Normalized defining polynomial

\( x^{24} - 10 x^{23} + 38 x^{22} - 56 x^{21} - 24 x^{20} + 160 x^{19} - 70 x^{18} - 240 x^{17} + 80 x^{16} + \cdots - 10 \) x^24 - 10*x^23 + 38*x^22 - 56*x^21 - 24*x^20 + 160*x^19 - 70*x^18 - 240*x^17 + 80*x^16 + 890*x^15 - 1180*x^14 - 940*x^13 + 3410*x^12 - 2300*x^11 - 1700*x^10 + 4040*x^9 - 2450*x^8 - 580*x^7 + 1810*x^6 - 1260*x^5 + 240*x^4 + 200*x^3 - 180*x^2 + 60*x - 10

Invariants

Degree:		$24$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[4, 10]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(1280000000000000000000000000000000\) 1280000000000000000000000000000000 \(\medspace = 2^{38}\cdot 5^{31}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(23.96\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$2^{19/12}5^{31/20}\approx 36.31067590725307$
Ramified primes:		\(2\), \(5\) 2, 5	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:		$4$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{52\!\cdots\!41}a^{23}+\frac{11\!\cdots\!86}{52\!\cdots\!41}a^{22}-\frac{11\!\cdots\!67}{52\!\cdots\!41}a^{21}-\frac{13\!\cdots\!97}{52\!\cdots\!41}a^{20}-\frac{14\!\cdots\!48}{52\!\cdots\!41}a^{19}+\frac{24\!\cdots\!15}{52\!\cdots\!41}a^{18}-\frac{35\!\cdots\!87}{52\!\cdots\!41}a^{17}-\frac{71\!\cdots\!15}{52\!\cdots\!41}a^{16}-\frac{11\!\cdots\!67}{52\!\cdots\!41}a^{15}+\frac{17\!\cdots\!11}{52\!\cdots\!41}a^{14}-\frac{25\!\cdots\!77}{13\!\cdots\!79}a^{13}+\frac{13\!\cdots\!99}{52\!\cdots\!41}a^{12}-\frac{65\!\cdots\!91}{52\!\cdots\!41}a^{11}+\frac{35\!\cdots\!25}{52\!\cdots\!41}a^{10}-\frac{61\!\cdots\!85}{52\!\cdots\!41}a^{9}-\frac{21\!\cdots\!04}{52\!\cdots\!41}a^{8}-\frac{14\!\cdots\!12}{52\!\cdots\!41}a^{7}-\frac{22\!\cdots\!11}{52\!\cdots\!41}a^{6}+\frac{38\!\cdots\!45}{52\!\cdots\!41}a^{5}+\frac{60\!\cdots\!26}{52\!\cdots\!41}a^{4}+\frac{25\!\cdots\!17}{52\!\cdots\!41}a^{3}-\frac{16\!\cdots\!38}{52\!\cdots\!41}a^{2}+\frac{18\!\cdots\!04}{52\!\cdots\!41}a-\frac{19\!\cdots\!58}{52\!\cdots\!41}$ 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, a^21, a^22, 1/520479354110640911431833864098941*a^23 + 110673789410603700659666807805486/520479354110640911431833864098941*a^22 - 113673048953253682663931812850867/520479354110640911431833864098941*a^21 - 136829500776182009252434418079997/520479354110640911431833864098941*a^20 - 145894597599634356704667889223248/520479354110640911431833864098941*a^19 + 24837975506230685218345948331315/520479354110640911431833864098941*a^18 - 35247616123044560524956033467687/520479354110640911431833864098941*a^17 - 71631023966402027690452695288115/520479354110640911431833864098941*a^16 - 111793159587115991489739559618667/520479354110640911431833864098941*a^15 + 173072984832492959341276153545011/520479354110640911431833864098941*a^14 - 256738440600114026945506986377/1373296448840741191113018111079*a^13 + 13731700844623205666771249744499/520479354110640911431833864098941*a^12 - 65325665619756602966394491979691/520479354110640911431833864098941*a^11 + 35291965851479829436329951392625/520479354110640911431833864098941*a^10 - 61009415072036903708861465220385/520479354110640911431833864098941*a^9 - 210926065553162559742655297038304/520479354110640911431833864098941*a^8 - 144131101515619538639533187683712/520479354110640911431833864098941*a^7 - 220267973719210821816664682219711/520479354110640911431833864098941*a^6 + 38809070681690952162800513184945/520479354110640911431833864098941*a^5 + 60939277046380092753665393622026/520479354110640911431833864098941*a^4 + 252370687381087304542709306003417/520479354110640911431833864098941*a^3 - 162706669463372993400085290136938/520479354110640911431833864098941*a^2 + 180749189713796623059332718874604/520479354110640911431833864098941*a - 19764697556471588146064815636158/520479354110640911431833864098941

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$13$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$\frac{24\!\cdots\!09}{52\!\cdots\!41}a^{23}+\frac{25\!\cdots\!36}{52\!\cdots\!41}a^{22}-\frac{10\!\cdots\!91}{52\!\cdots\!41}a^{21}+\frac{15\!\cdots\!81}{52\!\cdots\!41}a^{20}+\frac{73\!\cdots\!95}{52\!\cdots\!41}a^{19}-\frac{52\!\cdots\!84}{52\!\cdots\!41}a^{18}+\frac{28\!\cdots\!06}{52\!\cdots\!41}a^{17}+\frac{80\!\cdots\!68}{52\!\cdots\!41}a^{16}-\frac{48\!\cdots\!16}{52\!\cdots\!41}a^{15}-\frac{26\!\cdots\!24}{52\!\cdots\!41}a^{14}+\frac{10\!\cdots\!82}{13\!\cdots\!79}a^{13}+\frac{29\!\cdots\!84}{52\!\cdots\!41}a^{12}-\frac{11\!\cdots\!93}{52\!\cdots\!41}a^{11}+\frac{69\!\cdots\!10}{52\!\cdots\!41}a^{10}+\frac{75\!\cdots\!84}{52\!\cdots\!41}a^{9}-\frac{14\!\cdots\!84}{52\!\cdots\!41}a^{8}+\frac{66\!\cdots\!26}{52\!\cdots\!41}a^{7}+\frac{41\!\cdots\!62}{52\!\cdots\!41}a^{6}-\frac{69\!\cdots\!80}{52\!\cdots\!41}a^{5}+\frac{33\!\cdots\!36}{52\!\cdots\!41}a^{4}-\frac{19\!\cdots\!22}{52\!\cdots\!41}a^{3}-\frac{94\!\cdots\!21}{52\!\cdots\!41}a^{2}+\frac{54\!\cdots\!70}{52\!\cdots\!41}a-\frac{15\!\cdots\!41}{52\!\cdots\!41}$, $\frac{13\!\cdots\!25}{52\!\cdots\!41}a^{23}+\frac{12\!\cdots\!68}{52\!\cdots\!41}a^{22}-\frac{41\!\cdots\!88}{52\!\cdots\!41}a^{21}+\frac{41\!\cdots\!21}{52\!\cdots\!41}a^{20}+\frac{77\!\cdots\!12}{52\!\cdots\!41}a^{19}-\frac{18\!\cdots\!06}{52\!\cdots\!41}a^{18}-\frac{34\!\cdots\!84}{52\!\cdots\!41}a^{17}+\frac{35\!\cdots\!11}{52\!\cdots\!41}a^{16}+\frac{10\!\cdots\!77}{52\!\cdots\!41}a^{15}-\frac{11\!\cdots\!61}{52\!\cdots\!41}a^{14}+\frac{20\!\cdots\!57}{13\!\cdots\!79}a^{13}+\frac{21\!\cdots\!01}{52\!\cdots\!41}a^{12}-\frac{36\!\cdots\!51}{52\!\cdots\!41}a^{11}+\frac{36\!\cdots\!38}{52\!\cdots\!41}a^{10}+\frac{38\!\cdots\!35}{52\!\cdots\!41}a^{9}-\frac{39\!\cdots\!27}{52\!\cdots\!41}a^{8}+\frac{28\!\cdots\!53}{52\!\cdots\!41}a^{7}+\frac{23\!\cdots\!34}{52\!\cdots\!41}a^{6}-\frac{17\!\cdots\!15}{52\!\cdots\!41}a^{5}+\frac{32\!\cdots\!98}{52\!\cdots\!41}a^{4}+\frac{49\!\cdots\!28}{52\!\cdots\!41}a^{3}-\frac{25\!\cdots\!36}{52\!\cdots\!41}a^{2}+\frac{26\!\cdots\!77}{52\!\cdots\!41}a+\frac{25\!\cdots\!27}{52\!\cdots\!41}$, $\frac{10\!\cdots\!65}{52\!\cdots\!41}a^{23}-\frac{10\!\cdots\!71}{52\!\cdots\!41}a^{22}+\frac{38\!\cdots\!19}{52\!\cdots\!41}a^{21}-\frac{51\!\cdots\!92}{52\!\cdots\!41}a^{20}-\frac{38\!\cdots\!28}{52\!\cdots\!41}a^{19}+\frac{16\!\cdots\!55}{52\!\cdots\!41}a^{18}-\frac{32\!\cdots\!80}{52\!\cdots\!41}a^{17}-\frac{27\!\cdots\!36}{52\!\cdots\!41}a^{16}+\frac{18\!\cdots\!06}{52\!\cdots\!41}a^{15}+\frac{97\!\cdots\!22}{52\!\cdots\!41}a^{14}-\frac{27\!\cdots\!63}{13\!\cdots\!79}a^{13}-\frac{12\!\cdots\!02}{52\!\cdots\!41}a^{12}+\frac{33\!\cdots\!66}{52\!\cdots\!41}a^{11}-\frac{15\!\cdots\!78}{52\!\cdots\!41}a^{10}-\frac{22\!\cdots\!60}{52\!\cdots\!41}a^{9}+\frac{37\!\cdots\!97}{52\!\cdots\!41}a^{8}-\frac{15\!\cdots\!13}{52\!\cdots\!41}a^{7}-\frac{11\!\cdots\!04}{52\!\cdots\!41}a^{6}+\frac{16\!\cdots\!91}{52\!\cdots\!41}a^{5}-\frac{86\!\cdots\!71}{52\!\cdots\!41}a^{4}-\frac{21\!\cdots\!19}{52\!\cdots\!41}a^{3}+\frac{24\!\cdots\!42}{52\!\cdots\!41}a^{2}-\frac{11\!\cdots\!73}{52\!\cdots\!41}a+\frac{26\!\cdots\!39}{52\!\cdots\!41}$, $\frac{73\!\cdots\!18}{52\!\cdots\!41}a^{23}+\frac{59\!\cdots\!16}{52\!\cdots\!41}a^{22}-\frac{15\!\cdots\!91}{52\!\cdots\!41}a^{21}+\frac{44\!\cdots\!24}{52\!\cdots\!41}a^{20}+\frac{40\!\cdots\!93}{52\!\cdots\!41}a^{19}-\frac{37\!\cdots\!88}{52\!\cdots\!41}a^{18}-\frac{48\!\cdots\!66}{52\!\cdots\!41}a^{17}+\frac{40\!\cdots\!77}{52\!\cdots\!41}a^{16}+\frac{15\!\cdots\!66}{52\!\cdots\!41}a^{15}-\frac{29\!\cdots\!35}{52\!\cdots\!41}a^{14}-\frac{21\!\cdots\!40}{13\!\cdots\!79}a^{13}+\frac{70\!\cdots\!69}{52\!\cdots\!41}a^{12}-\frac{56\!\cdots\!77}{52\!\cdots\!41}a^{11}+\frac{16\!\cdots\!48}{52\!\cdots\!41}a^{10}+\frac{37\!\cdots\!77}{52\!\cdots\!41}a^{9}-\frac{13\!\cdots\!11}{52\!\cdots\!41}a^{8}+\frac{10\!\cdots\!24}{52\!\cdots\!41}a^{7}+\frac{52\!\cdots\!88}{52\!\cdots\!41}a^{6}-\frac{94\!\cdots\!25}{52\!\cdots\!41}a^{5}+\frac{10\!\cdots\!10}{52\!\cdots\!41}a^{4}+\frac{43\!\cdots\!87}{52\!\cdots\!41}a^{3}-\frac{86\!\cdots\!89}{52\!\cdots\!41}a^{2}+\frac{29\!\cdots\!95}{52\!\cdots\!41}a-\frac{29\!\cdots\!13}{52\!\cdots\!41}$, $\frac{57\!\cdots\!61}{52\!\cdots\!41}a^{23}-\frac{50\!\cdots\!79}{52\!\cdots\!41}a^{22}+\frac{14\!\cdots\!14}{52\!\cdots\!41}a^{21}-\frac{75\!\cdots\!61}{52\!\cdots\!41}a^{20}-\frac{43\!\cdots\!88}{52\!\cdots\!41}a^{19}+\frac{60\!\cdots\!40}{52\!\cdots\!41}a^{18}+\frac{63\!\cdots\!72}{52\!\cdots\!41}a^{17}-\frac{13\!\cdots\!89}{52\!\cdots\!41}a^{16}-\frac{13\!\cdots\!36}{52\!\cdots\!41}a^{15}+\frac{48\!\cdots\!59}{52\!\cdots\!41}a^{14}-\frac{58\!\cdots\!46}{13\!\cdots\!79}a^{13}-\frac{11\!\cdots\!61}{52\!\cdots\!41}a^{12}+\frac{98\!\cdots\!02}{52\!\cdots\!41}a^{11}+\frac{74\!\cdots\!56}{52\!\cdots\!41}a^{10}-\frac{16\!\cdots\!28}{52\!\cdots\!41}a^{9}+\frac{66\!\cdots\!81}{52\!\cdots\!41}a^{8}+\frac{72\!\cdots\!65}{52\!\cdots\!41}a^{7}-\frac{92\!\cdots\!22}{52\!\cdots\!41}a^{6}+\frac{20\!\cdots\!59}{52\!\cdots\!41}a^{5}+\frac{13\!\cdots\!49}{52\!\cdots\!41}a^{4}-\frac{21\!\cdots\!01}{52\!\cdots\!41}a^{3}+\frac{36\!\cdots\!02}{52\!\cdots\!41}a^{2}+\frac{17\!\cdots\!63}{52\!\cdots\!41}a-\frac{12\!\cdots\!39}{52\!\cdots\!41}$, $\frac{79\!\cdots\!16}{52\!\cdots\!41}a^{23}-\frac{75\!\cdots\!81}{52\!\cdots\!41}a^{22}+\frac{26\!\cdots\!35}{52\!\cdots\!41}a^{21}-\frac{29\!\cdots\!28}{52\!\cdots\!41}a^{20}-\frac{42\!\cdots\!43}{52\!\cdots\!41}a^{19}+\frac{12\!\cdots\!88}{52\!\cdots\!41}a^{18}-\frac{93\!\cdots\!40}{52\!\cdots\!41}a^{17}-\frac{21\!\cdots\!91}{52\!\cdots\!41}a^{16}-\frac{20\!\cdots\!70}{52\!\cdots\!41}a^{15}+\frac{74\!\cdots\!71}{52\!\cdots\!41}a^{14}-\frac{16\!\cdots\!77}{13\!\cdots\!79}a^{13}-\frac{12\!\cdots\!33}{52\!\cdots\!41}a^{12}+\frac{24\!\cdots\!50}{52\!\cdots\!41}a^{11}-\frac{61\!\cdots\!59}{52\!\cdots\!41}a^{10}-\frac{22\!\cdots\!76}{52\!\cdots\!41}a^{9}+\frac{27\!\cdots\!73}{52\!\cdots\!41}a^{8}-\frac{64\!\cdots\!49}{52\!\cdots\!41}a^{7}-\frac{13\!\cdots\!46}{52\!\cdots\!41}a^{6}+\frac{13\!\cdots\!37}{52\!\cdots\!41}a^{5}-\frac{49\!\cdots\!47}{52\!\cdots\!41}a^{4}-\frac{13\!\cdots\!79}{52\!\cdots\!41}a^{3}+\frac{20\!\cdots\!02}{52\!\cdots\!41}a^{2}-\frac{83\!\cdots\!13}{52\!\cdots\!41}a+\frac{97\!\cdots\!41}{52\!\cdots\!41}$, $\frac{10\!\cdots\!07}{52\!\cdots\!41}a^{23}+\frac{99\!\cdots\!09}{52\!\cdots\!41}a^{22}-\frac{32\!\cdots\!25}{52\!\cdots\!41}a^{21}+\frac{29\!\cdots\!03}{52\!\cdots\!41}a^{20}+\frac{62\!\cdots\!15}{52\!\cdots\!41}a^{19}-\frac{12\!\cdots\!02}{52\!\cdots\!41}a^{18}-\frac{69\!\cdots\!62}{52\!\cdots\!41}a^{17}+\frac{26\!\cdots\!93}{52\!\cdots\!41}a^{16}+\frac{17\!\cdots\!27}{52\!\cdots\!41}a^{15}-\frac{94\!\cdots\!65}{52\!\cdots\!41}a^{14}+\frac{10\!\cdots\!78}{13\!\cdots\!79}a^{13}+\frac{17\!\cdots\!39}{52\!\cdots\!41}a^{12}-\frac{22\!\cdots\!97}{52\!\cdots\!41}a^{11}-\frac{39\!\cdots\!20}{52\!\cdots\!41}a^{10}+\frac{26\!\cdots\!84}{52\!\cdots\!41}a^{9}-\frac{17\!\cdots\!51}{52\!\cdots\!41}a^{8}-\frac{52\!\cdots\!80}{52\!\cdots\!41}a^{7}+\frac{14\!\cdots\!23}{52\!\cdots\!41}a^{6}-\frac{55\!\cdots\!62}{52\!\cdots\!41}a^{5}-\frac{71\!\cdots\!77}{52\!\cdots\!41}a^{4}+\frac{31\!\cdots\!28}{52\!\cdots\!41}a^{3}-\frac{10\!\cdots\!91}{52\!\cdots\!41}a^{2}-\frac{39\!\cdots\!41}{52\!\cdots\!41}a+\frac{18\!\cdots\!59}{52\!\cdots\!41}$, $\frac{54\!\cdots\!90}{52\!\cdots\!41}a^{23}-\frac{59\!\cdots\!07}{52\!\cdots\!41}a^{22}+\frac{25\!\cdots\!65}{52\!\cdots\!41}a^{21}-\frac{46\!\cdots\!84}{52\!\cdots\!41}a^{20}+\frac{89\!\cdots\!87}{52\!\cdots\!41}a^{19}+\frac{12\!\cdots\!48}{52\!\cdots\!41}a^{18}-\frac{10\!\cdots\!04}{52\!\cdots\!41}a^{17}-\frac{15\!\cdots\!66}{52\!\cdots\!41}a^{16}+\frac{18\!\cdots\!23}{52\!\cdots\!41}a^{15}+\frac{55\!\cdots\!86}{52\!\cdots\!41}a^{14}-\frac{29\!\cdots\!93}{13\!\cdots\!79}a^{13}-\frac{28\!\cdots\!95}{52\!\cdots\!41}a^{12}+\frac{27\!\cdots\!49}{52\!\cdots\!41}a^{11}-\frac{25\!\cdots\!57}{52\!\cdots\!41}a^{10}-\frac{10\!\cdots\!51}{52\!\cdots\!41}a^{9}+\frac{37\!\cdots\!92}{52\!\cdots\!41}a^{8}-\frac{26\!\cdots\!89}{52\!\cdots\!41}a^{7}-\frac{33\!\cdots\!77}{52\!\cdots\!41}a^{6}+\frac{18\!\cdots\!84}{52\!\cdots\!41}a^{5}-\frac{13\!\cdots\!57}{52\!\cdots\!41}a^{4}+\frac{29\!\cdots\!11}{52\!\cdots\!41}a^{3}+\frac{26\!\cdots\!01}{52\!\cdots\!41}a^{2}-\frac{20\!\cdots\!81}{52\!\cdots\!41}a+\frac{61\!\cdots\!43}{52\!\cdots\!41}$, $\frac{71\!\cdots\!17}{52\!\cdots\!41}a^{23}+\frac{66\!\cdots\!89}{52\!\cdots\!41}a^{22}-\frac{22\!\cdots\!05}{52\!\cdots\!41}a^{21}+\frac{23\!\cdots\!32}{52\!\cdots\!41}a^{20}+\frac{38\!\cdots\!42}{52\!\cdots\!41}a^{19}-\frac{94\!\cdots\!44}{52\!\cdots\!41}a^{18}-\frac{26\!\cdots\!50}{52\!\cdots\!41}a^{17}+\frac{18\!\cdots\!58}{52\!\cdots\!41}a^{16}+\frac{69\!\cdots\!32}{52\!\cdots\!41}a^{15}-\frac{65\!\cdots\!36}{52\!\cdots\!41}a^{14}+\frac{10\!\cdots\!68}{13\!\cdots\!79}a^{13}+\frac{11\!\cdots\!80}{52\!\cdots\!41}a^{12}-\frac{18\!\cdots\!41}{52\!\cdots\!41}a^{11}+\frac{43\!\cdots\!17}{52\!\cdots\!41}a^{10}+\frac{19\!\cdots\!04}{52\!\cdots\!41}a^{9}-\frac{16\!\cdots\!47}{52\!\cdots\!41}a^{8}-\frac{17\!\cdots\!58}{52\!\cdots\!41}a^{7}+\frac{10\!\cdots\!80}{52\!\cdots\!41}a^{6}-\frac{54\!\cdots\!42}{52\!\cdots\!41}a^{5}-\frac{10\!\cdots\!82}{52\!\cdots\!41}a^{4}+\frac{19\!\cdots\!88}{52\!\cdots\!41}a^{3}-\frac{78\!\cdots\!71}{52\!\cdots\!41}a^{2}+\frac{73\!\cdots\!70}{52\!\cdots\!41}a+\frac{81\!\cdots\!27}{52\!\cdots\!41}$, $\frac{34\!\cdots\!12}{52\!\cdots\!41}a^{23}-\frac{34\!\cdots\!32}{52\!\cdots\!41}a^{22}+\frac{13\!\cdots\!90}{52\!\cdots\!41}a^{21}-\frac{21\!\cdots\!23}{52\!\cdots\!41}a^{20}-\frac{45\!\cdots\!28}{52\!\cdots\!41}a^{19}+\frac{50\!\cdots\!10}{52\!\cdots\!41}a^{18}-\frac{21\!\cdots\!39}{52\!\cdots\!41}a^{17}-\frac{81\!\cdots\!35}{52\!\cdots\!41}a^{16}+\frac{22\!\cdots\!74}{52\!\cdots\!41}a^{15}+\frac{30\!\cdots\!53}{52\!\cdots\!41}a^{14}-\frac{11\!\cdots\!86}{13\!\cdots\!79}a^{13}-\frac{27\!\cdots\!63}{52\!\cdots\!41}a^{12}+\frac{10\!\cdots\!83}{52\!\cdots\!41}a^{11}-\frac{77\!\cdots\!89}{52\!\cdots\!41}a^{10}-\frac{43\!\cdots\!23}{52\!\cdots\!41}a^{9}+\frac{11\!\cdots\!68}{52\!\cdots\!41}a^{8}-\frac{70\!\cdots\!67}{52\!\cdots\!41}a^{7}-\frac{12\!\cdots\!38}{52\!\cdots\!41}a^{6}+\frac{43\!\cdots\!35}{52\!\cdots\!41}a^{5}-\frac{28\!\cdots\!72}{52\!\cdots\!41}a^{4}+\frac{42\!\cdots\!35}{52\!\cdots\!41}a^{3}+\frac{57\!\cdots\!56}{52\!\cdots\!41}a^{2}-\frac{33\!\cdots\!10}{52\!\cdots\!41}a+\frac{11\!\cdots\!17}{52\!\cdots\!41}$, $\frac{88\!\cdots\!05}{52\!\cdots\!41}a^{23}-\frac{88\!\cdots\!68}{52\!\cdots\!41}a^{22}+\frac{33\!\cdots\!22}{52\!\cdots\!41}a^{21}-\frac{46\!\cdots\!63}{52\!\cdots\!41}a^{20}-\frac{28\!\cdots\!41}{52\!\cdots\!41}a^{19}+\frac{14\!\cdots\!54}{52\!\cdots\!41}a^{18}-\frac{46\!\cdots\!32}{52\!\cdots\!41}a^{17}-\frac{23\!\cdots\!02}{52\!\cdots\!41}a^{16}+\frac{46\!\cdots\!55}{52\!\cdots\!41}a^{15}+\frac{83\!\cdots\!09}{52\!\cdots\!41}a^{14}-\frac{25\!\cdots\!83}{13\!\cdots\!79}a^{13}-\frac{10\!\cdots\!34}{52\!\cdots\!41}a^{12}+\frac{30\!\cdots\!12}{52\!\cdots\!41}a^{11}-\frac{16\!\cdots\!87}{52\!\cdots\!41}a^{10}-\frac{18\!\cdots\!22}{52\!\cdots\!41}a^{9}+\frac{34\!\cdots\!68}{52\!\cdots\!41}a^{8}-\frac{16\!\cdots\!37}{52\!\cdots\!41}a^{7}-\frac{78\!\cdots\!55}{52\!\cdots\!41}a^{6}+\frac{14\!\cdots\!72}{52\!\cdots\!41}a^{5}-\frac{85\!\cdots\!62}{52\!\cdots\!41}a^{4}+\frac{12\!\cdots\!14}{52\!\cdots\!41}a^{3}+\frac{16\!\cdots\!17}{52\!\cdots\!41}a^{2}-\frac{91\!\cdots\!13}{52\!\cdots\!41}a+\frac{23\!\cdots\!77}{52\!\cdots\!41}$, $\frac{25\!\cdots\!77}{52\!\cdots\!41}a^{23}-\frac{19\!\cdots\!91}{52\!\cdots\!41}a^{22}+\frac{43\!\cdots\!95}{52\!\cdots\!41}a^{21}+\frac{47\!\cdots\!56}{52\!\cdots\!41}a^{20}-\frac{29\!\cdots\!95}{52\!\cdots\!41}a^{19}+\frac{19\!\cdots\!35}{52\!\cdots\!41}a^{18}+\frac{58\!\cdots\!52}{52\!\cdots\!41}a^{17}-\frac{65\!\cdots\!65}{52\!\cdots\!41}a^{16}-\frac{11\!\cdots\!39}{52\!\cdots\!41}a^{15}+\frac{21\!\cdots\!32}{52\!\cdots\!41}a^{14}+\frac{50\!\cdots\!88}{13\!\cdots\!79}a^{13}-\frac{69\!\cdots\!13}{52\!\cdots\!41}a^{12}+\frac{17\!\cdots\!06}{52\!\cdots\!41}a^{11}+\frac{97\!\cdots\!73}{52\!\cdots\!41}a^{10}-\frac{10\!\cdots\!36}{52\!\cdots\!41}a^{9}-\frac{12\!\cdots\!43}{52\!\cdots\!41}a^{8}+\frac{99\!\cdots\!92}{52\!\cdots\!41}a^{7}-\frac{64\!\cdots\!12}{52\!\cdots\!41}a^{6}-\frac{12\!\cdots\!92}{52\!\cdots\!41}a^{5}+\frac{31\!\cdots\!75}{52\!\cdots\!41}a^{4}-\frac{20\!\cdots\!45}{52\!\cdots\!41}a^{3}-\frac{10\!\cdots\!96}{52\!\cdots\!41}a^{2}+\frac{35\!\cdots\!37}{52\!\cdots\!41}a-\frac{16\!\cdots\!63}{52\!\cdots\!41}$, $\frac{22\!\cdots\!59}{52\!\cdots\!41}a^{23}-\frac{21\!\cdots\!46}{52\!\cdots\!41}a^{22}+\frac{76\!\cdots\!72}{52\!\cdots\!41}a^{21}-\frac{93\!\cdots\!95}{52\!\cdots\!41}a^{20}-\frac{98\!\cdots\!68}{52\!\cdots\!41}a^{19}+\frac{33\!\cdots\!91}{52\!\cdots\!41}a^{18}-\frac{24\!\cdots\!84}{52\!\cdots\!41}a^{17}-\frac{58\!\cdots\!16}{52\!\cdots\!41}a^{16}-\frac{29\!\cdots\!29}{52\!\cdots\!41}a^{15}+\frac{20\!\cdots\!14}{52\!\cdots\!41}a^{14}-\frac{49\!\cdots\!48}{13\!\cdots\!79}a^{13}-\frac{30\!\cdots\!04}{52\!\cdots\!41}a^{12}+\frac{67\!\cdots\!56}{52\!\cdots\!41}a^{11}-\frac{23\!\cdots\!72}{52\!\cdots\!41}a^{10}-\frac{55\!\cdots\!80}{52\!\cdots\!41}a^{9}+\frac{73\!\cdots\!45}{52\!\cdots\!41}a^{8}-\frac{22\!\cdots\!34}{52\!\cdots\!41}a^{7}-\frac{29\!\cdots\!07}{52\!\cdots\!41}a^{6}+\frac{32\!\cdots\!26}{52\!\cdots\!41}a^{5}-\frac{13\!\cdots\!53}{52\!\cdots\!41}a^{4}-\frac{26\!\cdots\!76}{52\!\cdots\!41}a^{3}+\frac{47\!\cdots\!32}{52\!\cdots\!41}a^{2}-\frac{19\!\cdots\!70}{52\!\cdots\!41}a+\frac{17\!\cdots\!61}{52\!\cdots\!41}$ 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305469597290992037094557335166731504/520479354110640911431833864098941*a^12 + 676786210988552287129899478234584056/520479354110640911431833864098941*a^11 - 231074495161000517060378421680176872/520479354110640911431833864098941*a^10 - 550562295001099814399791979140591380/520479354110640911431833864098941*a^9 + 738416561642461504873159947101839645/520479354110640911431833864098941*a^8 - 220815229957250248150596027516017834/520479354110640911431833864098941*a^7 - 295027584305014247170490422697141207/520479354110640911431833864098941*a^6 + 326419712696332963087898259632190426/520479354110640911431833864098941*a^5 - 134445129929812681386714869487518753/520479354110640911431833864098941*a^4 - 26606082217139191518088618818797076/520479354110640911431833864098941*a^3 + 47354437468279032630314851040364632/520479354110640911431833864098941*a^2 - 19096815887415767795413840265176070/520479354110640911431833864098941*a + 1756666768161166970688108085926761/520479354110640911431833864098941 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 20547515.71936001 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 20547515.71936001 \cdot 1}{2\cdot\sqrt{1280000000000000000000000000000000}}\cr\approx \mathstrut & 0.440598492731830 \end{aligned}\] (assuming GRH)

Galois group

$\GL(2,5)$ (as 24T1353):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Frobenius cycle types

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
\(2\) 2		Deg $24$	$24$	$1$	$38$
\(5\) 5	5.4.0.1	$x^{4} + 4 x^{2} + 4 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $20$	$20$	$1$	$31$