25}a^{15}+\frac{46\cdots 68}{22\cdots 85}a^{14}+\frac{72\cdots 82}{22\cdots 85}a^{13}-\frac{65\cdots 28}{36\cdots 75}a^{12}+\frac{18\cdots 98}{11\cdots 25}a^{11}+\frac{40\cdots 48}{11\cdots 25}a^{10}-\frac{25\cdots 62}{22\cdots 85}a^{9}+\frac{36\cdots 50}{44\cdots 17}a^{8}+\frac{17\cdots 71}{11\cdots 25}a^{7}-\frac{38\cdots 46}{11\cdots 25}a^{6}+\frac{12\cdots 91}{59\cdots 75}a^{5}+\frac{15\cdots 86}{44\cdots 17}a^{4}-\frac{16\cdots 61}{22\cdots 85}a^{3}+\frac{12\cdots 08}{11\cdots 25}a^{2}+\frac{14\cdots 93}{59\cdots 75}a-\frac{15\cdots 28}{11\cdots 25}$, $\frac{13\cdots 04}{56\cdots 25}a^{23}-\frac{17\cdots 81}{56\cdots 25}a^{22}+\frac{56\cdots 02}{29\cdots 75}a^{21}-\frac{42\cdots 38}{56\cdots 25}a^{20}+\frac{14\cdots 99}{11\cdots 25}a^{19}+\frac{16\cdots 88}{29\cdots 75}a^{18}-\frac{10\cdots 93}{56\cdots 25}a^{17}-\frac{52\cdots 66}{56\cdots 25}a^{16}+\frac{31\cdots 01}{56\cdots 25}a^{15}-\frac{71\cdots 91}{11\cdots 25}a^{14}-\frac{11\cdots 19}{56\cdots 25}a^{13}+\frac{41\cdots 06}{56\cdots 25}a^{12}-\frac{47\cdots 93}{56\cdots 25}a^{11}-\frac{29\cdots 47}{56\cdots 25}a^{10}+\frac{26\cdots 43}{11\cdots 25}a^{9}-\frac{61\cdots 37}{56\cdots 25}a^{8}-\frac{35\cdots 02}{56\cdots 25}a^{7}+\frac{34\cdots 86}{56\cdots 25}a^{6}+\frac{69\cdots 34}{56\cdots 25}a^{5}-\frac{26\cdots 82}{59\cdots 75}a^{4}+\frac{25\cdots 94}{56\cdots 25}a^{3}+\frac{24\cdots 79}{56\cdots 25}a^{2}-\frac{29\cdots 32}{56\cdots 25}a+\frac{43\cdots 62}{56\cdots 25}$, $\frac{16\cdots 58}{56\cdots 25}a^{23}+\frac{82\cdots 98}{29\cdots 75}a^{22}-\frac{15\cdots 07}{13\cdots 75}a^{21}+\frac{18\cdots 76}{56\cdots 25}a^{20}+\frac{78\cdots 02}{11\cdots 25}a^{19}-\frac{25\cdots 51}{29\cdots 75}a^{18}-\frac{16\cdots 14}{56\cdots 25}a^{17}+\frac{84\cdots 82}{56\cdots 25}a^{16}-\frac{12\cdots 27}{56\cdots 25}a^{15}-\frac{87\cdots 68}{11\cdots 25}a^{14}+\frac{13\cdots 63}{56\cdots 25}a^{13}-\frac{76\cdots 62}{56\cdots 25}a^{12}-\frac{27\cdots 39}{56\cdots 25}a^{11}+\frac{80\cdots 44}{56\cdots 25}a^{10}-\frac{65\cdots 11}{11\cdots 25}a^{9}-\frac{14\cdots 26}{56\cdots 25}a^{8}+\frac{65\cdots 04}{56\cdots 25}a^{7}+\frac{21\cdots 03}{56\cdots 25}a^{6}-\frac{10\cdots 68}{56\cdots 25}a^{5}+\frac{15\cdots 16}{11\cdots 25}a^{4}+\frac{21\cdots 37}{56\cdots 25}a^{3}-\frac{24\cdots 58}{56\cdots 25}a^{2}+\frac{50\cdots 39}{56\cdots 25}a+\frac{18\cdots 76}{56\cdots 25}$, $\frac{16\cdots 58}{56\cdots 25}a^{23}-\frac{23\cdots 72}{56\cdots 25}a^{22}+\frac{79\cdots 69}{29\cdots 75}a^{21}-\frac{63\cdots 41}{56\cdots 25}a^{20}+\frac{24\cdots 58}{11\cdots 25}a^{19}+\frac{19\cdots 01}{29\cdots 75}a^{18}-\frac{17\cdots 91}{56\cdots 25}a^{17}-\frac{61\cdots 52}{56\cdots 25}a^{16}+\frac{46\cdots 07}{56\cdots 25}a^{15}-\frac{13\cdots 17}{11\cdots 25}a^{14}-\frac{13\cdots 38}{56\cdots 25}a^{13}+\frac{65\cdots 22}{56\cdots 25}a^{12}-\frac{86\cdots 96}{56\cdots 25}a^{11}-\frac{33\cdots 54}{56\cdots 25}a^{10}+\frac{43\cdots 76}{11\cdots 25}a^{9}-\frac{48\cdots 74}{56\cdots 25}a^{8}-\frac{54\cdots 99}{56\cdots 25}a^{7}+\frac{11\cdots 78}{95\cdots 25}a^{6}+\frac{91\cdots 13}{56\cdots 25}a^{5}-\frac{85\cdots 16}{11\cdots 25}a^{4}+\frac{41\cdots 63}{56\cdots 25}a^{3}-\frac{49\cdots 77}{56\cdots 25}a^{2}-\frac{51\cdots 29}{56\cdots 25}a+\frac{64\cdots 59}{56\cdots 25}$, $\frac{67\cdots 56}{56\cdots 25}a^{23}-\frac{82\cdots 39}{56\cdots 25}a^{22}+\frac{47\cdots 87}{56\cdots 25}a^{21}-\frac{57\cdots 17}{18\cdots 75}a^{20}+\frac{43\cdots 16}{11\cdots 25}a^{19}+\frac{89\cdots 07}{29\cdots 75}a^{18}-\frac{36\cdots 17}{56\cdots 25}a^{17}-\frac{76\cdots 63}{13\cdots 75}a^{16}+\frac{13\cdots 54}{56\cdots 25}a^{15}-\frac{62\cdots 04}{11\cdots 25}a^{14}-\frac{63\cdots 41}{56\cdots 25}a^{13}+\frac{11\cdots 89}{56\cdots 25}a^{12}-\frac{42\cdots 32}{56\cdots 25}a^{11}-\frac{59\cdots 38}{56\cdots 25}a^{10}+\frac{20\cdots 47}{11\cdots 25}a^{9}+\frac{63\cdots 32}{56\cdots 25}a^{8}-\frac{10\cdots 13}{56\cdots 25}a^{7}-\frac{92\cdots 61}{56\cdots 25}a^{6}+\frac{31\cdots 86}{56\cdots 25}a^{5}-\frac{94\cdots 62}{11\cdots 25}a^{4}+\frac{50\cdots 91}{56\cdots 25}a^{3}-\frac{11\cdots 49}{56\cdots 25}a^{2}+\frac{63\cdots 82}{56\cdots 25}a-\frac{40\cdots 77}{56\cdots 25}$, $\frac{87\cdots 48}{11\cdots 25}a^{23}-\frac{10\cdots 76}{22\cdots 85}a^{22}+\frac{31\cdots 99}{36\cdots 75}a^{21}+\frac{13\cdots 07}{11\cdots 25}a^{20}-\frac{95\cdots 94}{22\cdots 85}a^{19}+\frac{93\cdots 01}{59\cdots 75}a^{18}+\frac{16\cdots 16}{22\cdots 85}a^{17}-\frac{35\cdots 93}{11\cdots 25}a^{16}-\frac{83\cdots 09}{11\cdots 25}a^{15}+\frac{14\cdots 23}{44\cdots 17}a^{14}+\frac{14\cdots 12}{11\cdots 25}a^{13}-\frac{25\cdots 84}{22\cdots 85}a^{12}+\frac{47\cdots 46}{11\cdots 25}a^{11}+\frac{19\cdots 08}{11\cdots 25}a^{10}-\frac{11\cdots 42}{22\cdots 85}a^{9}-\frac{59\cdots 43}{84\cdots 25}a^{8}+\frac{98\cdots 13}{44\cdots 17}a^{7}+\frac{49\cdots 83}{11\cdots 25}a^{6}-\frac{19\cdots 11}{11\cdots 25}a^{5}+\frac{10\cdots 84}{22\cdots 85}a^{4}+\frac{36\cdots 33}{11\cdots 25}a^{3}-\frac{18\cdots 99}{23\cdots 43}a^{2}+\frac{30\cdots 14}{11\cdots 25}a+\frac{11\cdots 02}{11\cdots 25}$, $\frac{72\cdots 98}{56\cdots 25}a^{23}-\frac{67\cdots 07}{56\cdots 25}a^{22}+\frac{32\cdots 66}{56\cdots 25}a^{21}-\frac{52\cdots 09}{29\cdots 75}a^{20}-\frac{13\cdots 02}{11\cdots 25}a^{19}+\frac{92\cdots 06}{29\cdots 75}a^{18}+\frac{53\cdots 79}{56\cdots 25}a^{17}-\frac{32\cdots 87}{56\cdots 25}a^{16}+\frac{58\cdots 92}{56\cdots 25}a^{15}+\frac{22\cdots 48}{11\cdots 25}a^{14}-\frac{46\cdots 28}{56\cdots 25}a^{13}+\frac{41\cdots 32}{56\cdots 25}a^{12}+\frac{59\cdots 74}{56\cdots 25}a^{11}-\frac{19\cdots 74}{56\cdots 25}a^{10}-\frac{14\cdots 94}{11\cdots 25}a^{9}+\frac{61\cdots 31}{56\cdots 25}a^{8}-\frac{31\cdots 19}{56\cdots 25}a^{7}-\frac{91\cdots 98}{56\cdots 25}a^{6}+\frac{15\cdots 37}{29\cdots 75}a^{5}-\frac{10\cdots 91}{36\cdots 75}a^{4}-\frac{42\cdots 97}{56\cdots 25}a^{3}+\frac{86\cdots 88}{56\cdots 25}a^{2}-\frac{40\cdots 24}{56\cdots 25}a-\frac{11\cdots 21}{56\cdots 25}$, $\frac{16\cdots 24}{56\cdots 25}a^{23}-\frac{14\cdots 11}{56\cdots 25}a^{22}+\frac{63\cdots 03}{56\cdots 25}a^{21}-\frac{20\cdots 28}{56\cdots 25}a^{20}-\frac{33\cdots 56}{11\cdots 25}a^{19}+\frac{17\cdots 78}{29\cdots 75}a^{18}+\frac{48\cdots 42}{56\cdots 25}a^{17}-\frac{14\cdots 97}{13\cdots 75}a^{16}+\frac{67\cdots 81}{56\cdots 25}a^{15}+\frac{33\cdots 04}{11\cdots 25}a^{14}-\frac{47\cdots 39}{56\cdots 25}a^{13}+\frac{69\cdots 11}{56\cdots 25}a^{12}+\frac{40\cdots 17}{56\cdots 25}a^{11}-\frac{85\cdots 82}{56\cdots 25}a^{10}-\frac{75\cdots 42}{11\cdots 25}a^{9}+\frac{59\cdots 53}{56\cdots 25}a^{8}-\frac{96\cdots 62}{56\cdots 25}a^{7}-\frac{28\cdots 09}{56\cdots 25}a^{6}+\frac{23\cdots 79}{56\cdots 25}a^{5}+\frac{31\cdots 77}{11\cdots 25}a^{4}+\frac{16\cdots 64}{56\cdots 25}a^{3}+\frac{69\cdots 99}{56\cdots 25}a^{2}+\frac{32\cdots 58}{56\cdots 25}a+\frac{14\cdots 22}{56\cdots 25}$, $\frac{10\cdots 04}{22\cdots 85}a^{22}-\frac{11\cdots 44}{22\cdots 85}a^{21}+\frac{11\cdots 64}{44\cdots 17}a^{20}-\frac{31\cdots 32}{44\cdots 17}a^{19}-\frac{19\cdots 44}{23\cdots 43}a^{18}+\frac{19\cdots 08}{11\cdots 15}a^{17}-\frac{34\cdots 02}{22\cdots 85}a^{16}-\frac{12\cdots 24}{44\cdots 17}a^{15}+\frac{32\cdots 35}{44\cdots 17}a^{14}+\frac{57\cdots 35}{44\cdots 17}a^{13}-\frac{49\cdots 74}{72\cdots 35}a^{12}+\frac{14\cdots 89}{22\cdots 85}a^{11}+\frac{71\cdots 32}{44\cdots 17}a^{10}-\frac{26\cdots 81}{44\cdots 17}a^{9}+\frac{16\cdots 31}{44\cdots 17}a^{8}+\frac{30\cdots 23}{22\cdots 85}a^{7}-\frac{45\cdots 03}{22\cdots 85}a^{6}-\frac{11\cdots 60}{23\cdots 43}a^{5}+\frac{14\cdots 42}{44\cdots 17}a^{4}-\frac{15\cdots 36}{44\cdots 17}a^{3}-\frac{97\cdots 91}{22\cdots 85}a^{2}+\frac{66\cdots 64}{11\cdots 15}a+\frac{21\cdots 38}{44\cdots 17}$
|
| |
| Regulator: | \( 143751634415414750 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 143751634415414750 \cdot 40}{2\cdot\sqrt{54372494083060640194576311797602218575775623321533203125}}\cr\approx \mathstrut & 0.598235010646652 \end{aligned}\] (assuming GRH)
Galois group
$\GL(2,5)$ (as 24T1353):
| A non-solvable group of order 480 |
| The 24 conjugacy class representatives for $\GL(2,5)$ |
| Character table for $\GL(2,5)$ |
Intermediate fields
| 6.2.267093828125.1, 12.4.356695565112335205078125.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $24$ | $24$ | R | $24$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{10}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/29.2.0.1}{2} }^{10}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }^{8}$ | R | ${\href{/padicField/47.8.0.1}{8} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $20{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 5.1.20.31a4.25 | $x^{20} + 10 x^{12} + 20$ | $20$ | $1$ | $31$ | 20T9 | $$[\frac{7}{4}]_{4}^{2}$$ | |
|
\(43\)
| 43.2.2.2a1.1 | $x^{4} + 84 x^{3} + 1770 x^{2} + 295 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 43.1.10.9a1.2 | $x^{10} + 129$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ | |
| 43.1.10.9a1.2 | $x^{10} + 129$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |