Normalized defining polynomial
\( x^{24} - 6 x^{23} + 47 x^{22} + 257 x^{21} + 4820 x^{20} + 13841 x^{19} - 139051 x^{18} + \cdots + 31122013545 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
| |
| Discriminant: |
\(270761008401829353605241639483649123576469719409942626953125\)
\(\medspace = 5^{39}\cdot 29^{22}\)
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| |
| Root discriminant: | \(299.47\) |
| |
| Galois root discriminant: | $5^{39/20}29^{19/20}\approx 565.287791942645$ | ||
| Ramified primes: |
\(5\), \(29\)
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{12}+\frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{13}+\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{17}-\frac{1}{3}a^{14}+\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{19}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{20}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{21}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{885}a^{22}+\frac{28}{885}a^{21}+\frac{25}{177}a^{20}+\frac{19}{177}a^{19}-\frac{1}{59}a^{18}-\frac{79}{885}a^{17}-\frac{7}{885}a^{16}-\frac{29}{177}a^{15}+\frac{29}{59}a^{14}+\frac{46}{177}a^{13}+\frac{57}{295}a^{12}-\frac{74}{295}a^{11}-\frac{71}{177}a^{10}+\frac{43}{177}a^{9}+\frac{73}{177}a^{8}+\frac{26}{885}a^{7}+\frac{51}{295}a^{6}-\frac{52}{177}a^{5}-\frac{24}{59}a^{4}-\frac{9}{59}a^{3}+\frac{176}{885}a^{2}-\frac{392}{885}a$, $\frac{1}{23\cdots 75}a^{23}-\frac{38\cdots 29}{23\cdots 75}a^{22}-\frac{97\cdots 98}{33\cdots 25}a^{21}+\frac{52\cdots 02}{46\cdots 35}a^{20}-\frac{49\cdots 92}{46\cdots 35}a^{19}+\frac{21\cdots 07}{77\cdots 25}a^{18}-\frac{14\cdots 59}{23\cdots 75}a^{17}+\frac{17\cdots 99}{23\cdots 75}a^{16}-\frac{30\cdots 64}{46\cdots 35}a^{15}-\frac{22\cdots 07}{66\cdots 05}a^{14}+\frac{30\cdots 61}{23\cdots 75}a^{13}-\frac{72\cdots 44}{23\cdots 75}a^{12}-\frac{44\cdots 62}{77\cdots 25}a^{11}-\frac{10\cdots 26}{31\cdots 09}a^{10}+\frac{19\cdots 64}{46\cdots 35}a^{9}-\frac{13\cdots 12}{33\cdots 25}a^{8}-\frac{20\cdots 32}{13\cdots 75}a^{7}-\frac{74\cdots 48}{33\cdots 25}a^{6}+\frac{27\cdots 74}{46\cdots 35}a^{5}+\frac{71\cdots 52}{46\cdots 35}a^{4}+\frac{57\cdots 06}{23\cdots 75}a^{3}+\frac{15\cdots 81}{11\cdots 75}a^{2}-\frac{36\cdots 32}{77\cdots 25}a+\frac{18\cdots 97}{37\cdots 65}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
| |
| Narrow class group: | $C_{4}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{31\cdots 98}{26\cdots 25}a^{23}+\frac{20\cdots 07}{26\cdots 25}a^{22}-\frac{22\cdots 61}{38\cdots 75}a^{21}-\frac{14\cdots 31}{53\cdots 45}a^{20}-\frac{28\cdots 49}{53\cdots 45}a^{19}-\frac{33\cdots 58}{26\cdots 25}a^{18}+\frac{15\cdots 99}{89\cdots 75}a^{17}-\frac{11\cdots 82}{26\cdots 25}a^{16}-\frac{27\cdots 03}{53\cdots 45}a^{15}-\frac{88\cdots 34}{76\cdots 35}a^{14}+\frac{51\cdots 99}{89\cdots 75}a^{13}+\frac{92\cdots 27}{26\cdots 25}a^{12}-\frac{51\cdots 84}{89\cdots 75}a^{11}-\frac{39\cdots 74}{35\cdots 43}a^{10}-\frac{29\cdots 57}{53\cdots 45}a^{9}-\frac{21\cdots 33}{12\cdots 25}a^{8}-\frac{91\cdots 38}{26\cdots 25}a^{7}-\frac{63\cdots 37}{12\cdots 25}a^{6}-\frac{82\cdots 89}{17\cdots 15}a^{5}-\frac{10\cdots 56}{53\cdots 45}a^{4}+\frac{11\cdots 62}{26\cdots 25}a^{3}-\frac{43\cdots 98}{12\cdots 25}a^{2}+\frac{18\cdots 26}{89\cdots 75}a+\frac{46\cdots 99}{43\cdots 55}$, $\frac{33\cdots 88}{53\cdots 45}a^{23}-\frac{24\cdots 67}{53\cdots 45}a^{22}+\frac{26\cdots 61}{76\cdots 35}a^{21}+\frac{43\cdots 43}{35\cdots 43}a^{20}+\frac{28\cdots 31}{10\cdots 29}a^{19}+\frac{23\cdots 98}{53\cdots 45}a^{18}-\frac{53\cdots 82}{53\cdots 45}a^{17}+\frac{30\cdots 28}{91\cdots 55}a^{16}+\frac{91\cdots 24}{35\cdots 43}a^{15}+\frac{14\cdots 50}{51\cdots 49}a^{14}-\frac{18\cdots 22}{53\cdots 45}a^{13}-\frac{26\cdots 89}{17\cdots 15}a^{12}+\frac{86\cdots 94}{17\cdots 15}a^{11}+\frac{19\cdots 23}{35\cdots 43}a^{10}+\frac{24\cdots 73}{10\cdots 29}a^{9}+\frac{16\cdots 43}{25\cdots 45}a^{8}+\frac{21\cdots 86}{17\cdots 15}a^{7}+\frac{12\cdots 46}{76\cdots 35}a^{6}+\frac{18\cdots 40}{10\cdots 29}a^{5}+\frac{12\cdots 01}{10\cdots 29}a^{4}+\frac{27\cdots 86}{17\cdots 15}a^{3}+\frac{35\cdots 38}{25\cdots 45}a^{2}-\frac{13\cdots 51}{17\cdots 15}a+\frac{55\cdots 37}{86\cdots 11}$, $\frac{30\cdots 08}{76\cdots 35}a^{23}+\frac{20\cdots 82}{76\cdots 35}a^{22}-\frac{15\cdots 62}{76\cdots 35}a^{21}-\frac{14\cdots 16}{15\cdots 47}a^{20}-\frac{29\cdots 33}{15\cdots 47}a^{19}-\frac{11\cdots 61}{25\cdots 45}a^{18}+\frac{45\cdots 82}{76\cdots 35}a^{17}-\frac{11\cdots 57}{76\cdots 35}a^{16}-\frac{26\cdots 60}{15\cdots 47}a^{15}-\frac{68\cdots 24}{15\cdots 47}a^{14}+\frac{14\cdots 27}{76\cdots 35}a^{13}+\frac{96\cdots 72}{76\cdots 35}a^{12}-\frac{45\cdots 19}{25\cdots 45}a^{11}-\frac{19\cdots 20}{51\cdots 49}a^{10}-\frac{30\cdots 16}{15\cdots 47}a^{9}-\frac{15\cdots 66}{25\cdots 45}a^{8}-\frac{96\cdots 38}{76\cdots 35}a^{7}-\frac{46\cdots 49}{25\cdots 45}a^{6}-\frac{86\cdots 60}{51\cdots 49}a^{5}-\frac{10\cdots 89}{15\cdots 47}a^{4}+\frac{68\cdots 84}{25\cdots 45}a^{3}-\frac{45\cdots 98}{76\cdots 35}a^{2}+\frac{97\cdots 36}{25\cdots 45}a+\frac{36\cdots 34}{86\cdots 11}$, $\frac{30\cdots 02}{26\cdots 25}a^{23}+\frac{22\cdots 18}{26\cdots 25}a^{22}-\frac{23\cdots 14}{38\cdots 75}a^{21}-\frac{12\cdots 64}{53\cdots 45}a^{20}-\frac{26\cdots 76}{53\cdots 45}a^{19}-\frac{66\cdots 89}{89\cdots 75}a^{18}+\frac{49\cdots 53}{26\cdots 25}a^{17}-\frac{15\cdots 43}{26\cdots 25}a^{16}-\frac{26\cdots 67}{53\cdots 45}a^{15}-\frac{43\cdots 96}{76\cdots 35}a^{14}+\frac{65\cdots 76}{89\cdots 75}a^{13}+\frac{75\cdots 23}{26\cdots 25}a^{12}-\frac{90\cdots 16}{89\cdots 75}a^{11}-\frac{11\cdots 19}{10\cdots 29}a^{10}-\frac{22\cdots 08}{53\cdots 45}a^{9}-\frac{12\cdots 17}{12\cdots 25}a^{8}-\frac{43\cdots 62}{26\cdots 25}a^{7}-\frac{24\cdots 88}{12\cdots 25}a^{6}-\frac{92\cdots 83}{53\cdots 45}a^{5}-\frac{35\cdots 94}{53\cdots 45}a^{4}+\frac{44\cdots 38}{26\cdots 25}a^{3}+\frac{71\cdots 19}{38\cdots 75}a^{2}-\frac{77\cdots 76}{89\cdots 75}a+\frac{31\cdots 01}{43\cdots 55}$, $\frac{26\cdots 73}{89\cdots 75}a^{23}+\frac{13\cdots 07}{89\cdots 75}a^{22}-\frac{42\cdots 58}{38\cdots 75}a^{21}-\frac{50\cdots 58}{53\cdots 45}a^{20}-\frac{25\cdots 54}{17\cdots 15}a^{19}-\frac{42\cdots 33}{89\cdots 75}a^{18}+\frac{11\cdots 16}{26\cdots 25}a^{17}-\frac{32\cdots 32}{89\cdots 75}a^{16}-\frac{86\cdots 09}{53\cdots 45}a^{15}-\frac{35\cdots 87}{76\cdots 35}a^{14}+\frac{44\cdots 16}{26\cdots 25}a^{13}+\frac{32\cdots 56}{26\cdots 25}a^{12}-\frac{25\cdots 31}{26\cdots 25}a^{11}-\frac{36\cdots 15}{10\cdots 29}a^{10}-\frac{92\cdots 56}{53\cdots 45}a^{9}-\frac{18\cdots 22}{38\cdots 75}a^{8}-\frac{23\cdots 14}{26\cdots 25}a^{7}-\frac{38\cdots 33}{38\cdots 75}a^{6}-\frac{44\cdots 66}{53\cdots 45}a^{5}-\frac{46\cdots 48}{53\cdots 45}a^{4}-\frac{88\cdots 63}{89\cdots 75}a^{3}-\frac{32\cdots 94}{12\cdots 25}a^{2}+\frac{38\cdots 17}{15\cdots 25}a-\frac{88\cdots 78}{43\cdots 55}$, $\frac{10\cdots 72}{77\cdots 25}a^{23}+\frac{13\cdots 73}{77\cdots 25}a^{22}-\frac{47\cdots 12}{33\cdots 25}a^{21}+\frac{42\cdots 81}{15\cdots 45}a^{20}-\frac{90\cdots 41}{15\cdots 45}a^{19}+\frac{45\cdots 64}{23\cdots 75}a^{18}+\frac{43\cdots 74}{23\cdots 75}a^{17}-\frac{41\cdots 19}{23\cdots 75}a^{16}+\frac{10\cdots 83}{15\cdots 45}a^{15}+\frac{58\cdots 57}{66\cdots 05}a^{14}+\frac{15\cdots 24}{23\cdots 75}a^{13}-\frac{30\cdots 66}{23\cdots 75}a^{12}-\frac{12\cdots 53}{77\cdots 25}a^{11}-\frac{11\cdots 91}{31\cdots 09}a^{10}-\frac{28\cdots 04}{46\cdots 35}a^{9}-\frac{86\cdots 11}{11\cdots 75}a^{8}-\frac{99\cdots 21}{23\cdots 75}a^{7}-\frac{35\cdots 87}{33\cdots 25}a^{6}-\frac{76\cdots 49}{46\cdots 35}a^{5}-\frac{38\cdots 67}{46\cdots 35}a^{4}+\frac{17\cdots 93}{77\cdots 25}a^{3}-\frac{12\cdots 73}{33\cdots 25}a^{2}+\frac{11\cdots 67}{77\cdots 25}a-\frac{44\cdots 17}{37\cdots 65}$, $\frac{76\cdots 11}{33\cdots 25}a^{23}+\frac{15\cdots 53}{11\cdots 75}a^{22}-\frac{11\cdots 53}{11\cdots 75}a^{21}-\frac{39\cdots 67}{66\cdots 05}a^{20}-\frac{73\cdots 63}{66\cdots 05}a^{19}-\frac{36\cdots 52}{11\cdots 75}a^{18}+\frac{10\cdots 39}{33\cdots 25}a^{17}-\frac{22\cdots 44}{33\cdots 25}a^{16}-\frac{69\cdots 96}{66\cdots 05}a^{15}-\frac{20\cdots 66}{66\cdots 05}a^{14}+\frac{30\cdots 79}{33\cdots 25}a^{13}+\frac{24\cdots 99}{33\cdots 25}a^{12}-\frac{19\cdots 59}{33\cdots 25}a^{11}-\frac{29\cdots 78}{13\cdots 61}a^{10}-\frac{82\cdots 24}{66\cdots 05}a^{9}-\frac{46\cdots 17}{11\cdots 75}a^{8}-\frac{33\cdots 06}{33\cdots 25}a^{7}-\frac{58\cdots 09}{33\cdots 25}a^{6}-\frac{15\cdots 04}{66\cdots 05}a^{5}-\frac{15\cdots 92}{66\cdots 05}a^{4}-\frac{63\cdots 41}{33\cdots 25}a^{3}-\frac{61\cdots 46}{33\cdots 25}a^{2}-\frac{57\cdots 49}{33\cdots 25}a+\frac{25\cdots 31}{37\cdots 65}$, $\frac{23\cdots 81}{23\cdots 75}a^{23}-\frac{14\cdots 54}{23\cdots 75}a^{22}+\frac{14\cdots 67}{33\cdots 25}a^{21}+\frac{44\cdots 74}{15\cdots 45}a^{20}+\frac{21\cdots 08}{46\cdots 35}a^{19}+\frac{30\cdots 26}{23\cdots 75}a^{18}-\frac{12\cdots 53}{77\cdots 25}a^{17}+\frac{20\cdots 18}{77\cdots 25}a^{16}+\frac{24\cdots 56}{46\cdots 35}a^{15}+\frac{78\cdots 93}{66\cdots 05}a^{14}-\frac{14\cdots 84}{23\cdots 75}a^{13}-\frac{84\cdots 94}{23\cdots 75}a^{12}+\frac{12\cdots 94}{23\cdots 75}a^{11}+\frac{10\cdots 42}{93\cdots 27}a^{10}+\frac{23\cdots 49}{46\cdots 35}a^{9}+\frac{45\cdots 53}{33\cdots 25}a^{8}+\frac{54\cdots 61}{23\cdots 75}a^{7}+\frac{90\cdots 17}{33\cdots 25}a^{6}+\frac{38\cdots 43}{15\cdots 45}a^{5}+\frac{43\cdots 39}{15\cdots 45}a^{4}+\frac{64\cdots 36}{23\cdots 75}a^{3}+\frac{18\cdots 43}{33\cdots 25}a^{2}+\frac{26\cdots 53}{77\cdots 25}a-\frac{82\cdots 53}{37\cdots 65}$, $\frac{31\cdots 47}{23\cdots 75}a^{23}+\frac{18\cdots 23}{23\cdots 75}a^{22}-\frac{21\cdots 54}{33\cdots 25}a^{21}-\frac{55\cdots 88}{15\cdots 45}a^{20}-\frac{10\cdots 52}{15\cdots 45}a^{19}-\frac{45\cdots 62}{23\cdots 75}a^{18}+\frac{14\cdots 36}{77\cdots 25}a^{17}-\frac{90\cdots 98}{23\cdots 75}a^{16}-\frac{29\cdots 77}{46\cdots 35}a^{15}-\frac{40\cdots 57}{22\cdots 35}a^{14}+\frac{12\cdots 58}{23\cdots 75}a^{13}+\frac{10\cdots 03}{23\cdots 75}a^{12}-\frac{80\cdots 78}{23\cdots 75}a^{11}-\frac{12\cdots 41}{93\cdots 27}a^{10}-\frac{11\cdots 91}{15\cdots 45}a^{9}-\frac{83\cdots 36}{33\cdots 25}a^{8}-\frac{13\cdots 57}{23\cdots 75}a^{7}-\frac{34\cdots 79}{33\cdots 25}a^{6}-\frac{21\cdots 26}{15\cdots 45}a^{5}-\frac{63\cdots 29}{46\cdots 35}a^{4}-\frac{87\cdots 69}{77\cdots 25}a^{3}-\frac{12\cdots 22}{11\cdots 75}a^{2}-\frac{23\cdots 58}{23\cdots 75}a+\frac{15\cdots 91}{37\cdots 65}$, $\frac{45\cdots 48}{13\cdots 61}a^{23}-\frac{40\cdots 71}{22\cdots 35}a^{22}+\frac{86\cdots 11}{66\cdots 05}a^{21}+\frac{14\cdots 15}{13\cdots 61}a^{20}+\frac{22\cdots 28}{13\cdots 61}a^{19}+\frac{73\cdots 80}{13\cdots 61}a^{18}-\frac{34\cdots 38}{66\cdots 05}a^{17}+\frac{29\cdots 61}{66\cdots 05}a^{16}+\frac{25\cdots 60}{13\cdots 61}a^{15}+\frac{23\cdots 56}{44\cdots 87}a^{14}-\frac{26\cdots 39}{13\cdots 61}a^{13}-\frac{95\cdots 93}{66\cdots 05}a^{12}+\frac{27\cdots 67}{22\cdots 35}a^{11}+\frac{53\cdots 31}{13\cdots 61}a^{10}+\frac{26\cdots 59}{13\cdots 61}a^{9}+\frac{73\cdots 25}{13\cdots 61}a^{8}+\frac{64\cdots 92}{66\cdots 05}a^{7}+\frac{72\cdots 96}{66\cdots 05}a^{6}+\frac{11\cdots 39}{13\cdots 61}a^{5}+\frac{12\cdots 98}{13\cdots 61}a^{4}+\frac{47\cdots 85}{44\cdots 87}a^{3}+\frac{45\cdots 79}{22\cdots 35}a^{2}-\frac{74\cdots 18}{22\cdots 35}a+\frac{35\cdots 00}{75\cdots 93}$, $\frac{10\cdots 87}{23\cdots 75}a^{23}+\frac{25\cdots 46}{77\cdots 25}a^{22}-\frac{80\cdots 39}{33\cdots 25}a^{21}-\frac{45\cdots 49}{46\cdots 35}a^{20}-\frac{31\cdots 72}{15\cdots 45}a^{19}-\frac{82\cdots 27}{23\cdots 75}a^{18}+\frac{17\cdots 98}{23\cdots 75}a^{17}-\frac{47\cdots 18}{23\cdots 75}a^{16}-\frac{31\cdots 94}{15\cdots 45}a^{15}-\frac{21\cdots 21}{66\cdots 05}a^{14}+\frac{70\cdots 68}{23\cdots 75}a^{13}+\frac{32\cdots 18}{23\cdots 75}a^{12}-\frac{26\cdots 41}{77\cdots 25}a^{11}-\frac{14\cdots 24}{31\cdots 09}a^{10}-\frac{29\cdots 26}{15\cdots 45}a^{9}-\frac{14\cdots 06}{33\cdots 25}a^{8}-\frac{36\cdots 89}{77\cdots 25}a^{7}+\frac{15\cdots 11}{33\cdots 25}a^{6}+\frac{13\cdots 37}{46\cdots 35}a^{5}+\frac{26\cdots 56}{46\cdots 35}a^{4}+\frac{39\cdots 76}{77\cdots 25}a^{3}-\frac{16\cdots 96}{33\cdots 25}a^{2}-\frac{40\cdots 51}{77\cdots 25}a+\frac{20\cdots 56}{37\cdots 65}$, $\frac{34\cdots 43}{23\cdots 75}a^{23}+\frac{12\cdots 02}{23\cdots 75}a^{22}-\frac{95\cdots 91}{33\cdots 25}a^{21}-\frac{32\cdots 11}{46\cdots 35}a^{20}-\frac{31\cdots 49}{46\cdots 35}a^{19}-\frac{26\cdots 51}{77\cdots 25}a^{18}+\frac{55\cdots 17}{23\cdots 75}a^{17}+\frac{91\cdots 11}{77\cdots 25}a^{16}-\frac{16\cdots 61}{15\cdots 45}a^{15}-\frac{48\cdots 23}{22\cdots 35}a^{14}+\frac{64\cdots 09}{77\cdots 25}a^{13}+\frac{51\cdots 99}{77\cdots 25}a^{12}-\frac{84\cdots 62}{23\cdots 75}a^{11}-\frac{17\cdots 92}{93\cdots 27}a^{10}-\frac{14\cdots 49}{15\cdots 45}a^{9}-\frac{96\cdots 09}{33\cdots 25}a^{8}-\frac{14\cdots 18}{23\cdots 75}a^{7}-\frac{29\cdots 66}{33\cdots 25}a^{6}-\frac{15\cdots 09}{15\cdots 45}a^{5}-\frac{99\cdots 77}{15\cdots 45}a^{4}-\frac{13\cdots 83}{23\cdots 75}a^{3}-\frac{14\cdots 28}{11\cdots 75}a^{2}+\frac{56\cdots 31}{77\cdots 25}a-\frac{36\cdots 36}{37\cdots 65}$, $\frac{16\cdots 01}{26\cdots 25}a^{23}+\frac{44\cdots 28}{89\cdots 75}a^{22}-\frac{14\cdots 82}{38\cdots 75}a^{21}-\frac{47\cdots 07}{53\cdots 45}a^{20}-\frac{15\cdots 83}{53\cdots 45}a^{19}-\frac{86\cdots 71}{26\cdots 25}a^{18}+\frac{25\cdots 64}{26\cdots 25}a^{17}-\frac{33\cdots 03}{89\cdots 75}a^{16}-\frac{38\cdots 07}{17\cdots 15}a^{15}-\frac{28\cdots 68}{76\cdots 35}a^{14}+\frac{89\cdots 39}{26\cdots 25}a^{13}+\frac{37\cdots 24}{26\cdots 25}a^{12}-\frac{41\cdots 33}{89\cdots 75}a^{11}-\frac{54\cdots 50}{10\cdots 29}a^{10}-\frac{41\cdots 73}{17\cdots 15}a^{9}-\frac{82\cdots 21}{12\cdots 25}a^{8}-\frac{34\cdots 06}{26\cdots 25}a^{7}-\frac{71\cdots 82}{38\cdots 75}a^{6}-\frac{10\cdots 04}{53\cdots 45}a^{5}-\frac{88\cdots 92}{53\cdots 45}a^{4}-\frac{27\cdots 31}{26\cdots 25}a^{3}-\frac{94\cdots 28}{38\cdots 75}a^{2}+\frac{10\cdots 62}{89\cdots 75}a-\frac{42\cdots 62}{43\cdots 55}$
|
| |
| Regulator: | \( 257526267927115430000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 257526267927115430000 \cdot 4}{2\cdot\sqrt{270761008401829353605241639483649123576469719409942626953125}}\cr\approx \mathstrut & 1.51871797997169 \end{aligned}\] (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
| 6.2.1381408203125.4, 12.4.8024353662494678497314453125.3 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | $24$ | R | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.8.0.1}{8} }^{3}$ | $24$ | $24$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 5.1.20.39a1.5 | $x^{20} + 50 x^{4} + 5$ | $20$ | $1$ | $39$ | 20T5 | $$[\frac{9}{4}]_{4}$$ | |
|
\(29\)
| 29.1.4.3a1.1 | $x^{4} + 29$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 29.1.20.19a1.3 | $x^{20} + 116$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |