Normalized defining polynomial
\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} + 446 x^{20} + 12994 x^{19} + 25454 x^{18} + \cdots - 8058266421349 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
| |
| Discriminant: |
\(2295251366305746200663037028252313182580209076404571533203125\)
\(\medspace = 5^{25}\cdot 89^{22}\)
|
| |
| Root discriminant: | \(327.37\) |
| |
| Galois root discriminant: | $5^{23/20}89^{19/20}\approx 452.62236684313837$ | ||
| Ramified primes: |
\(5\), \(89\)
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| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{3}+\frac{2}{5}a$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{5}+\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{8}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{25}a^{18}+\frac{1}{25}a^{16}+\frac{1}{25}a^{14}+\frac{2}{25}a^{13}+\frac{1}{25}a^{12}+\frac{2}{25}a^{11}+\frac{1}{25}a^{10}+\frac{2}{25}a^{9}+\frac{6}{25}a^{8}-\frac{8}{25}a^{7}-\frac{9}{25}a^{6}+\frac{7}{25}a^{5}+\frac{1}{25}a^{4}+\frac{1}{5}a^{3}-\frac{9}{25}a^{2}-\frac{2}{5}a+\frac{6}{25}$, $\frac{1}{25}a^{19}+\frac{1}{25}a^{17}+\frac{1}{25}a^{15}+\frac{2}{25}a^{14}+\frac{1}{25}a^{13}+\frac{2}{25}a^{12}+\frac{1}{25}a^{11}+\frac{2}{25}a^{10}+\frac{1}{25}a^{9}-\frac{8}{25}a^{8}+\frac{1}{25}a^{7}+\frac{7}{25}a^{6}-\frac{4}{25}a^{5}-\frac{9}{25}a^{3}+\frac{6}{25}a-\frac{1}{5}$, $\frac{1}{25}a^{20}+\frac{2}{25}a^{15}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{3}{25}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{4}{25}$, $\frac{1}{25}a^{21}+\frac{2}{25}a^{16}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{3}{25}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{4}{25}a+\frac{1}{5}$, $\frac{1}{25}a^{22}+\frac{2}{25}a^{17}-\frac{1}{5}a^{8}-\frac{7}{25}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}-\frac{6}{25}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{66\cdots 75}a^{23}+\frac{16\cdots 84}{13\cdots 55}a^{22}-\frac{96\cdots 92}{66\cdots 75}a^{21}+\frac{50\cdots 39}{66\cdots 75}a^{20}+\frac{86\cdots 86}{66\cdots 75}a^{19}+\frac{85\cdots 62}{66\cdots 75}a^{18}-\frac{62\cdots 49}{66\cdots 75}a^{17}-\frac{47\cdots 19}{66\cdots 75}a^{16}-\frac{23\cdots 11}{66\cdots 75}a^{15}-\frac{58\cdots 93}{66\cdots 75}a^{14}-\frac{25\cdots 44}{66\cdots 75}a^{13}+\frac{16\cdots 07}{66\cdots 75}a^{12}-\frac{56\cdots 74}{66\cdots 75}a^{11}-\frac{56\cdots 84}{94\cdots 25}a^{10}-\frac{71\cdots 12}{94\cdots 25}a^{9}-\frac{54\cdots 96}{13\cdots 55}a^{8}+\frac{51\cdots 36}{66\cdots 75}a^{7}+\frac{16\cdots 01}{66\cdots 75}a^{6}+\frac{17\cdots 83}{66\cdots 75}a^{5}-\frac{58\cdots 96}{13\cdots 55}a^{4}-\frac{85\cdots 67}{26\cdots 31}a^{3}-\frac{18\cdots 68}{13\cdots 55}a^{2}-\frac{22\cdots 27}{66\cdots 75}a-\frac{23\cdots 54}{66\cdots 75}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{53\cdots 17}{41\cdots 75}a^{23}+\frac{28\cdots 14}{41\cdots 75}a^{22}-\frac{67\cdots 66}{41\cdots 75}a^{21}+\frac{10\cdots 73}{41\cdots 75}a^{20}-\frac{50\cdots 04}{82\cdots 55}a^{19}-\frac{66\cdots 49}{41\cdots 75}a^{18}-\frac{53\cdots 77}{41\cdots 75}a^{17}-\frac{17\cdots 12}{41\cdots 75}a^{16}+\frac{31\cdots 21}{41\cdots 75}a^{15}+\frac{25\cdots 43}{16\cdots 31}a^{14}-\frac{45\cdots 77}{16\cdots 31}a^{13}-\frac{13\cdots 49}{16\cdots 31}a^{12}-\frac{15\cdots 11}{82\cdots 55}a^{11}+\frac{14\cdots 21}{11\cdots 65}a^{10}+\frac{13\cdots 46}{11\cdots 65}a^{9}+\frac{32\cdots 04}{41\cdots 75}a^{8}-\frac{13\cdots 63}{41\cdots 75}a^{7}-\frac{57\cdots 38}{41\cdots 75}a^{6}-\frac{47\cdots 51}{41\cdots 75}a^{5}+\frac{97\cdots 28}{82\cdots 55}a^{4}+\frac{11\cdots 72}{41\cdots 75}a^{3}+\frac{96\cdots 71}{41\cdots 75}a^{2}+\frac{48\cdots 96}{41\cdots 75}a+\frac{34\cdots 12}{41\cdots 75}$, $\frac{69\cdots 94}{11\cdots 25}a^{23}-\frac{10\cdots 84}{11\cdots 25}a^{22}+\frac{47\cdots 67}{11\cdots 25}a^{21}+\frac{88\cdots 77}{11\cdots 25}a^{20}-\frac{17\cdots 13}{23\cdots 05}a^{19}-\frac{86\cdots 88}{11\cdots 25}a^{18}-\frac{78\cdots 88}{11\cdots 25}a^{17}-\frac{21\cdots 31}{11\cdots 25}a^{16}+\frac{25\cdots 19}{11\cdots 25}a^{15}+\frac{14\cdots 81}{23\cdots 05}a^{14}+\frac{94\cdots 54}{23\cdots 05}a^{13}-\frac{14\cdots 88}{47\cdots 41}a^{12}-\frac{19\cdots 83}{47\cdots 41}a^{11}-\frac{14\cdots 43}{23\cdots 05}a^{10}+\frac{21\cdots 11}{23\cdots 05}a^{9}+\frac{59\cdots 23}{11\cdots 25}a^{8}-\frac{59\cdots 87}{11\cdots 25}a^{7}-\frac{21\cdots 54}{11\cdots 25}a^{6}-\frac{83\cdots 84}{11\cdots 25}a^{5}+\frac{11\cdots 37}{23\cdots 05}a^{4}+\frac{80\cdots 34}{11\cdots 25}a^{3}+\frac{18\cdots 34}{11\cdots 25}a^{2}+\frac{13\cdots 48}{11\cdots 25}a+\frac{55\cdots 48}{11\cdots 25}$, $\frac{18\cdots 48}{41\cdots 75}a^{23}-\frac{40\cdots 76}{16\cdots 31}a^{22}-\frac{69\cdots 59}{41\cdots 75}a^{21}+\frac{23\cdots 92}{41\cdots 75}a^{20}-\frac{70\cdots 62}{41\cdots 75}a^{19}+\frac{59\cdots 53}{82\cdots 55}a^{18}-\frac{14\cdots 97}{41\cdots 75}a^{17}-\frac{14\cdots 29}{41\cdots 75}a^{16}+\frac{11\cdots 27}{41\cdots 75}a^{15}-\frac{56\cdots 07}{82\cdots 55}a^{14}+\frac{61\cdots 76}{41\cdots 75}a^{13}+\frac{29\cdots 83}{82\cdots 55}a^{12}+\frac{38\cdots 61}{41\cdots 75}a^{11}-\frac{71\cdots 66}{11\cdots 65}a^{10}-\frac{19\cdots 57}{59\cdots 25}a^{9}+\frac{66\cdots 84}{41\cdots 75}a^{8}+\frac{56\cdots 91}{41\cdots 75}a^{7}+\frac{14\cdots 23}{41\cdots 75}a^{6}-\frac{41\cdots 03}{41\cdots 75}a^{5}-\frac{15\cdots 01}{41\cdots 75}a^{4}-\frac{29\cdots 65}{16\cdots 31}a^{3}+\frac{91\cdots 54}{41\cdots 75}a^{2}+\frac{62\cdots 42}{41\cdots 75}a+\frac{20\cdots 17}{41\cdots 75}$, $\frac{56\cdots 88}{47\cdots 41}a^{23}+\frac{11\cdots 43}{11\cdots 25}a^{22}-\frac{61\cdots 88}{11\cdots 25}a^{21}+\frac{28\cdots 02}{11\cdots 25}a^{20}-\frac{18\cdots 14}{11\cdots 25}a^{19}-\frac{10\cdots 04}{11\cdots 25}a^{18}+\frac{89\cdots 82}{11\cdots 25}a^{17}-\frac{31\cdots 61}{23\cdots 05}a^{16}+\frac{24\cdots 29}{47\cdots 41}a^{15}+\frac{14\cdots 43}{11\cdots 25}a^{14}-\frac{69\cdots 07}{11\cdots 25}a^{13}-\frac{58\cdots 17}{11\cdots 25}a^{12}-\frac{65\cdots 07}{11\cdots 25}a^{11}+\frac{10\cdots 98}{11\cdots 25}a^{10}+\frac{85\cdots 53}{11\cdots 25}a^{9}-\frac{83\cdots 12}{11\cdots 25}a^{8}-\frac{26\cdots 13}{11\cdots 25}a^{7}-\frac{79\cdots 31}{11\cdots 25}a^{6}+\frac{22\cdots 44}{11\cdots 25}a^{5}+\frac{61\cdots 96}{11\cdots 25}a^{4}+\frac{18\cdots 41}{11\cdots 25}a^{3}+\frac{32\cdots 43}{11\cdots 25}a^{2}+\frac{23\cdots 59}{11\cdots 25}a+\frac{11\cdots 69}{11\cdots 25}$, $\frac{85\cdots 66}{41\cdots 75}a^{23}+\frac{99\cdots 09}{82\cdots 55}a^{22}-\frac{12\cdots 58}{41\cdots 75}a^{21}+\frac{15\cdots 07}{41\cdots 75}a^{20}-\frac{38\cdots 27}{41\cdots 75}a^{19}-\frac{10\cdots 56}{41\cdots 75}a^{18}-\frac{18\cdots 87}{41\cdots 75}a^{17}-\frac{18\cdots 02}{82\cdots 55}a^{16}+\frac{61\cdots 52}{41\cdots 75}a^{15}+\frac{10\cdots 97}{41\cdots 75}a^{14}-\frac{47\cdots 17}{82\cdots 55}a^{13}-\frac{54\cdots 28}{41\cdots 75}a^{12}-\frac{17\cdots 07}{82\cdots 55}a^{11}+\frac{13\cdots 06}{59\cdots 25}a^{10}+\frac{20\cdots 98}{11\cdots 65}a^{9}-\frac{31\cdots 06}{41\cdots 75}a^{8}-\frac{44\cdots 53}{82\cdots 55}a^{7}-\frac{79\cdots 32}{41\cdots 75}a^{6}+\frac{74\cdots 76}{41\cdots 75}a^{5}+\frac{87\cdots 81}{41\cdots 75}a^{4}+\frac{12\cdots 64}{41\cdots 75}a^{3}-\frac{51\cdots 44}{41\cdots 75}a^{2}-\frac{13\cdots 74}{41\cdots 75}a-\frac{10\cdots 71}{41\cdots 75}$, $\frac{48\cdots 63}{41\cdots 75}a^{23}+\frac{19\cdots 17}{41\cdots 75}a^{22}+\frac{30\cdots 44}{41\cdots 75}a^{21}-\frac{53\cdots 96}{41\cdots 75}a^{20}+\frac{66\cdots 59}{41\cdots 75}a^{19}-\frac{14\cdots 98}{82\cdots 55}a^{18}-\frac{76\cdots 22}{41\cdots 75}a^{17}+\frac{35\cdots 19}{41\cdots 75}a^{16}-\frac{22\cdots 03}{41\cdots 75}a^{15}+\frac{67\cdots 39}{41\cdots 75}a^{14}-\frac{53\cdots 44}{41\cdots 75}a^{13}-\frac{38\cdots 16}{41\cdots 75}a^{12}-\frac{65\cdots 74}{41\cdots 75}a^{11}+\frac{83\cdots 22}{59\cdots 25}a^{10}+\frac{62\cdots 28}{59\cdots 25}a^{9}+\frac{77\cdots 19}{82\cdots 55}a^{8}-\frac{14\cdots 98}{41\cdots 75}a^{7}-\frac{57\cdots 44}{41\cdots 75}a^{6}+\frac{38\cdots 13}{41\cdots 75}a^{5}+\frac{50\cdots 66}{41\cdots 75}a^{4}+\frac{83\cdots 87}{41\cdots 75}a^{3}+\frac{62\cdots 04}{41\cdots 75}a^{2}+\frac{52\cdots 66}{82\cdots 55}a+\frac{18\cdots 77}{41\cdots 75}$, $\frac{27\cdots 73}{66\cdots 75}a^{23}+\frac{19\cdots 93}{66\cdots 75}a^{22}-\frac{77\cdots 49}{66\cdots 75}a^{21}+\frac{24\cdots 93}{66\cdots 75}a^{20}-\frac{19\cdots 22}{66\cdots 75}a^{19}-\frac{29\cdots 18}{66\cdots 75}a^{18}+\frac{21\cdots 04}{66\cdots 75}a^{17}-\frac{38\cdots 22}{13\cdots 55}a^{16}+\frac{63\cdots 94}{66\cdots 75}a^{15}+\frac{31\cdots 89}{66\cdots 75}a^{14}-\frac{11\cdots 51}{66\cdots 75}a^{13}-\frac{14\cdots 26}{66\cdots 75}a^{12}-\frac{15\cdots 56}{66\cdots 75}a^{11}+\frac{37\cdots 62}{94\cdots 25}a^{10}+\frac{27\cdots 42}{94\cdots 25}a^{9}-\frac{50\cdots 48}{26\cdots 31}a^{8}-\frac{62\cdots 12}{66\cdots 75}a^{7}-\frac{19\cdots 88}{66\cdots 75}a^{6}+\frac{19\cdots 58}{66\cdots 75}a^{5}+\frac{18\cdots 08}{66\cdots 75}a^{4}+\frac{31\cdots 06}{66\cdots 75}a^{3}+\frac{98\cdots 60}{26\cdots 31}a^{2}+\frac{10\cdots 57}{66\cdots 75}a+\frac{28\cdots 59}{26\cdots 31}$, $\frac{97\cdots 81}{66\cdots 75}a^{23}-\frac{51\cdots 61}{66\cdots 75}a^{22}+\frac{12\cdots 17}{66\cdots 75}a^{21}-\frac{76\cdots 54}{26\cdots 31}a^{20}+\frac{46\cdots 84}{66\cdots 75}a^{19}+\frac{12\cdots 51}{66\cdots 75}a^{18}+\frac{97\cdots 42}{66\cdots 75}a^{17}+\frac{31\cdots 03}{66\cdots 75}a^{16}-\frac{57\cdots 06}{66\cdots 75}a^{15}-\frac{11\cdots 73}{66\cdots 75}a^{14}+\frac{20\cdots 22}{66\cdots 75}a^{13}+\frac{62\cdots 22}{66\cdots 75}a^{12}+\frac{13\cdots 77}{66\cdots 75}a^{11}-\frac{13\cdots 84}{94\cdots 25}a^{10}-\frac{12\cdots 74}{94\cdots 25}a^{9}-\frac{23\cdots 27}{26\cdots 31}a^{8}+\frac{24\cdots 94}{66\cdots 75}a^{7}+\frac{10\cdots 18}{66\cdots 75}a^{6}+\frac{77\cdots 32}{66\cdots 75}a^{5}-\frac{89\cdots 86}{66\cdots 75}a^{4}-\frac{20\cdots 52}{66\cdots 75}a^{3}-\frac{69\cdots 62}{26\cdots 31}a^{2}-\frac{89\cdots 73}{66\cdots 75}a-\frac{63\cdots 61}{66\cdots 75}$, $\frac{34\cdots 58}{94\cdots 25}a^{23}+\frac{25\cdots 03}{94\cdots 25}a^{22}-\frac{10\cdots 63}{94\cdots 25}a^{21}+\frac{34\cdots 86}{94\cdots 25}a^{20}-\frac{10\cdots 28}{37\cdots 33}a^{19}-\frac{36\cdots 99}{94\cdots 25}a^{18}+\frac{35\cdots 81}{94\cdots 25}a^{17}-\frac{25\cdots 14}{94\cdots 25}a^{16}+\frac{88\cdots 12}{94\cdots 25}a^{15}+\frac{39\cdots 37}{94\cdots 25}a^{14}-\frac{15\cdots 41}{94\cdots 25}a^{13}-\frac{18\cdots 33}{94\cdots 25}a^{12}-\frac{14\cdots 06}{94\cdots 25}a^{11}+\frac{32\cdots 77}{94\cdots 25}a^{10}+\frac{23\cdots 34}{94\cdots 25}a^{9}-\frac{21\cdots 17}{94\cdots 25}a^{8}-\frac{77\cdots 37}{94\cdots 25}a^{7}-\frac{22\cdots 12}{94\cdots 25}a^{6}+\frac{28\cdots 02}{94\cdots 25}a^{5}+\frac{22\cdots 72}{94\cdots 25}a^{4}+\frac{35\cdots 58}{94\cdots 25}a^{3}+\frac{26\cdots 49}{94\cdots 25}a^{2}+\frac{10\cdots 18}{94\cdots 25}a+\frac{71\cdots 26}{94\cdots 25}$, $\frac{14\cdots 80}{26\cdots 31}a^{23}-\frac{27\cdots 57}{66\cdots 75}a^{22}+\frac{14\cdots 54}{66\cdots 75}a^{21}+\frac{34\cdots 74}{66\cdots 75}a^{20}+\frac{17\cdots 86}{66\cdots 75}a^{19}+\frac{10\cdots 86}{13\cdots 55}a^{18}+\frac{27\cdots 12}{66\cdots 75}a^{17}+\frac{10\cdots 78}{66\cdots 75}a^{16}+\frac{35\cdots 24}{66\cdots 75}a^{15}-\frac{33\cdots 18}{66\cdots 75}a^{14}-\frac{85\cdots 94}{66\cdots 75}a^{13}+\frac{21\cdots 32}{66\cdots 75}a^{12}+\frac{15\cdots 66}{66\cdots 75}a^{11}+\frac{31\cdots 91}{94\cdots 25}a^{10}-\frac{41\cdots 92}{94\cdots 25}a^{9}-\frac{15\cdots 58}{66\cdots 75}a^{8}+\frac{74\cdots 98}{13\cdots 55}a^{7}+\frac{63\cdots 59}{66\cdots 75}a^{6}+\frac{26\cdots 23}{66\cdots 75}a^{5}+\frac{10\cdots 17}{13\cdots 55}a^{4}+\frac{59\cdots 16}{66\cdots 75}a^{3}+\frac{38\cdots 27}{66\cdots 75}a^{2}+\frac{13\cdots 42}{66\cdots 75}a+\frac{87\cdots 11}{66\cdots 75}$, $\frac{12\cdots 77}{82\cdots 55}a^{23}+\frac{31\cdots 26}{41\cdots 75}a^{22}-\frac{13\cdots 36}{82\cdots 55}a^{21}+\frac{72\cdots 54}{41\cdots 75}a^{20}-\frac{28\cdots 59}{41\cdots 75}a^{19}-\frac{77\cdots 86}{41\cdots 75}a^{18}-\frac{69\cdots 62}{41\cdots 75}a^{17}-\frac{12\cdots 51}{41\cdots 75}a^{16}+\frac{34\cdots 34}{41\cdots 75}a^{15}+\frac{74\cdots 11}{41\cdots 75}a^{14}-\frac{12\cdots 16}{41\cdots 75}a^{13}-\frac{40\cdots 14}{41\cdots 75}a^{12}-\frac{90\cdots 06}{41\cdots 75}a^{11}+\frac{88\cdots 08}{59\cdots 25}a^{10}+\frac{78\cdots 72}{59\cdots 25}a^{9}+\frac{37\cdots 56}{41\cdots 75}a^{8}-\frac{15\cdots 58}{41\cdots 75}a^{7}-\frac{69\cdots 34}{41\cdots 75}a^{6}+\frac{32\cdots 31}{41\cdots 75}a^{5}+\frac{62\cdots 34}{41\cdots 75}a^{4}+\frac{13\cdots 96}{41\cdots 75}a^{3}+\frac{67\cdots 93}{41\cdots 75}a^{2}-\frac{20\cdots 99}{41\cdots 75}a-\frac{42\cdots 29}{82\cdots 55}$, $\frac{98\cdots 99}{66\cdots 75}a^{23}-\frac{74\cdots 56}{66\cdots 75}a^{22}+\frac{55\cdots 57}{13\cdots 55}a^{21}-\frac{53\cdots 92}{13\cdots 55}a^{20}+\frac{61\cdots 12}{66\cdots 75}a^{19}+\frac{82\cdots 47}{66\cdots 75}a^{18}+\frac{72\cdots 38}{13\cdots 55}a^{17}+\frac{16\cdots 24}{66\cdots 75}a^{16}-\frac{46\cdots 03}{66\cdots 75}a^{15}-\frac{78\cdots 32}{66\cdots 75}a^{14}+\frac{95\cdots 78}{13\cdots 55}a^{13}+\frac{45\cdots 23}{66\cdots 75}a^{12}+\frac{12\cdots 83}{13\cdots 55}a^{11}-\frac{13\cdots 21}{94\cdots 25}a^{10}-\frac{14\cdots 09}{18\cdots 65}a^{9}+\frac{22\cdots 56}{13\cdots 55}a^{8}+\frac{19\cdots 07}{66\cdots 75}a^{7}+\frac{51\cdots 38}{66\cdots 75}a^{6}-\frac{15\cdots 91}{13\cdots 55}a^{5}-\frac{57\cdots 96}{66\cdots 75}a^{4}-\frac{92\cdots 37}{66\cdots 75}a^{3}-\frac{11\cdots 99}{13\cdots 55}a^{2}-\frac{18\cdots 38}{66\cdots 75}a+\frac{57\cdots 74}{66\cdots 75}$, $\frac{30\cdots 79}{13\cdots 55}a^{23}+\frac{12\cdots 74}{66\cdots 75}a^{22}-\frac{66\cdots 81}{66\cdots 75}a^{21}+\frac{30\cdots 21}{66\cdots 75}a^{20}-\frac{19\cdots 28}{66\cdots 75}a^{19}-\frac{21\cdots 91}{13\cdots 55}a^{18}+\frac{44\cdots 10}{26\cdots 31}a^{17}-\frac{11\cdots 87}{66\cdots 75}a^{16}+\frac{56\cdots 99}{66\cdots 75}a^{15}+\frac{15\cdots 29}{66\cdots 75}a^{14}-\frac{79\cdots 88}{66\cdots 75}a^{13}-\frac{64\cdots 86}{66\cdots 75}a^{12}-\frac{38\cdots 48}{66\cdots 75}a^{11}+\frac{19\cdots 47}{94\cdots 25}a^{10}+\frac{12\cdots 76}{94\cdots 25}a^{9}-\frac{14\cdots 96}{66\cdots 75}a^{8}-\frac{29\cdots 46}{66\cdots 75}a^{7}-\frac{74\cdots 49}{66\cdots 75}a^{6}+\frac{24\cdots 81}{13\cdots 55}a^{5}+\frac{32\cdots 79}{26\cdots 31}a^{4}+\frac{12\cdots 82}{66\cdots 75}a^{3}+\frac{94\cdots 41}{66\cdots 75}a^{2}+\frac{38\cdots 43}{66\cdots 75}a+\frac{26\cdots 79}{66\cdots 75}$
|
| |
| Regulator: | \( 90673756995467580000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 90673756995467580000 \cdot 8}{2\cdot\sqrt{2295251366305746200663037028252313182580209076404571533203125}}\cr\approx \mathstrut & 0.367320478188184 \end{aligned}\] (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
| 6.2.196069503125.2, 12.4.1522544918455380058642578125.2 |
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 | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | $24$ | $24$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | $24$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.2.2a1.1 | $x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 5.1.20.23a2.2 | $x^{20} + 20 x^{4} + 10$ | $20$ | $1$ | $23$ | 20T20 | $not computed$ | |
|
\(89\)
| 89.1.4.3a1.1 | $x^{4} + 89$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 89.1.20.19a1.3 | $x^{20} + 801$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |