Normalized defining polynomial
\( x^{24} - 10 x^{23} + 25 x^{22} + 1935 x^{21} - 10030 x^{20} - 34087 x^{19} + 1112160 x^{18} + \cdots - 119337300832275 \)
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: |
\(5\), \(89\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{12}-\frac{1}{5}a^{2}$, $\frac{1}{15}a^{18}+\frac{1}{15}a^{17}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{15}a^{3}-\frac{1}{15}a^{2}+\frac{1}{3}a$, $\frac{1}{15}a^{19}-\frac{1}{15}a^{17}-\frac{1}{5}a^{14}+\frac{1}{5}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{2}-\frac{1}{3}a$, $\frac{1}{15}a^{20}+\frac{1}{15}a^{17}-\frac{1}{5}a^{15}-\frac{1}{5}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{4}{15}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{2}{5}a^{2}+\frac{1}{3}a$, $\frac{1}{15}a^{21}-\frac{1}{15}a^{17}-\frac{1}{5}a^{16}-\frac{2}{15}a^{12}-\frac{1}{3}a^{11}+\frac{4}{15}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{2}{5}a^{2}-\frac{1}{3}a$, $\frac{1}{3075}a^{22}+\frac{38}{3075}a^{21}+\frac{31}{1025}a^{20}-\frac{59}{3075}a^{19}-\frac{1}{41}a^{18}+\frac{112}{3075}a^{17}-\frac{148}{1025}a^{16}-\frac{68}{1025}a^{15}+\frac{209}{1025}a^{14}-\frac{44}{123}a^{13}-\frac{19}{205}a^{12}-\frac{46}{123}a^{11}-\frac{61}{615}a^{10}-\frac{22}{123}a^{9}+\frac{236}{615}a^{8}-\frac{161}{3075}a^{7}+\frac{1447}{3075}a^{6}+\frac{607}{3075}a^{5}+\frac{498}{1025}a^{4}+\frac{221}{615}a^{3}+\frac{77}{205}a^{2}-\frac{16}{123}a-\frac{15}{41}$, $\frac{1}{49\cdots 75}a^{23}-\frac{75\cdots 72}{49\cdots 75}a^{22}-\frac{46\cdots 07}{49\cdots 75}a^{21}-\frac{47\cdots 23}{16\cdots 25}a^{20}+\frac{19\cdots 93}{19\cdots 63}a^{19}-\frac{95\cdots 48}{49\cdots 75}a^{18}+\frac{59\cdots 32}{16\cdots 25}a^{17}-\frac{16\cdots 06}{55\cdots 75}a^{16}+\frac{64\cdots 69}{16\cdots 25}a^{15}+\frac{27\cdots 51}{19\cdots 63}a^{14}-\frac{98\cdots 82}{33\cdots 05}a^{13}-\frac{44\cdots 49}{11\cdots 35}a^{12}-\frac{29\cdots 71}{99\cdots 15}a^{11}-\frac{51\cdots 16}{22\cdots 07}a^{10}+\frac{45\cdots 41}{99\cdots 15}a^{9}-\frac{68\cdots 62}{16\cdots 25}a^{8}-\frac{10\cdots 68}{49\cdots 75}a^{7}-\frac{61\cdots 06}{16\cdots 25}a^{6}-\frac{34\cdots 32}{16\cdots 25}a^{5}-\frac{52\cdots 09}{99\cdots 15}a^{4}-\frac{48\cdots 72}{99\cdots 15}a^{3}+\frac{45\cdots 18}{99\cdots 15}a^{2}+\frac{52\cdots 15}{66\cdots 21}a+\frac{10\cdots 68}{73\cdots 69}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{15\cdots 46}{81\cdots 05}a^{23}-\frac{34\cdots 96}{16\cdots 81}a^{22}+\frac{60\cdots 23}{81\cdots 05}a^{21}+\frac{96\cdots 84}{27\cdots 35}a^{20}-\frac{19\cdots 42}{81\cdots 05}a^{19}-\frac{27\cdots 17}{81\cdots 05}a^{18}+\frac{11\cdots 28}{54\cdots 27}a^{17}-\frac{41\cdots 31}{90\cdots 45}a^{16}-\frac{15\cdots 17}{27\cdots 35}a^{15}+\frac{24\cdots 16}{81\cdots 05}a^{14}+\frac{14\cdots 99}{27\cdots 35}a^{13}-\frac{59\cdots 58}{54\cdots 27}a^{12}-\frac{64\cdots 17}{16\cdots 81}a^{11}-\frac{18\cdots 67}{18\cdots 09}a^{10}-\frac{63\cdots 33}{16\cdots 81}a^{9}-\frac{41\cdots 22}{27\cdots 35}a^{8}-\frac{94\cdots 14}{16\cdots 81}a^{7}-\frac{27\cdots 26}{27\cdots 35}a^{6}+\frac{94\cdots 42}{90\cdots 45}a^{5}-\frac{77\cdots 08}{81\cdots 05}a^{4}+\frac{11\cdots 44}{81\cdots 05}a^{3}-\frac{11\cdots 76}{16\cdots 81}a^{2}-\frac{47\cdots 79}{54\cdots 27}a-\frac{31\cdots 57}{18\cdots 09}$, $\frac{40\cdots 58}{29\cdots 05}a^{23}+\frac{85\cdots 59}{58\cdots 81}a^{22}-\frac{36\cdots 01}{88\cdots 15}a^{21}-\frac{15\cdots 83}{58\cdots 81}a^{20}+\frac{13\cdots 17}{88\cdots 15}a^{19}+\frac{11\cdots 29}{29\cdots 05}a^{18}-\frac{92\cdots 58}{58\cdots 81}a^{17}+\frac{62\cdots 21}{29\cdots 05}a^{16}+\frac{27\cdots 92}{58\cdots 81}a^{15}-\frac{55\cdots 62}{29\cdots 05}a^{14}-\frac{35\cdots 32}{58\cdots 81}a^{13}+\frac{14\cdots 63}{17\cdots 43}a^{12}+\frac{63\cdots 83}{17\cdots 43}a^{11}+\frac{16\cdots 06}{17\cdots 43}a^{10}+\frac{56\cdots 69}{17\cdots 43}a^{9}+\frac{11\cdots 99}{88\cdots 15}a^{8}+\frac{85\cdots 28}{17\cdots 43}a^{7}+\frac{92\cdots 67}{29\cdots 05}a^{6}-\frac{60\cdots 98}{58\cdots 81}a^{5}+\frac{19\cdots 56}{29\cdots 05}a^{4}-\frac{80\cdots 91}{17\cdots 43}a^{3}-\frac{54\cdots 47}{58\cdots 81}a^{2}+\frac{12\cdots 25}{58\cdots 81}a+\frac{68\cdots 64}{58\cdots 81}$, $\frac{32\cdots 54}{81\cdots 05}a^{23}+\frac{41\cdots 38}{81\cdots 05}a^{22}-\frac{17\cdots 24}{81\cdots 05}a^{21}-\frac{65\cdots 38}{90\cdots 45}a^{20}+\frac{49\cdots 76}{81\cdots 05}a^{19}+\frac{43\cdots 12}{81\cdots 05}a^{18}-\frac{12\cdots 78}{27\cdots 35}a^{17}+\frac{14\cdots 88}{90\cdots 45}a^{16}+\frac{30\cdots 47}{27\cdots 35}a^{15}-\frac{67\cdots 68}{81\cdots 05}a^{14}-\frac{60\cdots 83}{18\cdots 09}a^{13}+\frac{14\cdots 98}{54\cdots 27}a^{12}+\frac{78\cdots 84}{16\cdots 81}a^{11}+\frac{28\cdots 56}{54\cdots 27}a^{10}+\frac{68\cdots 95}{16\cdots 81}a^{9}+\frac{49\cdots 43}{27\cdots 35}a^{8}+\frac{53\cdots 22}{81\cdots 05}a^{7}-\frac{17\cdots 99}{90\cdots 45}a^{6}-\frac{94\cdots 26}{27\cdots 35}a^{5}+\frac{20\cdots 64}{81\cdots 05}a^{4}-\frac{97\cdots 56}{16\cdots 81}a^{3}+\frac{44\cdots 87}{16\cdots 81}a^{2}+\frac{51\cdots 15}{54\cdots 27}a-\frac{23\cdots 49}{18\cdots 09}$, $\frac{12\cdots 49}{58\cdots 81}a^{23}+\frac{13\cdots 56}{58\cdots 81}a^{22}-\frac{20\cdots 17}{29\cdots 05}a^{21}-\frac{12\cdots 83}{29\cdots 05}a^{20}+\frac{72\cdots 43}{29\cdots 05}a^{19}+\frac{36\cdots 80}{58\cdots 81}a^{18}-\frac{14\cdots 77}{58\cdots 81}a^{17}+\frac{10\cdots 91}{29\cdots 05}a^{16}+\frac{21\cdots 39}{29\cdots 05}a^{15}-\frac{88\cdots 84}{29\cdots 05}a^{14}-\frac{54\cdots 96}{58\cdots 81}a^{13}+\frac{74\cdots 89}{58\cdots 81}a^{12}+\frac{32\cdots 34}{58\cdots 81}a^{11}+\frac{82\cdots 28}{58\cdots 81}a^{10}+\frac{29\cdots 13}{58\cdots 81}a^{9}+\frac{11\cdots 86}{58\cdots 81}a^{8}+\frac{44\cdots 09}{58\cdots 81}a^{7}+\frac{12\cdots 02}{29\cdots 05}a^{6}-\frac{48\cdots 12}{29\cdots 05}a^{5}+\frac{30\cdots 07}{29\cdots 05}a^{4}-\frac{47\cdots 65}{58\cdots 81}a^{3}-\frac{80\cdots 89}{58\cdots 81}a^{2}+\frac{20\cdots 40}{58\cdots 81}a+\frac{85\cdots 26}{58\cdots 81}$, $\frac{52\cdots 54}{81\cdots 05}a^{23}-\frac{49\cdots 17}{16\cdots 81}a^{22}+\frac{71\cdots 18}{81\cdots 05}a^{21}-\frac{42\cdots 76}{27\cdots 35}a^{20}-\frac{93\cdots 21}{81\cdots 05}a^{19}+\frac{11\cdots 12}{81\cdots 05}a^{18}-\frac{27\cdots 67}{54\cdots 27}a^{17}-\frac{91\cdots 01}{90\cdots 45}a^{16}+\frac{11\cdots 13}{27\cdots 35}a^{15}+\frac{18\cdots 48}{81\cdots 05}a^{14}-\frac{94\cdots 11}{54\cdots 27}a^{13}+\frac{38\cdots 81}{18\cdots 09}a^{12}+\frac{12\cdots 93}{16\cdots 81}a^{11}+\frac{14\cdots 96}{54\cdots 27}a^{10}+\frac{11\cdots 45}{16\cdots 81}a^{9}+\frac{23\cdots 96}{90\cdots 45}a^{8}+\frac{16\cdots 58}{16\cdots 81}a^{7}+\frac{88\cdots 59}{27\cdots 35}a^{6}+\frac{20\cdots 22}{90\cdots 45}a^{5}-\frac{37\cdots 74}{81\cdots 05}a^{4}+\frac{78\cdots 65}{16\cdots 81}a^{3}-\frac{10\cdots 42}{16\cdots 81}a^{2}-\frac{15\cdots 60}{54\cdots 27}a+\frac{32\cdots 08}{18\cdots 09}$, $\frac{23\cdots 89}{15\cdots 55}a^{23}-\frac{16\cdots 49}{15\cdots 55}a^{22}-\frac{40\cdots 91}{15\cdots 55}a^{21}+\frac{43\cdots 46}{15\cdots 55}a^{20}-\frac{19\cdots 84}{52\cdots 85}a^{19}-\frac{15\cdots 88}{15\cdots 55}a^{18}+\frac{59\cdots 29}{52\cdots 85}a^{17}+\frac{30\cdots 26}{52\cdots 85}a^{16}-\frac{26\cdots 66}{52\cdots 85}a^{15}-\frac{24\cdots 79}{15\cdots 55}a^{14}+\frac{25\cdots 48}{15\cdots 55}a^{13}-\frac{60\cdots 83}{31\cdots 31}a^{12}-\frac{87\cdots 77}{10\cdots 77}a^{11}-\frac{30\cdots 06}{10\cdots 77}a^{10}-\frac{79\cdots 49}{31\cdots 31}a^{9}+\frac{14\cdots 21}{15\cdots 55}a^{8}+\frac{47\cdots 29}{15\cdots 55}a^{7}-\frac{14\cdots 79}{15\cdots 55}a^{6}+\frac{36\cdots 64}{15\cdots 55}a^{5}+\frac{56\cdots 82}{15\cdots 55}a^{4}-\frac{58\cdots 78}{52\cdots 85}a^{3}+\frac{91\cdots 88}{31\cdots 31}a^{2}+\frac{80\cdots 04}{10\cdots 77}a+\frac{12\cdots 07}{10\cdots 77}$, $\frac{12\cdots 29}{18\cdots 25}a^{23}+\frac{36\cdots 22}{55\cdots 75}a^{22}-\frac{73\cdots 32}{55\cdots 75}a^{21}-\frac{75\cdots 08}{55\cdots 75}a^{20}+\frac{36\cdots 28}{55\cdots 75}a^{19}+\frac{52\cdots 57}{18\cdots 25}a^{18}-\frac{14\cdots 87}{18\cdots 25}a^{17}+\frac{77\cdots 02}{18\cdots 25}a^{16}+\frac{46\cdots 58}{18\cdots 25}a^{15}-\frac{14\cdots 03}{18\cdots 25}a^{14}-\frac{86\cdots 65}{22\cdots 07}a^{13}+\frac{14\cdots 08}{36\cdots 45}a^{12}+\frac{23\cdots 97}{11\cdots 35}a^{11}+\frac{21\cdots 39}{36\cdots 45}a^{10}+\frac{20\cdots 28}{11\cdots 35}a^{9}+\frac{13\cdots 99}{18\cdots 25}a^{8}+\frac{16\cdots 28}{55\cdots 75}a^{7}+\frac{54\cdots 54}{18\cdots 25}a^{6}-\frac{93\cdots 54}{18\cdots 25}a^{5}+\frac{16\cdots 42}{55\cdots 75}a^{4}-\frac{19\cdots 97}{11\cdots 35}a^{3}-\frac{60\cdots 49}{90\cdots 45}a^{2}+\frac{49\cdots 43}{73\cdots 69}a+\frac{12\cdots 11}{73\cdots 69}$, $\frac{17\cdots 72}{16\cdots 25}a^{23}+\frac{57\cdots 41}{66\cdots 21}a^{22}-\frac{10\cdots 03}{16\cdots 25}a^{21}-\frac{11\cdots 13}{55\cdots 75}a^{20}+\frac{11\cdots 51}{16\cdots 25}a^{19}+\frac{95\cdots 81}{16\cdots 25}a^{18}-\frac{12\cdots 68}{11\cdots 35}a^{17}-\frac{21\cdots 44}{18\cdots 25}a^{16}+\frac{21\cdots 64}{55\cdots 75}a^{15}-\frac{86\cdots 78}{16\cdots 25}a^{14}-\frac{29\cdots 62}{36\cdots 45}a^{13}+\frac{34\cdots 64}{73\cdots 69}a^{12}+\frac{13\cdots 62}{33\cdots 05}a^{11}+\frac{14\cdots 23}{11\cdots 35}a^{10}+\frac{26\cdots 03}{81\cdots 05}a^{9}+\frac{19\cdots 58}{18\cdots 25}a^{8}+\frac{14\cdots 79}{33\cdots 05}a^{7}+\frac{14\cdots 82}{18\cdots 25}a^{6}-\frac{66\cdots 02}{55\cdots 75}a^{5}-\frac{21\cdots 51}{16\cdots 25}a^{4}+\frac{18\cdots 19}{66\cdots 21}a^{3}+\frac{30\cdots 58}{66\cdots 21}a^{2}+\frac{70\cdots 64}{22\cdots 07}a+\frac{31\cdots 83}{73\cdots 69}$, $\frac{75\cdots 56}{31\cdots 31}a^{23}-\frac{10\cdots 86}{52\cdots 85}a^{22}+\frac{15\cdots 24}{52\cdots 85}a^{21}+\frac{73\cdots 27}{15\cdots 55}a^{20}-\frac{27\cdots 66}{15\cdots 55}a^{19}-\frac{16\cdots 72}{15\cdots 55}a^{18}+\frac{13\cdots 63}{52\cdots 85}a^{17}+\frac{72\cdots 78}{52\cdots 85}a^{16}-\frac{42\cdots 97}{52\cdots 85}a^{15}+\frac{25\cdots 93}{15\cdots 55}a^{14}+\frac{22\cdots 36}{15\cdots 55}a^{13}-\frac{11\cdots 60}{10\cdots 77}a^{12}-\frac{26\cdots 09}{31\cdots 31}a^{11}-\frac{99\cdots 56}{31\cdots 31}a^{10}-\frac{33\cdots 95}{31\cdots 31}a^{9}-\frac{40\cdots 51}{10\cdots 77}a^{8}-\frac{22\cdots 52}{15\cdots 55}a^{7}-\frac{44\cdots 97}{15\cdots 55}a^{6}-\frac{25\cdots 77}{15\cdots 55}a^{5}-\frac{49\cdots 53}{52\cdots 85}a^{4}-\frac{85\cdots 48}{15\cdots 55}a^{3}+\frac{61\cdots 02}{31\cdots 31}a^{2}+\frac{26\cdots 18}{10\cdots 77}a+\frac{16\cdots 98}{10\cdots 77}$, $\frac{17\cdots 06}{16\cdots 25}a^{23}-\frac{53\cdots 24}{33\cdots 05}a^{22}+\frac{17\cdots 49}{16\cdots 25}a^{21}+\frac{88\cdots 74}{55\cdots 75}a^{20}-\frac{31\cdots 03}{16\cdots 25}a^{19}+\frac{92\cdots 32}{16\cdots 25}a^{18}+\frac{10\cdots 47}{11\cdots 35}a^{17}-\frac{11\cdots 73}{18\cdots 25}a^{16}-\frac{41\cdots 47}{55\cdots 75}a^{15}+\frac{30\cdots 34}{16\cdots 25}a^{14}-\frac{15\cdots 02}{36\cdots 45}a^{13}-\frac{45\cdots 77}{11\cdots 35}a^{12}-\frac{28\cdots 96}{33\cdots 05}a^{11}-\frac{11\cdots 04}{27\cdots 35}a^{10}-\frac{26\cdots 79}{33\cdots 05}a^{9}-\frac{37\cdots 27}{55\cdots 75}a^{8}-\frac{35\cdots 03}{33\cdots 05}a^{7}+\frac{39\cdots 94}{18\cdots 25}a^{6}-\frac{47\cdots 29}{55\cdots 75}a^{5}+\frac{71\cdots 53}{16\cdots 25}a^{4}+\frac{46\cdots 46}{33\cdots 05}a^{3}-\frac{47\cdots 92}{33\cdots 05}a^{2}-\frac{55\cdots 51}{22\cdots 07}a-\frac{16\cdots 10}{73\cdots 69}$, $\frac{59\cdots 13}{40\cdots 25}a^{23}-\frac{28\cdots 78}{16\cdots 25}a^{22}+\frac{10\cdots 03}{16\cdots 25}a^{21}+\frac{15\cdots 24}{55\cdots 75}a^{20}-\frac{32\cdots 42}{16\cdots 25}a^{19}-\frac{39\cdots 39}{16\cdots 25}a^{18}+\frac{94\cdots 38}{55\cdots 75}a^{17}-\frac{75\cdots 11}{18\cdots 25}a^{16}-\frac{25\cdots 47}{55\cdots 75}a^{15}+\frac{41\cdots 26}{16\cdots 25}a^{14}+\frac{87\cdots 25}{22\cdots 07}a^{13}-\frac{33\cdots 39}{36\cdots 45}a^{12}-\frac{94\cdots 48}{33\cdots 05}a^{11}-\frac{22\cdots 02}{36\cdots 45}a^{10}-\frac{84\cdots 52}{33\cdots 05}a^{9}-\frac{57\cdots 41}{55\cdots 75}a^{8}-\frac{64\cdots 72}{16\cdots 25}a^{7}+\frac{89\cdots 34}{55\cdots 75}a^{6}+\frac{66\cdots 86}{55\cdots 75}a^{5}-\frac{13\cdots 13}{16\cdots 25}a^{4}+\frac{42\cdots 83}{33\cdots 05}a^{3}+\frac{21\cdots 78}{33\cdots 05}a^{2}-\frac{64\cdots 07}{22\cdots 07}a+\frac{11\cdots 39}{73\cdots 69}$, $\frac{75\cdots 96}{49\cdots 75}a^{23}+\frac{67\cdots 64}{49\cdots 75}a^{22}-\frac{83\cdots 87}{49\cdots 75}a^{21}-\frac{10\cdots 63}{33\cdots 05}a^{20}+\frac{60\cdots 57}{49\cdots 75}a^{19}+\frac{39\cdots 38}{49\cdots 75}a^{18}-\frac{27\cdots 09}{16\cdots 25}a^{17}-\frac{40\cdots 86}{55\cdots 75}a^{16}+\frac{19\cdots 89}{33\cdots 05}a^{15}-\frac{53\cdots 71}{49\cdots 75}a^{14}-\frac{37\cdots 69}{33\cdots 05}a^{13}+\frac{82\cdots 77}{11\cdots 35}a^{12}+\frac{56\cdots 76}{99\cdots 15}a^{11}+\frac{17\cdots 17}{11\cdots 35}a^{10}+\frac{32\cdots 49}{99\cdots 15}a^{9}+\frac{18\cdots 97}{16\cdots 25}a^{8}+\frac{27\cdots 06}{49\cdots 75}a^{7}+\frac{11\cdots 19}{16\cdots 25}a^{6}-\frac{92\cdots 76}{33\cdots 05}a^{5}-\frac{33\cdots 42}{49\cdots 75}a^{4}+\frac{76\cdots 23}{99\cdots 15}a^{3}+\frac{83\cdots 11}{99\cdots 15}a^{2}+\frac{28\cdots 03}{66\cdots 21}a-\frac{35\cdots 84}{73\cdots 69}$, $\frac{14\cdots 69}{36\cdots 45}a^{23}-\frac{24\cdots 38}{55\cdots 75}a^{22}+\frac{23\cdots 72}{18\cdots 25}a^{21}+\frac{43\cdots 96}{55\cdots 75}a^{20}-\frac{25\cdots 58}{55\cdots 75}a^{19}-\frac{40\cdots 98}{36\cdots 45}a^{18}+\frac{85\cdots 63}{18\cdots 25}a^{17}-\frac{12\cdots 61}{18\cdots 25}a^{16}-\frac{25\cdots 71}{18\cdots 25}a^{15}+\frac{10\cdots 33}{18\cdots 25}a^{14}+\frac{18\cdots 04}{11\cdots 35}a^{13}-\frac{26\cdots 46}{11\cdots 35}a^{12}-\frac{22\cdots 66}{22\cdots 07}a^{11}-\frac{92\cdots 94}{36\cdots 45}a^{10}-\frac{66\cdots 33}{73\cdots 69}a^{9}-\frac{19\cdots 85}{54\cdots 27}a^{8}-\frac{76\cdots 07}{55\cdots 75}a^{7}-\frac{39\cdots 71}{55\cdots 75}a^{6}+\frac{16\cdots 79}{55\cdots 75}a^{5}-\frac{10\cdots 22}{55\cdots 75}a^{4}+\frac{37\cdots 06}{22\cdots 07}a^{3}+\frac{85\cdots 51}{36\cdots 45}a^{2}-\frac{48\cdots 29}{73\cdots 69}a-\frac{16\cdots 20}{73\cdots 69}$
|
| |
| Regulator: | \( 158466950947891900000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 158466950947891900000 \cdot 32}{2\cdot\sqrt{2295251366305746200663037028252313182580209076404571533203125}}\cr\approx \mathstrut & 2.56780608316970 \end{aligned}\] (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
| 6.2.196069503125.1, 12.4.1522544918455380058642578125.3 |
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 | $24$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | $24$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | $24$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.10.11a2.1 | $x^{10} + 5 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
| 5.1.10.11a2.1 | $x^{10} + 5 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
|
\(89\)
| 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 | $$[\ ]_{20}^{2}$$ |