Number field 24.4.91810054652229848026521481130092527303208363056182861328125.6

• Label: 24.4.918...125.6
• Degree: $24$
• Signature: $(4, 10)$
• Discriminant: $9.181\times 10^{58}$
• Root discriminant: \(286.28\)
• Ramified primes: $5,89$
• Class number: $16$ (GRH)
• Class group: [2, 2, 4] (GRH)
• Galois group: $\GL(2,5)$ (as 24T1353)

Normalized defining polynomial

x^24 - 120*x^22 - 10*x^21 + 6905*x^20 + 799*x^19 - 278060*x^18 + 173055*x^17 + 7887200*x^16 - 16327735*x^15 - 133024731*x^14 + 532379580*x^13 + 1177337010*x^12 - 9573340425*x^11 - 2946551855*x^10 + 125420329119*x^9 - 144462051275*x^8 - 959076041810*x^7 + 2708721649225*x^6 + 397551181590*x^5 - 9951527396947*x^4 + 12150299345835*x^3 + 5468893445760*x^2 - 21535626575215*x + 14471246844265

Invariants

Degree:		$24$
Signature:		$(4, 10)$
Discriminant:		\(91810054652229848026521481130092527303208363056182861328125\) \(\medspace = 5^{23}\cdot 89^{22}\)
Root discriminant:		\(286.28\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{14}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{15}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{18}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{15}a^{19}+\frac{1}{3}a^{15}+\frac{1}{15}a^{14}+\frac{1}{15}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{15}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{15}a^{20}+\frac{1}{15}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{11}+\frac{1}{15}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{15}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{15}a^{21}+\frac{1}{15}a^{16}-\frac{1}{3}a^{15}-\frac{1}{3}a^{12}+\frac{1}{15}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{15}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{675}a^{22}+\frac{1}{75}a^{21}+\frac{17}{675}a^{20}+\frac{17}{675}a^{19}-\frac{1}{15}a^{18}-\frac{49}{675}a^{17}+\frac{4}{675}a^{16}+\frac{22}{675}a^{15}+\frac{67}{675}a^{14}-\frac{61}{135}a^{13}-\frac{169}{675}a^{12}+\frac{179}{675}a^{11}-\frac{218}{675}a^{10}-\frac{143}{675}a^{9}-\frac{4}{15}a^{8}-\frac{4}{675}a^{7}-\frac{41}{675}a^{6}-\frac{103}{675}a^{5}-\frac{17}{75}a^{4}-\frac{52}{135}a^{3}+\frac{2}{27}a^{2}+\frac{2}{27}a+\frac{32}{135}$, $\frac{1}{17\cdots 25}a^{23}-\frac{16\cdots 47}{59\cdots 75}a^{22}+\frac{98\cdots 22}{17\cdots 25}a^{21}-\frac{56\cdots 73}{17\cdots 25}a^{20}+\frac{18\cdots 26}{11\cdots 15}a^{19}+\frac{97\cdots 51}{17\cdots 25}a^{18}-\frac{57\cdots 96}{17\cdots 25}a^{17}-\frac{26\cdots 48}{17\cdots 25}a^{16}+\frac{46\cdots 27}{17\cdots 25}a^{15}+\frac{13\cdots 21}{35\cdots 45}a^{14}-\frac{39\cdots 44}{17\cdots 25}a^{13}+\frac{60\cdots 54}{17\cdots 25}a^{12}+\frac{13\cdots 12}{17\cdots 25}a^{11}+\frac{81\cdots 17}{17\cdots 25}a^{10}+\frac{19\cdots 27}{11\cdots 15}a^{9}+\frac{71\cdots 46}{17\cdots 25}a^{8}-\frac{19\cdots 66}{17\cdots 25}a^{7}-\frac{72\cdots 98}{17\cdots 25}a^{6}+\frac{10\cdots 19}{59\cdots 75}a^{5}+\frac{29\cdots 34}{71\cdots 29}a^{4}-\frac{83\cdots 54}{71\cdots 29}a^{3}+\frac{10\cdots 65}{71\cdots 29}a^{2}+\frac{14\cdots 82}{35\cdots 45}a-\frac{75\cdots 07}{23\cdots 43}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH)
Narrow class group:		$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{25\cdots 06}{80\cdots 75}a^{23}+\frac{30\cdots 62}{16\cdots 35}a^{22}-\frac{27\cdots 79}{80\cdots 75}a^{21}-\frac{55\cdots 06}{80\cdots 75}a^{20}+\frac{14\cdots 37}{80\cdots 75}a^{19}-\frac{20\cdots 39}{80\cdots 75}a^{18}-\frac{36\cdots 88}{53\cdots 45}a^{17}+\frac{71\cdots 46}{80\cdots 75}a^{16}+\frac{14\cdots 24}{80\cdots 75}a^{15}-\frac{47\cdots 48}{80\cdots 75}a^{14}-\frac{20\cdots 91}{89\cdots 75}a^{13}+\frac{25\cdots 27}{16\cdots 35}a^{12}+\frac{34\cdots 09}{89\cdots 75}a^{11}-\frac{17\cdots 51}{80\cdots 75}a^{10}+\frac{22\cdots 67}{80\cdots 75}a^{9}+\frac{18\cdots 96}{80\cdots 75}a^{8}-\frac{40\cdots 03}{53\cdots 45}a^{7}-\frac{22\cdots 58}{26\cdots 25}a^{6}+\frac{67\cdots 79}{80\cdots 75}a^{5}-\frac{12\cdots 73}{80\cdots 75}a^{4}-\frac{70\cdots 54}{53\cdots 45}a^{3}+\frac{11\cdots 09}{32\cdots 27}a^{2}-\frac{27\cdots 36}{53\cdots 45}a+\frac{40\cdots 67}{16\cdots 35}$, $\frac{69\cdots 78}{10\cdots 75}a^{23}+\frac{66\cdots 03}{33\cdots 25}a^{22}-\frac{82\cdots 16}{10\cdots 75}a^{21}-\frac{48\cdots 19}{20\cdots 15}a^{20}+\frac{15\cdots 33}{33\cdots 25}a^{19}+\frac{13\cdots 38}{10\cdots 75}a^{18}-\frac{18\cdots 31}{10\cdots 75}a^{17}-\frac{41\cdots 11}{10\cdots 75}a^{16}+\frac{10\cdots 83}{20\cdots 15}a^{15}+\frac{37\cdots 19}{10\cdots 75}a^{14}-\frac{11\cdots 17}{10\cdots 75}a^{13}+\frac{10\cdots 04}{10\cdots 75}a^{12}+\frac{16\cdots 19}{10\cdots 75}a^{11}-\frac{75\cdots 99}{20\cdots 15}a^{10}-\frac{60\cdots 67}{33\cdots 25}a^{9}+\frac{68\cdots 03}{10\cdots 75}a^{8}+\frac{12\cdots 99}{10\cdots 75}a^{7}-\frac{76\cdots 01}{10\cdots 75}a^{6}+\frac{18\cdots 27}{67\cdots 05}a^{5}+\frac{37\cdots 64}{10\cdots 75}a^{4}-\frac{85\cdots 99}{20\cdots 15}a^{3}-\frac{11\cdots 57}{40\cdots 03}a^{2}+\frac{21\cdots 96}{20\cdots 15}a-\frac{29\cdots 07}{67\cdots 05}$, $\frac{15\cdots 36}{16\cdots 35}a^{23}-\frac{23\cdots 16}{26\cdots 25}a^{22}+\frac{90\cdots 18}{80\cdots 75}a^{21}+\frac{94\cdots 39}{80\cdots 75}a^{20}-\frac{17\cdots 37}{26\cdots 25}a^{19}-\frac{11\cdots 94}{16\cdots 35}a^{18}+\frac{20\cdots 77}{80\cdots 75}a^{17}+\frac{73\cdots 58}{80\cdots 75}a^{16}-\frac{60\cdots 76}{80\cdots 75}a^{15}+\frac{64\cdots 14}{80\cdots 75}a^{14}+\frac{22\cdots 62}{16\cdots 35}a^{13}-\frac{29\cdots 38}{80\cdots 75}a^{12}-\frac{12\cdots 67}{80\cdots 75}a^{11}+\frac{61\cdots 44}{80\cdots 75}a^{10}+\frac{29\cdots 23}{26\cdots 25}a^{9}-\frac{17\cdots 93}{16\cdots 35}a^{8}+\frac{16\cdots 67}{80\cdots 75}a^{7}+\frac{78\cdots 68}{80\cdots 75}a^{6}-\frac{42\cdots 67}{26\cdots 25}a^{5}-\frac{18\cdots 01}{80\cdots 75}a^{4}+\frac{11\cdots 71}{16\cdots 35}a^{3}-\frac{86\cdots 47}{32\cdots 27}a^{2}-\frac{31\cdots 54}{32\cdots 27}a+\frac{52\cdots 53}{53\cdots 45}$, $\frac{67\cdots 76}{35\cdots 45}a^{23}-\frac{26\cdots 69}{17\cdots 25}a^{22}+\frac{45\cdots 04}{17\cdots 25}a^{21}+\frac{24\cdots 02}{17\cdots 25}a^{20}-\frac{28\cdots 23}{17\cdots 25}a^{19}-\frac{43\cdots 19}{71\cdots 29}a^{18}+\frac{40\cdots 52}{59\cdots 75}a^{17}+\frac{31\cdots 74}{17\cdots 25}a^{16}-\frac{42\cdots 93}{17\cdots 25}a^{15}-\frac{18\cdots 48}{17\cdots 25}a^{14}+\frac{21\cdots 02}{39\cdots 05}a^{13}-\frac{18\cdots 14}{17\cdots 25}a^{12}-\frac{12\cdots 64}{19\cdots 25}a^{11}+\frac{47\cdots 42}{17\cdots 25}a^{10}+\frac{80\cdots 67}{17\cdots 25}a^{9}-\frac{13\cdots 32}{35\cdots 45}a^{8}-\frac{28\cdots 08}{59\cdots 75}a^{7}+\frac{25\cdots 18}{59\cdots 75}a^{6}-\frac{19\cdots 43}{17\cdots 25}a^{5}+\frac{10\cdots 82}{17\cdots 25}a^{4}+\frac{27\cdots 46}{11\cdots 15}a^{3}-\frac{37\cdots 94}{71\cdots 29}a^{2}+\frac{11\cdots 90}{23\cdots 43}a-\frac{63\cdots 13}{35\cdots 45}$, $\frac{17\cdots 24}{17\cdots 25}a^{23}-\frac{19\cdots 77}{17\cdots 25}a^{22}+\frac{20\cdots 68}{17\cdots 25}a^{21}+\frac{17\cdots 08}{11\cdots 15}a^{20}-\frac{11\cdots 87}{17\cdots 25}a^{19}-\frac{15\cdots 04}{17\cdots 25}a^{18}+\frac{46\cdots 68}{17\cdots 25}a^{17}+\frac{93\cdots 01}{59\cdots 75}a^{16}-\frac{90\cdots 79}{11\cdots 15}a^{15}+\frac{11\cdots 17}{19\cdots 25}a^{14}+\frac{25\cdots 36}{17\cdots 25}a^{13}-\frac{19\cdots 29}{59\cdots 75}a^{12}-\frac{29\cdots 62}{17\cdots 25}a^{11}+\frac{83\cdots 08}{11\cdots 15}a^{10}+\frac{23\cdots 38}{17\cdots 25}a^{9}-\frac{18\cdots 99}{17\cdots 25}a^{8}-\frac{29\cdots 22}{17\cdots 25}a^{7}+\frac{16\cdots 98}{17\cdots 25}a^{6}-\frac{49\cdots 76}{35\cdots 45}a^{5}-\frac{41\cdots 32}{17\cdots 25}a^{4}+\frac{24\cdots 72}{35\cdots 45}a^{3}-\frac{57\cdots 67}{23\cdots 43}a^{2}-\frac{31\cdots 93}{35\cdots 45}a+\frac{33\cdots 88}{35\cdots 45}$, $\frac{55\cdots 19}{67\cdots 05}a^{23}+\frac{50\cdots 04}{37\cdots 25}a^{22}+\frac{57\cdots 39}{33\cdots 25}a^{21}-\frac{56\cdots 58}{33\cdots 25}a^{20}-\frac{16\cdots 61}{11\cdots 75}a^{19}+\frac{64\cdots 92}{67\cdots 05}a^{18}+\frac{26\cdots 11}{33\cdots 25}a^{17}-\frac{12\cdots 91}{33\cdots 25}a^{16}-\frac{79\cdots 53}{33\cdots 25}a^{15}+\frac{42\cdots 92}{33\cdots 25}a^{14}+\frac{44\cdots 84}{13\cdots 01}a^{13}-\frac{11\cdots 84}{33\cdots 25}a^{12}+\frac{83\cdots 34}{33\cdots 25}a^{11}+\frac{20\cdots 57}{33\cdots 25}a^{10}-\frac{70\cdots 02}{37\cdots 25}a^{9}-\frac{88\cdots 49}{13\cdots 01}a^{8}+\frac{11\cdots 81}{33\cdots 25}a^{7}+\frac{99\cdots 89}{33\cdots 25}a^{6}-\frac{35\cdots 51}{11\cdots 75}a^{5}+\frac{48\cdots 22}{33\cdots 25}a^{4}+\frac{73\cdots 33}{67\cdots 05}a^{3}-\frac{14\cdots 24}{13\cdots 01}a^{2}-\frac{15\cdots 22}{13\cdots 01}a+\frac{38\cdots 34}{22\cdots 35}$, $\frac{11\cdots 08}{89\cdots 75}a^{23}-\frac{49\cdots 31}{80\cdots 75}a^{22}-\frac{14\cdots 88}{89\cdots 75}a^{21}+\frac{57\cdots 16}{80\cdots 75}a^{20}+\frac{80\cdots 92}{80\cdots 75}a^{19}-\frac{35\cdots 47}{89\cdots 75}a^{18}-\frac{34\cdots 66}{80\cdots 75}a^{17}+\frac{14\cdots 63}{80\cdots 75}a^{16}+\frac{93\cdots 71}{80\cdots 75}a^{15}-\frac{53\cdots 53}{80\cdots 75}a^{14}-\frac{11\cdots 23}{80\cdots 75}a^{13}+\frac{12\cdots 89}{80\cdots 75}a^{12}-\frac{16\cdots 92}{80\cdots 75}a^{11}-\frac{17\cdots 89}{80\cdots 75}a^{10}+\frac{27\cdots 87}{80\cdots 75}a^{9}+\frac{74\cdots 91}{29\cdots 25}a^{8}-\frac{61\cdots 36}{80\cdots 75}a^{7}-\frac{11\cdots 82}{80\cdots 75}a^{6}+\frac{69\cdots 06}{80\cdots 75}a^{5}-\frac{13\cdots 29}{29\cdots 25}a^{4}-\frac{39\cdots 37}{16\cdots 35}a^{3}+\frac{95\cdots 75}{32\cdots 27}a^{2}+\frac{18\cdots 54}{16\cdots 35}a-\frac{46\cdots 93}{16\cdots 35}$, $\frac{16\cdots 01}{59\cdots 75}a^{23}-\frac{12\cdots 23}{17\cdots 25}a^{22}+\frac{61\cdots 17}{23\cdots 43}a^{21}+\frac{12\cdots 77}{17\cdots 25}a^{20}+\frac{24\cdots 68}{17\cdots 25}a^{19}-\frac{19\cdots 86}{59\cdots 75}a^{18}-\frac{15\cdots 83}{17\cdots 25}a^{17}+\frac{39\cdots 18}{35\cdots 45}a^{16}+\frac{42\cdots 97}{17\cdots 25}a^{15}-\frac{48\cdots 02}{17\cdots 25}a^{14}-\frac{31\cdots 88}{17\cdots 25}a^{13}+\frac{84\cdots 37}{17\cdots 25}a^{12}-\frac{16\cdots 18}{35\cdots 45}a^{11}-\frac{11\cdots 58}{17\cdots 25}a^{10}+\frac{24\cdots 58}{17\cdots 25}a^{9}+\frac{44\cdots 14}{59\cdots 75}a^{8}-\frac{39\cdots 18}{17\cdots 25}a^{7}-\frac{15\cdots 53}{35\cdots 45}a^{6}+\frac{30\cdots 32}{17\cdots 25}a^{5}+\frac{50\cdots 11}{59\cdots 75}a^{4}-\frac{20\cdots 23}{35\cdots 45}a^{3}+\frac{25\cdots 55}{71\cdots 29}a^{2}+\frac{28\cdots 64}{35\cdots 45}a-\frac{30\cdots 22}{35\cdots 45}$, $\frac{20\cdots 37}{59\cdots 75}a^{23}+\frac{25\cdots 66}{59\cdots 75}a^{22}-\frac{23\cdots 34}{59\cdots 75}a^{21}-\frac{62\cdots 38}{11\cdots 15}a^{20}+\frac{13\cdots 86}{59\cdots 75}a^{19}+\frac{18\cdots 77}{59\cdots 75}a^{18}-\frac{17\cdots 73}{19\cdots 25}a^{17}-\frac{30\cdots 14}{59\cdots 75}a^{16}+\frac{61\cdots 42}{23\cdots 43}a^{15}-\frac{13\cdots 59}{59\cdots 75}a^{14}-\frac{31\cdots 52}{66\cdots 75}a^{13}+\frac{71\cdots 96}{59\cdots 75}a^{12}+\frac{10\cdots 02}{19\cdots 25}a^{11}-\frac{30\cdots 03}{11\cdots 15}a^{10}-\frac{24\cdots 64}{59\cdots 75}a^{9}+\frac{21\cdots 62}{59\cdots 75}a^{8}-\frac{90\cdots 36}{66\cdots 75}a^{7}-\frac{64\cdots 08}{19\cdots 25}a^{6}+\frac{57\cdots 32}{11\cdots 15}a^{5}+\frac{48\cdots 71}{59\cdots 75}a^{4}-\frac{31\cdots 69}{13\cdots 35}a^{3}+\frac{18\cdots 02}{23\cdots 43}a^{2}+\frac{12\cdots 38}{39\cdots 05}a-\frac{36\cdots 09}{11\cdots 15}$, $\frac{10\cdots 21}{17\cdots 25}a^{23}-\frac{49\cdots 03}{59\cdots 75}a^{22}-\frac{32\cdots 64}{35\cdots 45}a^{21}+\frac{17\cdots 91}{17\cdots 25}a^{20}+\frac{38\cdots 43}{59\cdots 75}a^{19}-\frac{93\cdots 69}{17\cdots 25}a^{18}-\frac{54\cdots 64}{17\cdots 25}a^{17}+\frac{15\cdots 23}{71\cdots 29}a^{16}+\frac{15\cdots 76}{17\cdots 25}a^{15}-\frac{12\cdots 81}{17\cdots 25}a^{14}-\frac{17\cdots 84}{17\cdots 25}a^{13}+\frac{26\cdots 46}{17\cdots 25}a^{12}-\frac{26\cdots 43}{35\cdots 45}a^{11}-\frac{38\cdots 64}{17\cdots 25}a^{10}+\frac{24\cdots 83}{59\cdots 75}a^{9}+\frac{42\cdots 81}{17\cdots 25}a^{8}-\frac{14\cdots 94}{17\cdots 25}a^{7}-\frac{54\cdots 03}{35\cdots 45}a^{6}+\frac{51\cdots 02}{59\cdots 75}a^{5}-\frac{42\cdots 01}{17\cdots 25}a^{4}-\frac{11\cdots 64}{35\cdots 45}a^{3}+\frac{22\cdots 00}{71\cdots 29}a^{2}+\frac{79\cdots 02}{35\cdots 45}a-\frac{84\cdots 02}{11\cdots 15}$, $\frac{79\cdots 41}{26\cdots 25}a^{23}-\frac{19\cdots 01}{80\cdots 75}a^{22}-\frac{11\cdots 82}{26\cdots 25}a^{21}+\frac{45\cdots 02}{16\cdots 35}a^{20}+\frac{22\cdots 64}{80\cdots 75}a^{19}-\frac{42\cdots 64}{26\cdots 25}a^{18}-\frac{10\cdots 66}{80\cdots 75}a^{17}+\frac{55\cdots 34}{80\cdots 75}a^{16}+\frac{56\cdots 84}{16\cdots 35}a^{15}-\frac{18\cdots 91}{80\cdots 75}a^{14}-\frac{37\cdots 72}{80\cdots 75}a^{13}+\frac{42\cdots 69}{80\cdots 75}a^{12}-\frac{69\cdots 61}{80\cdots 75}a^{11}-\frac{12\cdots 73}{16\cdots 35}a^{10}+\frac{10\cdots 64}{80\cdots 75}a^{9}+\frac{24\cdots 66}{26\cdots 25}a^{8}-\frac{21\cdots 61}{80\cdots 75}a^{7}-\frac{49\cdots 06}{80\cdots 75}a^{6}+\frac{50\cdots 52}{16\cdots 35}a^{5}-\frac{76\cdots 32}{26\cdots 25}a^{4}-\frac{19\cdots 99}{16\cdots 35}a^{3}+\frac{31\cdots 98}{32\cdots 27}a^{2}+\frac{21\cdots 86}{16\cdots 35}a-\frac{29\cdots 41}{16\cdots 35}$, $\frac{82\cdots 46}{59\cdots 75}a^{23}+\frac{73\cdots 73}{19\cdots 25}a^{22}-\frac{38\cdots 09}{23\cdots 43}a^{21}-\frac{24\cdots 31}{59\cdots 75}a^{20}+\frac{16\cdots 24}{66\cdots 75}a^{19}+\frac{12\cdots 69}{59\cdots 75}a^{18}-\frac{59\cdots 51}{59\cdots 75}a^{17}-\frac{95\cdots 04}{11\cdots 15}a^{16}+\frac{68\cdots 34}{59\cdots 75}a^{15}+\frac{11\cdots 01}{59\cdots 75}a^{14}-\frac{39\cdots 41}{59\cdots 75}a^{13}-\frac{11\cdots 61}{59\cdots 75}a^{12}+\frac{16\cdots 64}{11\cdots 15}a^{11}+\frac{40\cdots 74}{59\cdots 75}a^{10}-\frac{38\cdots 68}{19\cdots 25}a^{9}+\frac{84\cdots 19}{59\cdots 75}a^{8}+\frac{12\cdots 54}{59\cdots 75}a^{7}-\frac{70\cdots 51}{11\cdots 15}a^{6}-\frac{29\cdots 82}{19\cdots 25}a^{5}+\frac{14\cdots 46}{59\cdots 75}a^{4}-\frac{36\cdots 61}{11\cdots 15}a^{3}-\frac{32\cdots 48}{23\cdots 43}a^{2}+\frac{68\cdots 08}{11\cdots 15}a-\frac{15\cdots 58}{39\cdots 05}$, $\frac{86\cdots 12}{17\cdots 25}a^{23}+\frac{15\cdots 72}{59\cdots 75}a^{22}-\frac{79\cdots 24}{17\cdots 25}a^{21}-\frac{84\cdots 43}{35\cdots 45}a^{20}+\frac{12\cdots 32}{59\cdots 75}a^{19}+\frac{19\cdots 02}{17\cdots 25}a^{18}-\frac{13\cdots 69}{17\cdots 25}a^{17}-\frac{54\cdots 29}{17\cdots 25}a^{16}+\frac{15\cdots 80}{71\cdots 29}a^{15}+\frac{60\cdots 76}{17\cdots 25}a^{14}-\frac{80\cdots 18}{17\cdots 25}a^{13}+\frac{36\cdots 96}{17\cdots 25}a^{12}+\frac{11\cdots 16}{17\cdots 25}a^{11}-\frac{41\cdots 08}{35\cdots 45}a^{10}-\frac{43\cdots 68}{59\cdots 75}a^{9}+\frac{39\cdots 12}{17\cdots 25}a^{8}+\frac{75\cdots 76}{17\cdots 25}a^{7}-\frac{41\cdots 39}{17\cdots 25}a^{6}+\frac{12\cdots 94}{11\cdots 15}a^{5}+\frac{12\cdots 31}{17\cdots 25}a^{4}-\frac{42\cdots 76}{35\cdots 45}a^{3}-\frac{97\cdots 11}{71\cdots 29}a^{2}+\frac{62\cdots 94}{35\cdots 45}a-\frac{16\cdots 88}{11\cdots 15}$ (assuming GRH)
Regulator:		\( 13752419633095248000 \) (assuming GRH)
Unit signature rank:		\( 4 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 13752419633095248000 \cdot 16}{2\cdot\sqrt{91810054652229848026521481130092527303208363056182861328125}}\cr\approx \mathstrut & 0.557112170407324 \end{aligned}\] (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

Degree 24 siblings:	24.4.2295251366305746200663037028252313182580209076404571533203125.3, 24.4.2295251366305746200663037028252313182580209076404571533203125.5
Arithmetically equivalent sibling:	24.4.91810054652229848026521481130092527303208363056182861328125.4
Minimal sibling:	24.4.91810054652229848026521481130092527303208363056182861328125.4

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$24$	${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$	R	$24$	${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$	$24$	${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.4.0.1}{4} }^{6}$	$24$	${\href{/padicField/29.4.0.1}{4} }^{6}$	${\href{/padicField/31.12.0.1}{12} }^{2}$	${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$	$24$	${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	$24$	${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$$[\ ]_{4}$$
	5.1.5.5a1.1	$x^{5} + 5 x + 5$	$5$	$1$	$5$	$F_5$	$$[\frac{5}{4}]_{4}$$
	5.1.5.5a1.1	$x^{5} + 5 x + 5$	$5$	$1$	$5$	$F_5$	$$[\frac{5}{4}]_{4}$$
	5.1.5.5a1.1	$x^{5} + 5 x + 5$	$5$	$1$	$5$	$F_5$	$$[\frac{5}{4}]_{4}$$
	5.1.5.5a1.1	$x^{5} + 5 x + 5$	$5$	$1$	$5$	$F_5$	$$[\frac{5}{4}]_{4}$$
\(89\)	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	$$[\ ]_{20}^{2}$$

Spectrum of ring of integers