Number field 24.4.270761008401829353605241639483649123576469719409942626953125.9

• Label: 24.4.270...125.9
• Degree: $24$
• Signature: $(4, 10)$
• Discriminant: $2.708\times 10^{59}$
• Root discriminant: \(299.47\)
• Ramified primes: $5,29$
• Class number: $4$ (GRH)
• Class group: [4] (GRH)
• Galois group: $\GL(2,5)$ (as 24T1353)

Normalized defining polynomial

x^24 - 7*x^23 - 6*x^22 - 88*x^21 + 800*x^20 + 33972*x^19 - 33044*x^18 - 679237*x^17 - 7247986*x^16 - 18808685*x^15 + 240393964*x^14 + 1548866817*x^13 - 1226398054*x^12 - 64602797057*x^11 - 428838287440*x^10 - 1562629162232*x^9 - 3495074371706*x^8 - 4678953132493*x^7 - 4731483065509*x^6 - 12469668216145*x^5 - 33540673060979*x^4 - 37892792137502*x^3 - 4499980454036*x^2 + 18569747459402*x + 8710672685155

Invariants

Degree:		$24$
Signature:		$(4, 10)$
Discriminant:		\(270761008401829353605241639483649123576469719409942626953125\) \(\medspace = 5^{39}\cdot 29^{22}\)
Root discriminant:		\(299.47\)
Galois root discriminant:		$5^{39/20}29^{19/20}\approx 565.287791942645$
Ramified primes:		\(5\), \(29\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{3}a^{21}-\frac{1}{3}a^{20}+\frac{1}{3}a^{17}+\frac{1}{3}a^{15}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{10773}a^{22}+\frac{19}{567}a^{21}-\frac{1391}{10773}a^{20}+\frac{7}{19}a^{19}-\frac{467}{1539}a^{18}+\frac{2402}{10773}a^{17}+\frac{238}{1539}a^{16}-\frac{580}{1539}a^{15}-\frac{634}{10773}a^{14}-\frac{1550}{3591}a^{13}+\frac{209}{567}a^{12}+\frac{3091}{10773}a^{11}+\frac{659}{1539}a^{10}+\frac{121}{10773}a^{9}-\frac{2221}{10773}a^{8}-\frac{122}{567}a^{7}-\frac{3062}{10773}a^{6}+\frac{6}{19}a^{5}-\frac{4397}{10773}a^{4}-\frac{767}{10773}a^{3}-\frac{5020}{10773}a^{2}-\frac{4985}{10773}a-\frac{158}{1539}$, $\frac{1}{88\cdots 15}a^{23}+\frac{92\cdots 75}{21\cdots 03}a^{22}-\frac{43\cdots 42}{29\cdots 05}a^{21}-\frac{58\cdots 00}{17\cdots 43}a^{20}+\frac{50\cdots 08}{25\cdots 49}a^{19}-\frac{37\cdots 83}{88\cdots 15}a^{18}-\frac{57\cdots 45}{17\cdots 43}a^{17}-\frac{10\cdots 61}{12\cdots 45}a^{16}+\frac{19\cdots 68}{59\cdots 81}a^{15}-\frac{83\cdots 38}{17\cdots 43}a^{14}+\frac{25\cdots 64}{88\cdots 15}a^{13}-\frac{52\cdots 02}{17\cdots 43}a^{12}+\frac{35\cdots 46}{88\cdots 15}a^{11}+\frac{16\cdots 67}{17\cdots 43}a^{10}-\frac{13\cdots 13}{17\cdots 43}a^{9}+\frac{29\cdots 63}{88\cdots 15}a^{8}-\frac{24\cdots 71}{59\cdots 81}a^{7}-\frac{15\cdots 98}{88\cdots 15}a^{6}+\frac{70\cdots 60}{17\cdots 43}a^{5}+\frac{48\cdots 21}{59\cdots 81}a^{4}-\frac{19\cdots 24}{88\cdots 15}a^{3}-\frac{53\cdots 71}{25\cdots 49}a^{2}+\frac{20\cdots 26}{98\cdots 35}a-\frac{84\cdots 23}{25\cdots 49}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$3$

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:		$\frac{16\cdots 63}{10\cdots 35}a^{23}-\frac{12\cdots 42}{11\cdots 83}a^{22}-\frac{34\cdots 53}{49\cdots 35}a^{21}-\frac{33\cdots 07}{20\cdots 87}a^{20}+\frac{39\cdots 96}{29\cdots 41}a^{19}+\frac{57\cdots 36}{10\cdots 35}a^{18}-\frac{11\cdots 46}{20\cdots 87}a^{17}-\frac{15\cdots 08}{14\cdots 05}a^{16}-\frac{82\cdots 86}{69\cdots 29}a^{15}-\frac{92\cdots 87}{29\cdots 41}a^{14}+\frac{39\cdots 72}{10\cdots 35}a^{13}+\frac{51\cdots 09}{20\cdots 87}a^{12}-\frac{17\cdots 42}{10\cdots 35}a^{11}-\frac{21\cdots 51}{20\cdots 87}a^{10}-\frac{14\cdots 18}{20\cdots 87}a^{9}-\frac{38\cdots 18}{14\cdots 05}a^{8}-\frac{43\cdots 81}{69\cdots 29}a^{7}-\frac{10\cdots 54}{10\cdots 35}a^{6}-\frac{25\cdots 41}{20\cdots 87}a^{5}-\frac{25\cdots 31}{99\cdots 47}a^{4}-\frac{63\cdots 27}{10\cdots 35}a^{3}-\frac{16\cdots 30}{20\cdots 87}a^{2}-\frac{61\cdots 47}{11\cdots 15}a-\frac{39\cdots 43}{29\cdots 41}$, $\frac{38\cdots 84}{23\cdots 43}a^{23}+\frac{12\cdots 16}{85\cdots 09}a^{22}-\frac{10\cdots 98}{77\cdots 81}a^{21}+\frac{38\cdots 19}{23\cdots 43}a^{20}-\frac{52\cdots 94}{33\cdots 49}a^{19}-\frac{12\cdots 07}{23\cdots 43}a^{18}+\frac{33\cdots 94}{23\cdots 43}a^{17}+\frac{30\cdots 55}{33\cdots 49}a^{16}+\frac{81\cdots 81}{77\cdots 81}a^{15}+\frac{32\cdots 61}{23\cdots 43}a^{14}-\frac{98\cdots 67}{23\cdots 43}a^{13}-\frac{62\cdots 17}{33\cdots 49}a^{12}+\frac{12\cdots 39}{23\cdots 43}a^{11}+\frac{23\cdots 01}{23\cdots 43}a^{10}+\frac{12\cdots 86}{23\cdots 43}a^{9}+\frac{38\cdots 51}{23\cdots 43}a^{8}+\frac{23\cdots 34}{77\cdots 81}a^{7}+\frac{62\cdots 65}{23\cdots 43}a^{6}+\frac{74\cdots 35}{23\cdots 43}a^{5}+\frac{12\cdots 25}{77\cdots 81}a^{4}+\frac{10\cdots 93}{33\cdots 49}a^{3}+\frac{27\cdots 97}{23\cdots 43}a^{2}-\frac{34\cdots 78}{25\cdots 27}a-\frac{28\cdots 96}{33\cdots 49}$, $\frac{16\cdots 37}{34\cdots 45}a^{23}-\frac{11\cdots 44}{25\cdots 27}a^{22}+\frac{65\cdots 91}{11\cdots 15}a^{21}-\frac{36\cdots 92}{69\cdots 29}a^{20}+\frac{48\cdots 82}{99\cdots 47}a^{19}+\frac{52\cdots 69}{34\cdots 45}a^{18}-\frac{32\cdots 49}{69\cdots 29}a^{17}-\frac{11\cdots 87}{49\cdots 35}a^{16}-\frac{69\cdots 82}{23\cdots 43}a^{15}-\frac{20\cdots 00}{69\cdots 29}a^{14}+\frac{59\cdots 69}{49\cdots 35}a^{13}+\frac{34\cdots 72}{69\cdots 29}a^{12}-\frac{55\cdots 78}{34\cdots 45}a^{11}-\frac{19\cdots 19}{69\cdots 29}a^{10}-\frac{14\cdots 79}{99\cdots 47}a^{9}-\frac{15\cdots 24}{34\cdots 45}a^{8}-\frac{17\cdots 41}{23\cdots 43}a^{7}-\frac{23\cdots 21}{34\cdots 45}a^{6}-\frac{61\cdots 57}{69\cdots 29}a^{5}-\frac{96\cdots 55}{23\cdots 43}a^{4}-\frac{26\cdots 03}{34\cdots 45}a^{3}-\frac{19\cdots 02}{69\cdots 29}a^{2}+\frac{13\cdots 12}{38\cdots 05}a+\frac{20\cdots 65}{99\cdots 47}$, $\frac{18\cdots 91}{10\cdots 35}a^{23}-\frac{10\cdots 37}{77\cdots 81}a^{22}+\frac{30\cdots 78}{34\cdots 45}a^{21}-\frac{45\cdots 71}{29\cdots 41}a^{20}+\frac{44\cdots 99}{29\cdots 41}a^{19}+\frac{59\cdots 02}{10\cdots 35}a^{18}-\frac{31\cdots 70}{29\cdots 41}a^{17}-\frac{16\cdots 36}{14\cdots 05}a^{16}-\frac{80\cdots 32}{69\cdots 29}a^{15}-\frac{47\cdots 02}{20\cdots 87}a^{14}+\frac{45\cdots 59}{10\cdots 35}a^{13}+\frac{48\cdots 25}{20\cdots 87}a^{12}-\frac{60\cdots 37}{14\cdots 05}a^{11}-\frac{22\cdots 56}{20\cdots 87}a^{10}-\frac{13\cdots 48}{20\cdots 87}a^{9}-\frac{22\cdots 27}{10\cdots 35}a^{8}-\frac{29\cdots 29}{69\cdots 29}a^{7}-\frac{67\cdots 39}{14\cdots 05}a^{6}-\frac{91\cdots 20}{20\cdots 87}a^{5}-\frac{12\cdots 65}{69\cdots 29}a^{4}-\frac{44\cdots 49}{10\cdots 35}a^{3}-\frac{61\cdots 47}{20\cdots 87}a^{2}+\frac{28\cdots 73}{16\cdots 45}a+\frac{53\cdots 95}{29\cdots 41}$, $\frac{62\cdots 13}{34\cdots 45}a^{23}+\frac{16\cdots 70}{25\cdots 27}a^{22}+\frac{11\cdots 31}{11\cdots 15}a^{21}-\frac{34\cdots 61}{99\cdots 47}a^{20}+\frac{10\cdots 84}{99\cdots 47}a^{19}-\frac{26\cdots 66}{34\cdots 45}a^{18}-\frac{67\cdots 89}{99\cdots 47}a^{17}+\frac{13\cdots 98}{49\cdots 35}a^{16}+\frac{21\cdots 64}{23\cdots 43}a^{15}+\frac{52\cdots 97}{69\cdots 29}a^{14}-\frac{21\cdots 42}{34\cdots 45}a^{13}-\frac{26\cdots 43}{69\cdots 29}a^{12}+\frac{17\cdots 26}{49\cdots 35}a^{11}+\frac{10\cdots 23}{69\cdots 29}a^{10}+\frac{66\cdots 40}{69\cdots 29}a^{9}+\frac{10\cdots 36}{34\cdots 45}a^{8}+\frac{13\cdots 21}{23\cdots 43}a^{7}+\frac{26\cdots 42}{49\cdots 35}a^{6}+\frac{30\cdots 41}{69\cdots 29}a^{5}+\frac{67\cdots 87}{23\cdots 43}a^{4}+\frac{21\cdots 17}{34\cdots 45}a^{3}+\frac{17\cdots 06}{69\cdots 29}a^{2}-\frac{14\cdots 29}{55\cdots 15}a-\frac{16\cdots 02}{99\cdots 47}$, $\frac{47\cdots 92}{29\cdots 41}a^{23}-\frac{10\cdots 86}{77\cdots 81}a^{22}+\frac{10\cdots 01}{69\cdots 29}a^{21}-\frac{34\cdots 57}{20\cdots 87}a^{20}+\frac{46\cdots 70}{29\cdots 41}a^{19}+\frac{15\cdots 01}{29\cdots 41}a^{18}-\frac{30\cdots 20}{20\cdots 87}a^{17}-\frac{24\cdots 13}{29\cdots 41}a^{16}-\frac{10\cdots 93}{99\cdots 47}a^{15}-\frac{25\cdots 38}{20\cdots 87}a^{14}+\frac{85\cdots 60}{20\cdots 87}a^{13}+\frac{36\cdots 83}{20\cdots 87}a^{12}-\frac{10\cdots 86}{20\cdots 87}a^{11}-\frac{28\cdots 76}{29\cdots 41}a^{10}-\frac{10\cdots 27}{20\cdots 87}a^{9}-\frac{32\cdots 69}{20\cdots 87}a^{8}-\frac{19\cdots 97}{69\cdots 29}a^{7}-\frac{52\cdots 30}{20\cdots 87}a^{6}-\frac{91\cdots 79}{29\cdots 41}a^{5}-\frac{10\cdots 14}{69\cdots 29}a^{4}-\frac{58\cdots 88}{20\cdots 87}a^{3}-\frac{22\cdots 90}{20\cdots 87}a^{2}+\frac{28\cdots 85}{23\cdots 43}a+\frac{22\cdots 33}{29\cdots 41}$, $\frac{24\cdots 38}{93\cdots 97}a^{23}-\frac{10\cdots 12}{49\cdots 73}a^{22}+\frac{53\cdots 53}{44\cdots 57}a^{21}-\frac{21\cdots 51}{93\cdots 97}a^{20}+\frac{30\cdots 77}{13\cdots 71}a^{19}+\frac{82\cdots 33}{93\cdots 97}a^{18}-\frac{14\cdots 69}{93\cdots 97}a^{17}-\frac{22\cdots 94}{13\cdots 71}a^{16}-\frac{55\cdots 20}{31\cdots 99}a^{15}-\frac{47\cdots 91}{13\cdots 71}a^{14}+\frac{61\cdots 23}{93\cdots 97}a^{13}+\frac{33\cdots 10}{93\cdots 97}a^{12}-\frac{56\cdots 94}{93\cdots 97}a^{11}-\frac{81\cdots 49}{49\cdots 63}a^{10}-\frac{49\cdots 97}{49\cdots 63}a^{9}-\frac{44\cdots 05}{13\cdots 71}a^{8}-\frac{20\cdots 76}{31\cdots 99}a^{7}-\frac{66\cdots 65}{93\cdots 97}a^{6}-\frac{64\cdots 12}{93\cdots 97}a^{5}-\frac{12\cdots 60}{44\cdots 57}a^{4}-\frac{62\cdots 22}{93\cdots 97}a^{3}-\frac{44\cdots 69}{93\cdots 97}a^{2}+\frac{26\cdots 08}{10\cdots 33}a+\frac{38\cdots 18}{13\cdots 71}$, $\frac{11\cdots 90}{17\cdots 43}a^{23}-\frac{43\cdots 19}{65\cdots 09}a^{22}+\frac{92\cdots 13}{59\cdots 81}a^{21}-\frac{17\cdots 02}{17\cdots 43}a^{20}+\frac{20\cdots 71}{25\cdots 49}a^{19}+\frac{35\cdots 15}{17\cdots 43}a^{18}-\frac{14\cdots 80}{17\cdots 43}a^{17}-\frac{50\cdots 19}{25\cdots 49}a^{16}-\frac{23\cdots 72}{59\cdots 81}a^{15}-\frac{22\cdots 98}{17\cdots 43}a^{14}+\frac{40\cdots 90}{25\cdots 49}a^{13}+\frac{91\cdots 35}{17\cdots 43}a^{12}-\frac{43\cdots 71}{17\cdots 43}a^{11}-\frac{61\cdots 77}{17\cdots 43}a^{10}-\frac{43\cdots 28}{25\cdots 49}a^{9}-\frac{84\cdots 65}{17\cdots 43}a^{8}-\frac{44\cdots 47}{59\cdots 81}a^{7}-\frac{10\cdots 65}{17\cdots 43}a^{6}-\frac{19\cdots 93}{17\cdots 43}a^{5}-\frac{27\cdots 12}{59\cdots 81}a^{4}-\frac{12\cdots 99}{17\cdots 43}a^{3}-\frac{32\cdots 66}{17\cdots 43}a^{2}+\frac{62\cdots 46}{19\cdots 27}a+\frac{43\cdots 49}{25\cdots 49}$, $\frac{22\cdots 32}{88\cdots 15}a^{23}+\frac{15\cdots 09}{65\cdots 09}a^{22}-\frac{10\cdots 86}{29\cdots 05}a^{21}+\frac{52\cdots 05}{17\cdots 43}a^{20}-\frac{67\cdots 51}{25\cdots 49}a^{19}-\frac{70\cdots 99}{88\cdots 15}a^{18}+\frac{45\cdots 18}{17\cdots 43}a^{17}+\frac{14\cdots 42}{12\cdots 45}a^{16}+\frac{92\cdots 53}{59\cdots 81}a^{15}+\frac{22\cdots 10}{17\cdots 43}a^{14}-\frac{55\cdots 43}{88\cdots 15}a^{13}-\frac{44\cdots 05}{17\cdots 43}a^{12}+\frac{76\cdots 58}{88\cdots 15}a^{11}+\frac{25\cdots 78}{17\cdots 43}a^{10}+\frac{13\cdots 69}{17\cdots 43}a^{9}+\frac{19\cdots 54}{88\cdots 15}a^{8}+\frac{22\cdots 15}{59\cdots 81}a^{7}+\frac{29\cdots 86}{88\cdots 15}a^{6}+\frac{80\cdots 07}{17\cdots 43}a^{5}+\frac{12\cdots 77}{59\cdots 81}a^{4}+\frac{33\cdots 13}{88\cdots 15}a^{3}+\frac{31\cdots 42}{25\cdots 49}a^{2}-\frac{16\cdots 42}{98\cdots 35}a-\frac{24\cdots 40}{25\cdots 49}$, $\frac{98\cdots 12}{29\cdots 05}a^{23}+\frac{21\cdots 47}{10\cdots 43}a^{22}+\frac{99\cdots 72}{14\cdots 05}a^{21}+\frac{90\cdots 49}{59\cdots 81}a^{20}-\frac{48\cdots 36}{84\cdots 83}a^{19}-\frac{28\cdots 89}{29\cdots 05}a^{18}+\frac{72\cdots 07}{59\cdots 81}a^{17}+\frac{10\cdots 27}{42\cdots 15}a^{16}+\frac{52\cdots 85}{19\cdots 27}a^{15}+\frac{20\cdots 00}{84\cdots 83}a^{14}-\frac{28\cdots 43}{29\cdots 05}a^{13}-\frac{24\cdots 69}{59\cdots 81}a^{12}+\frac{32\cdots 18}{29\cdots 05}a^{11}+\frac{12\cdots 79}{59\cdots 81}a^{10}+\frac{74\cdots 42}{59\cdots 81}a^{9}+\frac{19\cdots 02}{42\cdots 15}a^{8}+\frac{21\cdots 47}{19\cdots 27}a^{7}+\frac{56\cdots 41}{29\cdots 05}a^{6}+\frac{13\cdots 26}{59\cdots 81}a^{5}+\frac{35\cdots 48}{28\cdots 61}a^{4}-\frac{13\cdots 17}{29\cdots 05}a^{3}-\frac{56\cdots 81}{59\cdots 81}a^{2}-\frac{10\cdots 07}{32\cdots 45}a+\frac{16\cdots 40}{84\cdots 83}$, $\frac{88\cdots 46}{88\cdots 15}a^{23}-\frac{57\cdots 18}{65\cdots 09}a^{22}+\frac{91\cdots 13}{29\cdots 05}a^{21}-\frac{45\cdots 62}{17\cdots 43}a^{20}+\frac{17\cdots 20}{25\cdots 49}a^{19}+\frac{29\cdots 37}{88\cdots 15}a^{18}-\frac{17\cdots 79}{17\cdots 43}a^{17}-\frac{89\cdots 06}{12\cdots 45}a^{16}-\frac{29\cdots 62}{59\cdots 81}a^{15}-\frac{12\cdots 31}{17\cdots 43}a^{14}+\frac{24\cdots 49}{88\cdots 15}a^{13}+\frac{18\cdots 48}{17\cdots 43}a^{12}-\frac{43\cdots 64}{88\cdots 15}a^{11}-\frac{10\cdots 10}{17\cdots 43}a^{10}-\frac{50\cdots 18}{17\cdots 43}a^{9}-\frac{65\cdots 22}{88\cdots 15}a^{8}-\frac{75\cdots 53}{84\cdots 83}a^{7}-\frac{48\cdots 68}{88\cdots 15}a^{6}-\frac{41\cdots 70}{17\cdots 43}a^{5}-\frac{42\cdots 51}{59\cdots 81}a^{4}-\frac{66\cdots 99}{88\cdots 15}a^{3}-\frac{65\cdots 25}{17\cdots 43}a^{2}+\frac{36\cdots 21}{98\cdots 35}a+\frac{39\cdots 15}{25\cdots 49}$, $\frac{43\cdots 97}{88\cdots 15}a^{23}-\frac{29\cdots 23}{65\cdots 09}a^{22}+\frac{18\cdots 31}{29\cdots 05}a^{21}-\frac{91\cdots 29}{17\cdots 43}a^{20}+\frac{12\cdots 97}{25\cdots 49}a^{19}+\frac{14\cdots 59}{88\cdots 15}a^{18}-\frac{89\cdots 24}{17\cdots 43}a^{17}-\frac{30\cdots 92}{12\cdots 45}a^{16}-\frac{17\cdots 57}{59\cdots 81}a^{15}-\frac{55\cdots 70}{17\cdots 43}a^{14}+\frac{11\cdots 13}{88\cdots 15}a^{13}+\frac{86\cdots 09}{17\cdots 43}a^{12}-\frac{15\cdots 28}{88\cdots 15}a^{11}-\frac{50\cdots 35}{17\cdots 43}a^{10}-\frac{26\cdots 06}{17\cdots 43}a^{9}-\frac{38\cdots 69}{88\cdots 15}a^{8}-\frac{62\cdots 95}{84\cdots 83}a^{7}-\frac{54\cdots 91}{88\cdots 15}a^{6}-\frac{15\cdots 96}{17\cdots 43}a^{5}-\frac{25\cdots 61}{59\cdots 81}a^{4}-\frac{63\cdots 93}{88\cdots 15}a^{3}-\frac{39\cdots 68}{17\cdots 43}a^{2}+\frac{31\cdots 07}{98\cdots 35}a+\frac{46\cdots 81}{25\cdots 49}$, $\frac{72\cdots 78}{29\cdots 05}a^{23}+\frac{38\cdots 20}{31\cdots 29}a^{22}+\frac{11\cdots 88}{14\cdots 05}a^{21}-\frac{39\cdots 30}{59\cdots 81}a^{20}-\frac{59\cdots 16}{84\cdots 83}a^{19}-\frac{27\cdots 71}{29\cdots 05}a^{18}-\frac{26\cdots 00}{59\cdots 81}a^{17}+\frac{11\cdots 53}{42\cdots 15}a^{16}+\frac{33\cdots 66}{19\cdots 27}a^{15}+\frac{61\cdots 77}{84\cdots 83}a^{14}-\frac{20\cdots 27}{29\cdots 05}a^{13}-\frac{28\cdots 21}{59\cdots 81}a^{12}+\frac{83\cdots 92}{29\cdots 05}a^{11}+\frac{10\cdots 03}{59\cdots 81}a^{10}+\frac{73\cdots 70}{59\cdots 81}a^{9}+\frac{18\cdots 88}{42\cdots 15}a^{8}+\frac{17\cdots 12}{19\cdots 27}a^{7}+\frac{27\cdots 39}{29\cdots 05}a^{6}+\frac{38\cdots 71}{59\cdots 81}a^{5}+\frac{98\cdots 86}{28\cdots 61}a^{4}+\frac{28\cdots 92}{29\cdots 05}a^{3}+\frac{30\cdots 84}{59\cdots 81}a^{2}-\frac{13\cdots 53}{32\cdots 45}a-\frac{26\cdots 04}{84\cdots 83}$ (assuming GRH)
Regulator:		\( 1278504647863611700000 \) (assuming GRH)
Unit signature rank:		\( 3 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1278504647863611700000 \cdot 4}{2\cdot\sqrt{270761008401829353605241639483649123576469719409942626953125}}\cr\approx \mathstrut & 7.53976676560766 \end{aligned}\] (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

Degree 24 siblings:	24.4.6769025210045733840131040987091228089411742985248565673828125.1, 24.4.6769025210045733840131040987091228089411742985248565673828125.6
Arithmetically equivalent sibling:	24.4.270761008401829353605241639483649123576469719409942626953125.4
Minimal sibling:	24.4.270761008401829353605241639483649123576469719409942626953125.4

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.8.0.1}{8} }^{3}$	${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$	R	${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$	${\href{/padicField/11.12.0.1}{12} }^{2}$	${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$	$24$	${\href{/padicField/19.3.0.1}{3} }^{8}$	$24$	R	${\href{/padicField/31.12.0.1}{12} }^{2}$	${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$	$20{,}\,{\href{/padicField/41.4.0.1}{4} }$	${\href{/padicField/43.8.0.1}{8} }^{3}$	$24$	$24$	${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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	$x^{20} + 116$	$20$	$1$	$19$	$C_4\times D_5$	$$[\ ]_{20}^{2}$$

Spectrum of ring of integers