Number field 24.4.2295251366305746200663037028252313182580209076404571533203125.2

• Label: 24.4.229...125.2
• Degree: $24$
• Signature: $(4, 10)$
• Discriminant: $2.295\times 10^{60}$
• Root discriminant: \(327.37\)
• Ramified primes: $5,89$
• Class number: $32$ (GRH)
• Class group: [2, 2, 2, 4] (GRH)
• Galois group: $\GL(2,5)$ (as 24T1353)

Normalized defining polynomial

x^24 - 11*x^23 - 114*x^22 + 824*x^21 + 12301*x^20 - 5277*x^19 - 1149493*x^18 - 598157*x^17 + 62457032*x^16 - 28337252*x^15 - 1893188568*x^14 + 3477178433*x^13 + 30807485227*x^12 - 143840039207*x^11 - 97275021018*x^10 + 3086068549814*x^9 - 6286206651279*x^8 - 28596060866796*x^7 + 105662214126501*x^6 + 51235342679184*x^5 - 580910749748916*x^4 + 441206727204676*x^3 + 114072323164339*x^2 + 541180287831136*x - 312927910098871

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{15}a^{20}+\frac{1}{3}a^{17}+\frac{1}{5}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{2}{15}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{4}{15}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{4}{15}$, $\frac{1}{15}a^{21}+\frac{1}{3}a^{18}+\frac{1}{5}a^{16}+\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{2}{15}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{4}{15}a^{6}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{4}{15}a$, $\frac{1}{15}a^{22}+\frac{1}{3}a^{19}+\frac{1}{5}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{2}{15}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{4}{15}a^{7}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{4}{15}a^{2}$, $\frac{1}{35\cdots 35}a^{23}-\frac{95\cdots 08}{35\cdots 35}a^{22}+\frac{21\cdots 33}{35\cdots 35}a^{21}-\frac{11\cdots 17}{35\cdots 35}a^{20}-\frac{74\cdots 46}{70\cdots 27}a^{19}-\frac{15\cdots 82}{35\cdots 35}a^{18}-\frac{97\cdots 14}{35\cdots 35}a^{17}-\frac{15\cdots 11}{35\cdots 35}a^{16}-\frac{13\cdots 96}{35\cdots 35}a^{15}-\frac{30\cdots 24}{23\cdots 09}a^{14}+\frac{11\cdots 34}{11\cdots 45}a^{13}+\frac{11\cdots 64}{35\cdots 35}a^{12}+\frac{42\cdots 56}{35\cdots 35}a^{11}-\frac{62\cdots 08}{11\cdots 45}a^{10}-\frac{38\cdots 95}{23\cdots 09}a^{9}-\frac{16\cdots 41}{35\cdots 35}a^{8}-\frac{15\cdots 37}{35\cdots 35}a^{7}-\frac{32\cdots 23}{35\cdots 35}a^{6}+\frac{11\cdots 87}{35\cdots 35}a^{5}-\frac{49\cdots 60}{23\cdots 09}a^{4}+\frac{14\cdots 39}{35\cdots 35}a^{3}+\frac{53\cdots 78}{35\cdots 35}a^{2}-\frac{57\cdots 88}{35\cdots 35}a-\frac{25\cdots 36}{11\cdots 45}$

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

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:		$\frac{36\cdots 81}{78\cdots 61}a^{23}+\frac{67\cdots 94}{11\cdots 15}a^{22}+\frac{18\cdots 71}{39\cdots 05}a^{21}-\frac{18\cdots 69}{39\cdots 05}a^{20}-\frac{12\cdots 00}{23\cdots 83}a^{19}+\frac{93\cdots 37}{78\cdots 61}a^{18}+\frac{21\cdots 74}{39\cdots 05}a^{17}-\frac{53\cdots 96}{11\cdots 15}a^{16}-\frac{37\cdots 11}{11\cdots 15}a^{15}+\frac{11\cdots 86}{23\cdots 83}a^{14}+\frac{76\cdots 40}{78\cdots 61}a^{13}-\frac{31\cdots 12}{11\cdots 15}a^{12}-\frac{18\cdots 39}{11\cdots 15}a^{11}+\frac{10\cdots 61}{11\cdots 15}a^{10}+\frac{92\cdots 81}{78\cdots 61}a^{9}-\frac{38\cdots 82}{23\cdots 83}a^{8}+\frac{51\cdots 61}{11\cdots 15}a^{7}+\frac{55\cdots 29}{39\cdots 05}a^{6}-\frac{27\cdots 56}{39\cdots 05}a^{5}-\frac{39\cdots 76}{23\cdots 83}a^{4}+\frac{90\cdots 76}{23\cdots 83}a^{3}-\frac{34\cdots 39}{11\cdots 15}a^{2}-\frac{13\cdots 61}{39\cdots 05}a-\frac{23\cdots 01}{39\cdots 05}$, $\frac{59\cdots 02}{10\cdots 55}a^{23}-\frac{24\cdots 49}{30\cdots 65}a^{22}-\frac{49\cdots 97}{10\cdots 55}a^{21}+\frac{13\cdots 88}{20\cdots 51}a^{20}+\frac{35\cdots 86}{61\cdots 53}a^{19}-\frac{21\cdots 74}{10\cdots 55}a^{18}-\frac{65\cdots 19}{10\cdots 55}a^{17}+\frac{42\cdots 22}{30\cdots 65}a^{16}+\frac{21\cdots 01}{61\cdots 53}a^{15}-\frac{68\cdots 73}{61\cdots 53}a^{14}-\frac{10\cdots 81}{10\cdots 55}a^{13}+\frac{14\cdots 02}{30\cdots 65}a^{12}+\frac{32\cdots 83}{30\cdots 65}a^{11}-\frac{78\cdots 03}{61\cdots 53}a^{10}+\frac{40\cdots 59}{20\cdots 51}a^{9}+\frac{55\cdots 44}{30\cdots 65}a^{8}-\frac{26\cdots 11}{30\cdots 65}a^{7}-\frac{39\cdots 53}{10\cdots 55}a^{6}+\frac{20\cdots 24}{20\cdots 51}a^{5}-\frac{10\cdots 51}{61\cdots 53}a^{4}-\frac{95\cdots 31}{30\cdots 65}a^{3}+\frac{38\cdots 69}{30\cdots 65}a^{2}-\frac{13\cdots 98}{10\cdots 55}a+\frac{77\cdots 26}{20\cdots 51}$, $\frac{65\cdots 18}{44\cdots 53}a^{23}-\frac{23\cdots 50}{14\cdots 51}a^{22}-\frac{25\cdots 48}{14\cdots 51}a^{21}+\frac{90\cdots 92}{74\cdots 55}a^{20}+\frac{27\cdots 33}{14\cdots 51}a^{19}-\frac{11\cdots 88}{14\cdots 51}a^{18}-\frac{75\cdots 64}{44\cdots 53}a^{17}-\frac{43\cdots 81}{44\cdots 53}a^{16}+\frac{20\cdots 43}{22\cdots 65}a^{15}-\frac{16\cdots 94}{44\cdots 53}a^{14}-\frac{42\cdots 03}{14\cdots 51}a^{13}+\frac{22\cdots 14}{44\cdots 53}a^{12}+\frac{21\cdots 84}{44\cdots 53}a^{11}-\frac{47\cdots 68}{22\cdots 65}a^{10}-\frac{29\cdots 48}{14\cdots 51}a^{9}+\frac{21\cdots 02}{44\cdots 53}a^{8}-\frac{13\cdots 92}{14\cdots 51}a^{7}-\frac{20\cdots 07}{44\cdots 53}a^{6}+\frac{12\cdots 73}{74\cdots 55}a^{5}+\frac{53\cdots 16}{44\cdots 53}a^{4}-\frac{46\cdots 63}{44\cdots 53}a^{3}+\frac{27\cdots 33}{44\cdots 53}a^{2}+\frac{51\cdots 05}{44\cdots 53}a-\frac{13\cdots 51}{22\cdots 65}$, $\frac{19\cdots 77}{30\cdots 65}a^{23}-\frac{29\cdots 08}{30\cdots 65}a^{22}-\frac{45\cdots 49}{10\cdots 55}a^{21}+\frac{24\cdots 72}{30\cdots 65}a^{20}+\frac{35\cdots 48}{61\cdots 53}a^{19}-\frac{33\cdots 38}{10\cdots 55}a^{18}-\frac{21\cdots 59}{30\cdots 65}a^{17}+\frac{76\cdots 19}{30\cdots 65}a^{16}+\frac{40\cdots 72}{10\cdots 55}a^{15}-\frac{10\cdots 81}{61\cdots 53}a^{14}-\frac{32\cdots 01}{30\cdots 65}a^{13}+\frac{21\cdots 14}{30\cdots 65}a^{12}+\frac{81\cdots 57}{10\cdots 55}a^{11}-\frac{50\cdots 21}{30\cdots 65}a^{10}+\frac{21\cdots 81}{61\cdots 53}a^{9}+\frac{61\cdots 38}{30\cdots 65}a^{8}-\frac{37\cdots 97}{30\cdots 65}a^{7}+\frac{23\cdots 09}{10\cdots 55}a^{6}+\frac{40\cdots 83}{30\cdots 65}a^{5}-\frac{57\cdots 53}{20\cdots 51}a^{4}-\frac{99\cdots 57}{30\cdots 65}a^{3}+\frac{57\cdots 18}{30\cdots 65}a^{2}-\frac{22\cdots 61}{10\cdots 55}a+\frac{88\cdots 21}{10\cdots 55}$, $\frac{13\cdots 81}{74\cdots 55}a^{23}+\frac{44\cdots 17}{22\cdots 65}a^{22}+\frac{44\cdots 43}{22\cdots 65}a^{21}-\frac{11\cdots 66}{74\cdots 55}a^{20}-\frac{97\cdots 65}{44\cdots 53}a^{19}+\frac{34\cdots 96}{22\cdots 65}a^{18}+\frac{15\cdots 72}{74\cdots 55}a^{17}+\frac{13\cdots 29}{22\cdots 65}a^{16}-\frac{25\cdots 89}{22\cdots 65}a^{15}+\frac{11\cdots 77}{14\cdots 51}a^{14}+\frac{77\cdots 59}{22\cdots 65}a^{13}-\frac{15\cdots 61}{22\cdots 65}a^{12}-\frac{12\cdots 19}{22\cdots 65}a^{11}+\frac{20\cdots 33}{74\cdots 55}a^{10}+\frac{68\cdots 69}{44\cdots 53}a^{9}-\frac{12\cdots 32}{22\cdots 65}a^{8}+\frac{27\cdots 33}{22\cdots 65}a^{7}+\frac{11\cdots 32}{22\cdots 65}a^{6}-\frac{15\cdots 94}{74\cdots 55}a^{5}-\frac{34\cdots 84}{44\cdots 53}a^{4}+\frac{90\cdots 86}{74\cdots 55}a^{3}-\frac{24\cdots 92}{22\cdots 65}a^{2}-\frac{17\cdots 53}{22\cdots 65}a+\frac{37\cdots 66}{74\cdots 55}$, $\frac{64\cdots 27}{22\cdots 65}a^{23}-\frac{34\cdots 74}{74\cdots 55}a^{22}-\frac{14\cdots 54}{44\cdots 53}a^{21}+\frac{42\cdots 81}{74\cdots 55}a^{20}+\frac{66\cdots 21}{14\cdots 51}a^{19}-\frac{64\cdots 79}{22\cdots 65}a^{18}-\frac{12\cdots 86}{22\cdots 65}a^{17}+\frac{49\cdots 97}{44\cdots 53}a^{16}+\frac{27\cdots 88}{74\cdots 55}a^{15}-\frac{76\cdots 77}{14\cdots 51}a^{14}-\frac{30\cdots 21}{22\cdots 65}a^{13}+\frac{50\cdots 91}{22\cdots 65}a^{12}+\frac{40\cdots 82}{14\cdots 51}a^{11}-\frac{59\cdots 88}{74\cdots 55}a^{10}-\frac{76\cdots 61}{44\cdots 53}a^{9}+\frac{45\cdots 08}{22\cdots 65}a^{8}-\frac{89\cdots 28}{22\cdots 65}a^{7}-\frac{45\cdots 91}{14\cdots 51}a^{6}+\frac{59\cdots 99}{74\cdots 55}a^{5}+\frac{75\cdots 35}{44\cdots 53}a^{4}-\frac{20\cdots 39}{74\cdots 55}a^{3}-\frac{36\cdots 98}{22\cdots 65}a^{2}-\frac{29\cdots 14}{14\cdots 51}a+\frac{34\cdots 62}{22\cdots 65}$, $\frac{51\cdots 07}{61\cdots 95}a^{23}+\frac{15\cdots 49}{18\cdots 85}a^{22}+\frac{64\cdots 64}{61\cdots 95}a^{21}-\frac{10\cdots 68}{18\cdots 85}a^{20}-\frac{40\cdots 07}{36\cdots 97}a^{19}-\frac{51\cdots 36}{61\cdots 95}a^{18}+\frac{17\cdots 97}{18\cdots 85}a^{17}+\frac{28\cdots 16}{18\cdots 85}a^{16}-\frac{91\cdots 54}{18\cdots 85}a^{15}-\frac{10\cdots 20}{36\cdots 97}a^{14}+\frac{27\cdots 38}{18\cdots 85}a^{13}-\frac{26\cdots 87}{18\cdots 85}a^{12}-\frac{45\cdots 31}{18\cdots 85}a^{11}+\frac{17\cdots 39}{18\cdots 85}a^{10}+\frac{47\cdots 39}{36\cdots 97}a^{9}-\frac{43\cdots 39}{18\cdots 85}a^{8}+\frac{60\cdots 31}{18\cdots 85}a^{7}+\frac{43\cdots 73}{18\cdots 85}a^{6}-\frac{11\cdots 32}{18\cdots 85}a^{5}-\frac{18\cdots 04}{36\cdots 97}a^{4}+\frac{67\cdots 86}{18\cdots 85}a^{3}-\frac{68\cdots 34}{18\cdots 85}a^{2}-\frac{18\cdots 52}{18\cdots 85}a+\frac{18\cdots 63}{18\cdots 85}$, $\frac{97\cdots 36}{11\cdots 45}a^{23}-\frac{40\cdots 54}{35\cdots 35}a^{22}-\frac{22\cdots 63}{35\cdots 35}a^{21}+\frac{10\cdots 08}{11\cdots 45}a^{20}+\frac{54\cdots 54}{70\cdots 27}a^{19}-\frac{91\cdots 31}{35\cdots 35}a^{18}-\frac{10\cdots 54}{11\cdots 45}a^{17}+\frac{68\cdots 76}{35\cdots 35}a^{16}+\frac{54\cdots 04}{11\cdots 45}a^{15}-\frac{10\cdots 92}{70\cdots 27}a^{14}-\frac{39\cdots 99}{35\cdots 35}a^{13}+\frac{21\cdots 02}{35\cdots 35}a^{12}+\frac{88\cdots 78}{11\cdots 45}a^{11}-\frac{49\cdots 22}{35\cdots 35}a^{10}+\frac{24\cdots 28}{70\cdots 27}a^{9}+\frac{54\cdots 52}{35\cdots 35}a^{8}-\frac{11\cdots 02}{11\cdots 45}a^{7}+\frac{22\cdots 73}{35\cdots 35}a^{6}+\frac{82\cdots 12}{11\cdots 45}a^{5}-\frac{12\cdots 75}{70\cdots 27}a^{4}+\frac{36\cdots 72}{35\cdots 35}a^{3}-\frac{16\cdots 47}{11\cdots 45}a^{2}+\frac{59\cdots 03}{35\cdots 35}a-\frac{73\cdots 93}{11\cdots 45}$, $\frac{81\cdots 18}{11\cdots 45}a^{23}-\frac{67\cdots 24}{11\cdots 45}a^{22}-\frac{62\cdots 37}{70\cdots 27}a^{21}+\frac{10\cdots 43}{35\cdots 35}a^{20}+\frac{19\cdots 88}{23\cdots 09}a^{19}+\frac{68\cdots 42}{35\cdots 35}a^{18}-\frac{23\cdots 21}{35\cdots 35}a^{17}-\frac{43\cdots 13}{23\cdots 09}a^{16}+\frac{11\cdots 44}{35\cdots 35}a^{15}+\frac{30\cdots 89}{70\cdots 27}a^{14}-\frac{11\cdots 29}{11\cdots 45}a^{13}+\frac{21\cdots 56}{35\cdots 35}a^{12}+\frac{12\cdots 35}{70\cdots 27}a^{11}-\frac{22\cdots 19}{35\cdots 35}a^{10}-\frac{30\cdots 35}{23\cdots 09}a^{9}+\frac{56\cdots 76}{35\cdots 35}a^{8}-\frac{59\cdots 18}{35\cdots 35}a^{7}-\frac{11\cdots 85}{70\cdots 27}a^{6}+\frac{14\cdots 57}{35\cdots 35}a^{5}+\frac{10\cdots 97}{23\cdots 09}a^{4}-\frac{87\cdots 69}{35\cdots 35}a^{3}+\frac{20\cdots 24}{11\cdots 45}a^{2}+\frac{88\cdots 33}{70\cdots 27}a-\frac{82\cdots 73}{35\cdots 35}$, $\frac{11\cdots 96}{35\cdots 35}a^{23}-\frac{24\cdots 02}{70\cdots 27}a^{22}-\frac{25\cdots 27}{70\cdots 27}a^{21}+\frac{16\cdots 91}{70\cdots 27}a^{20}+\frac{25\cdots 26}{70\cdots 27}a^{19}+\frac{32\cdots 58}{35\cdots 35}a^{18}-\frac{23\cdots 65}{70\cdots 27}a^{17}-\frac{28\cdots 70}{23\cdots 09}a^{16}+\frac{39\cdots 07}{23\cdots 09}a^{15}-\frac{52\cdots 22}{23\cdots 09}a^{14}-\frac{16\cdots 03}{35\cdots 35}a^{13}+\frac{38\cdots 33}{23\cdots 09}a^{12}+\frac{14\cdots 31}{23\cdots 09}a^{11}-\frac{12\cdots 10}{23\cdots 09}a^{10}+\frac{26\cdots 55}{70\cdots 27}a^{9}+\frac{97\cdots 98}{11\cdots 45}a^{8}-\frac{20\cdots 66}{70\cdots 27}a^{7}-\frac{24\cdots 59}{70\cdots 27}a^{6}+\frac{27\cdots 71}{70\cdots 27}a^{5}-\frac{36\cdots 54}{70\cdots 27}a^{4}-\frac{46\cdots 26}{35\cdots 35}a^{3}+\frac{32\cdots 19}{70\cdots 27}a^{2}-\frac{32\cdots 32}{70\cdots 27}a+\frac{31\cdots 34}{23\cdots 09}$, $\frac{81\cdots 67}{35\cdots 35}a^{23}+\frac{73\cdots 73}{35\cdots 35}a^{22}+\frac{71\cdots 37}{23\cdots 09}a^{21}-\frac{45\cdots 47}{35\cdots 35}a^{20}-\frac{21\cdots 35}{70\cdots 27}a^{19}-\frac{57\cdots 92}{11\cdots 45}a^{18}+\frac{30\cdots 93}{11\cdots 45}a^{17}+\frac{15\cdots 47}{23\cdots 09}a^{16}-\frac{46\cdots 46}{35\cdots 35}a^{15}-\frac{13\cdots 48}{70\cdots 27}a^{14}+\frac{14\cdots 01}{35\cdots 35}a^{13}-\frac{48\cdots 84}{35\cdots 35}a^{12}-\frac{50\cdots 80}{70\cdots 27}a^{11}+\frac{67\cdots 36}{35\cdots 35}a^{10}+\frac{42\cdots 23}{70\cdots 27}a^{9}-\frac{20\cdots 98}{35\cdots 35}a^{8}+\frac{99\cdots 42}{35\cdots 35}a^{7}+\frac{50\cdots 17}{70\cdots 27}a^{6}-\frac{36\cdots 33}{35\cdots 35}a^{5}-\frac{22\cdots 44}{70\cdots 27}a^{4}+\frac{83\cdots 19}{11\cdots 45}a^{3}+\frac{12\cdots 02}{35\cdots 35}a^{2}+\frac{31\cdots 88}{70\cdots 27}a-\frac{13\cdots 58}{35\cdots 35}$, $\frac{19\cdots 82}{35\cdots 35}a^{23}+\frac{25\cdots 07}{35\cdots 35}a^{22}+\frac{27\cdots 07}{70\cdots 27}a^{21}-\frac{15\cdots 69}{35\cdots 35}a^{20}-\frac{37\cdots 39}{70\cdots 27}a^{19}+\frac{26\cdots 59}{35\cdots 35}a^{18}+\frac{19\cdots 36}{35\cdots 35}a^{17}-\frac{15\cdots 49}{23\cdots 09}a^{16}-\frac{91\cdots 32}{35\cdots 35}a^{15}+\frac{35\cdots 39}{70\cdots 27}a^{14}+\frac{65\cdots 77}{11\cdots 45}a^{13}-\frac{17\cdots 07}{11\cdots 45}a^{12}-\frac{36\cdots 06}{70\cdots 27}a^{11}+\frac{10\cdots 22}{35\cdots 35}a^{10}-\frac{14\cdots 95}{23\cdots 09}a^{9}-\frac{39\cdots 66}{11\cdots 45}a^{8}+\frac{26\cdots 06}{11\cdots 45}a^{7}-\frac{12\cdots 64}{70\cdots 27}a^{6}-\frac{25\cdots 01}{35\cdots 35}a^{5}+\frac{35\cdots 92}{70\cdots 27}a^{4}+\frac{10\cdots 42}{35\cdots 35}a^{3}+\frac{21\cdots 71}{11\cdots 45}a^{2}+\frac{36\cdots 68}{70\cdots 27}a-\frac{12\cdots 82}{11\cdots 45}$, $\frac{93\cdots 05}{70\cdots 27}a^{23}+\frac{16\cdots 99}{11\cdots 45}a^{22}+\frac{37\cdots 91}{23\cdots 09}a^{21}-\frac{23\cdots 81}{23\cdots 09}a^{20}-\frac{39\cdots 05}{23\cdots 09}a^{19}-\frac{42\cdots 68}{23\cdots 09}a^{18}+\frac{53\cdots 61}{35\cdots 35}a^{17}+\frac{11\cdots 61}{70\cdots 27}a^{16}-\frac{57\cdots 44}{70\cdots 27}a^{15}-\frac{24\cdots 44}{70\cdots 27}a^{14}+\frac{58\cdots 52}{23\cdots 09}a^{13}-\frac{11\cdots 71}{35\cdots 35}a^{12}-\frac{29\cdots 34}{70\cdots 27}a^{11}+\frac{11\cdots 76}{70\cdots 27}a^{10}+\frac{49\cdots 18}{23\cdots 09}a^{9}-\frac{28\cdots 77}{70\cdots 27}a^{8}+\frac{74\cdots 26}{11\cdots 45}a^{7}+\frac{28\cdots 89}{70\cdots 27}a^{6}-\frac{28\cdots 98}{23\cdots 09}a^{5}-\frac{88\cdots 68}{70\cdots 27}a^{4}+\frac{49\cdots 76}{70\cdots 27}a^{3}-\frac{86\cdots 57}{35\cdots 35}a^{2}-\frac{18\cdots 20}{70\cdots 27}a-\frac{59\cdots 45}{70\cdots 27}$ (assuming GRH)
Regulator:		\( 39093008005448335000 \) (assuming GRH)
Unit signature rank:		\( 4 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 39093008005448335000 \cdot 32}{2\cdot\sqrt{2295251366305746200663037028252313182580209076404571533203125}}\cr\approx \mathstrut & 0.633464979071887 \end{aligned}\] (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:	24.4.91810054652229848026521481130092527303208363056182861328125.5, 24.4.91810054652229848026521481130092527303208363056182861328125.8
Arithmetically equivalent sibling:	24.4.2295251366305746200663037028252313182580209076404571533203125.1
Minimal sibling:	24.4.91810054652229848026521481130092527303208363056182861328125.5

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	$24$	${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$	$24$	${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$	${\href{/padicField/19.4.0.1}{4} }^{6}$	$24$	${\href{/padicField/29.4.0.1}{4} }^{6}$	${\href{/padicField/31.12.0.1}{12} }^{2}$	${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$	$24$	${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$	$24$	${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(5\)	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	$$[\ ]_{20}^{2}$$

Spectrum of ring of integers