Normalized defining polynomial

\( x^{24} - 6 x^{23} + 11 x^{22} - x^{21} - 42 x^{20} + 132 x^{19} - 157 x^{18} - 93 x^{17} + 463 x^{16} + \cdots + 83 \) x^24 - 6*x^23 + 11*x^22 - x^21 - 42*x^20 + 132*x^19 - 157*x^18 - 93*x^17 + 463*x^16 - 504*x^15 + 89*x^14 + 756*x^13 - 1626*x^12 + 841*x^11 + 2572*x^10 - 3447*x^9 - 1348*x^8 + 4593*x^7 - 688*x^6 - 3121*x^5 + 1191*x^4 + 1049*x^3 - 514*x^2 - 151*x + 83

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:		\(641953627807088196277618408203125\) 641953627807088196277618408203125 \(\medspace = 5^{31}\cdot 13^{10}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(23.28\)	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:		$5^{31/20}13^{1/2}\approx 43.68930970521314$
Ramified primes:		\(5\), \(13\) 5, 13	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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{13}+\frac{1}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{16}+\frac{2}{5}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{17}+\frac{2}{5}a^{13}-\frac{1}{5}a^{12}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{13}-\frac{2}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{5}a^{19}-\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{20}+\frac{1}{5}a^{12}+\frac{1}{5}a^{11}+\frac{1}{5}a^{10}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{2}-\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{5}a^{21}+\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{1}{5}a^{11}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{25}a^{22}-\frac{2}{25}a^{21}-\frac{1}{25}a^{20}-\frac{2}{25}a^{19}-\frac{1}{25}a^{18}+\frac{1}{25}a^{17}+\frac{1}{25}a^{16}+\frac{2}{25}a^{15}+\frac{2}{25}a^{14}+\frac{6}{25}a^{13}-\frac{1}{5}a^{12}-\frac{8}{25}a^{11}+\frac{7}{25}a^{10}+\frac{11}{25}a^{9}-\frac{2}{25}a^{8}+\frac{11}{25}a^{7}+\frac{9}{25}a^{6}-\frac{2}{5}a^{5}+\frac{11}{25}a^{4}-\frac{12}{25}a^{3}-\frac{1}{25}a^{2}-\frac{12}{25}a-\frac{8}{25}$, $\frac{1}{10\!\cdots\!25}a^{23}+\frac{19\!\cdots\!13}{10\!\cdots\!25}a^{22}-\frac{84\!\cdots\!51}{10\!\cdots\!25}a^{21}+\frac{82\!\cdots\!73}{10\!\cdots\!25}a^{20}-\frac{46\!\cdots\!36}{10\!\cdots\!25}a^{19}-\frac{34\!\cdots\!19}{10\!\cdots\!25}a^{18}+\frac{83\!\cdots\!76}{10\!\cdots\!25}a^{17}+\frac{83\!\cdots\!97}{10\!\cdots\!25}a^{16}-\frac{10\!\cdots\!53}{10\!\cdots\!25}a^{15}+\frac{82\!\cdots\!01}{10\!\cdots\!25}a^{14}-\frac{61\!\cdots\!26}{20\!\cdots\!85}a^{13}-\frac{37\!\cdots\!43}{10\!\cdots\!25}a^{12}-\frac{36\!\cdots\!18}{10\!\cdots\!25}a^{11}+\frac{27\!\cdots\!56}{10\!\cdots\!25}a^{10}-\frac{59\!\cdots\!87}{10\!\cdots\!25}a^{9}-\frac{19\!\cdots\!24}{10\!\cdots\!25}a^{8}-\frac{37\!\cdots\!56}{10\!\cdots\!25}a^{7}+\frac{15\!\cdots\!91}{40\!\cdots\!37}a^{6}-\frac{12\!\cdots\!09}{10\!\cdots\!25}a^{5}-\frac{22\!\cdots\!17}{10\!\cdots\!25}a^{4}+\frac{30\!\cdots\!79}{10\!\cdots\!25}a^{3}-\frac{10\!\cdots\!07}{10\!\cdots\!25}a^{2}-\frac{19\!\cdots\!63}{10\!\cdots\!25}a-\frac{45\!\cdots\!56}{40\!\cdots\!37}$ 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, 1/5*a^14 - 1/5*a^13 + 1/5*a^12 - 1/5*a^11 - 2/5*a^10 + 1/5*a^9 - 1/5*a^8 + 1/5*a^7 - 1/5*a^6 - 2/5*a^5 - 1/5*a^4 + 1/5*a^3 - 1/5*a^2 + 1/5*a + 2/5, 1/5*a^15 + 2/5*a^11 - 1/5*a^10 + 2/5*a^6 + 2/5*a^5 - 2/5*a + 2/5, 1/5*a^16 + 2/5*a^12 - 1/5*a^11 + 2/5*a^7 + 2/5*a^6 - 2/5*a^2 + 2/5*a, 1/5*a^17 + 2/5*a^13 - 1/5*a^12 + 2/5*a^8 + 2/5*a^7 - 2/5*a^3 + 2/5*a^2, 1/5*a^18 + 1/5*a^13 - 2/5*a^12 + 2/5*a^11 - 1/5*a^10 - 1/5*a^8 - 2/5*a^7 + 2/5*a^6 - 1/5*a^5 + 2/5*a^2 - 2/5*a + 1/5, 1/5*a^19 - 1/5*a^13 + 1/5*a^12 + 2/5*a^10 - 2/5*a^9 - 1/5*a^8 + 1/5*a^7 + 2/5*a^5 + 1/5*a^4 + 1/5*a^3 - 1/5*a^2 - 2/5, 1/5*a^20 + 1/5*a^12 + 1/5*a^11 + 1/5*a^10 + 1/5*a^7 + 1/5*a^6 - 1/5*a^5 - 1/5*a^2 - 1/5*a + 2/5, 1/5*a^21 + 1/5*a^13 + 1/5*a^12 + 1/5*a^11 + 1/5*a^8 + 1/5*a^7 - 1/5*a^6 - 1/5*a^3 - 1/5*a^2 + 2/5*a, 1/25*a^22 - 2/25*a^21 - 1/25*a^20 - 2/25*a^19 - 1/25*a^18 + 1/25*a^17 + 1/25*a^16 + 2/25*a^15 + 2/25*a^14 + 6/25*a^13 - 1/5*a^12 - 8/25*a^11 + 7/25*a^10 + 11/25*a^9 - 2/25*a^8 + 11/25*a^7 + 9/25*a^6 - 2/5*a^5 + 11/25*a^4 - 12/25*a^3 - 1/25*a^2 - 12/25*a - 8/25, 1/1014050943578398939313708163425*a^23 + 1987102169012311200135307913/1014050943578398939313708163425*a^22 - 84036942328064213803220892351/1014050943578398939313708163425*a^21 + 82006517465629168325170225773/1014050943578398939313708163425*a^20 - 46446250998117769010609394836/1014050943578398939313708163425*a^19 - 34000524131342533890715600719/1014050943578398939313708163425*a^18 + 83947510266550272921106945376/1014050943578398939313708163425*a^17 + 83715903037287019276355735997/1014050943578398939313708163425*a^16 - 101011533148188715707802017953/1014050943578398939313708163425*a^15 + 82501894148991327979699832201/1014050943578398939313708163425*a^14 - 61163048292993752721731477826/202810188715679787862741632685*a^13 - 379263878711700807326203159343/1014050943578398939313708163425*a^12 - 365512104674739038148459123118/1014050943578398939313708163425*a^11 + 279584528655565761687846801156/1014050943578398939313708163425*a^10 - 59869075910670747922092697687/1014050943578398939313708163425*a^9 - 191167633138827872174861342824/1014050943578398939313708163425*a^8 - 377446875134422810210099205956/1014050943578398939313708163425*a^7 + 15094939585988882810052810691/40562037743135957572548326537*a^6 - 12890438801603246481975610109/1014050943578398939313708163425*a^5 - 228063976867485331165754996217/1014050943578398939313708163425*a^4 + 305171643866889032194600426879/1014050943578398939313708163425*a^3 - 100501963764302003954055615107/1014050943578398939313708163425*a^2 - 199553791282672646554678613963/1014050943578398939313708163425*a - 4564165417551972975441744756/40562037743135957572548326537

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:		$\frac{29\!\cdots\!32}{37\!\cdots\!05}a^{23}-\frac{15\!\cdots\!92}{37\!\cdots\!05}a^{22}+\frac{22\!\cdots\!84}{37\!\cdots\!05}a^{21}+\frac{11\!\cdots\!48}{37\!\cdots\!05}a^{20}-\frac{11\!\cdots\!76}{37\!\cdots\!05}a^{19}+\frac{62\!\cdots\!23}{74\!\cdots\!81}a^{18}-\frac{26\!\cdots\!11}{37\!\cdots\!05}a^{17}-\frac{44\!\cdots\!19}{37\!\cdots\!05}a^{16}+\frac{10\!\cdots\!53}{37\!\cdots\!05}a^{15}-\frac{79\!\cdots\!98}{37\!\cdots\!05}a^{14}-\frac{24\!\cdots\!99}{37\!\cdots\!05}a^{13}+\frac{20\!\cdots\!74}{37\!\cdots\!05}a^{12}-\frac{34\!\cdots\!44}{37\!\cdots\!05}a^{11}+\frac{26\!\cdots\!88}{37\!\cdots\!05}a^{10}+\frac{77\!\cdots\!04}{37\!\cdots\!05}a^{9}-\frac{52\!\cdots\!13}{37\!\cdots\!05}a^{8}-\frac{14\!\cdots\!51}{74\!\cdots\!81}a^{7}+\frac{88\!\cdots\!31}{37\!\cdots\!05}a^{6}+\frac{36\!\cdots\!26}{37\!\cdots\!05}a^{5}-\frac{69\!\cdots\!73}{37\!\cdots\!05}a^{4}-\frac{17\!\cdots\!72}{74\!\cdots\!81}a^{3}+\frac{25\!\cdots\!84}{37\!\cdots\!05}a^{2}+\frac{10\!\cdots\!32}{37\!\cdots\!05}a-\frac{37\!\cdots\!31}{37\!\cdots\!05}$, $\frac{16\!\cdots\!56}{10\!\cdots\!25}a^{23}+\frac{76\!\cdots\!94}{10\!\cdots\!25}a^{22}-\frac{51\!\cdots\!73}{10\!\cdots\!25}a^{21}-\frac{43\!\cdots\!92}{20\!\cdots\!85}a^{20}+\frac{74\!\cdots\!67}{10\!\cdots\!25}a^{19}-\frac{13\!\cdots\!03}{10\!\cdots\!25}a^{18}-\frac{76\!\cdots\!84}{10\!\cdots\!25}a^{17}+\frac{96\!\cdots\!69}{20\!\cdots\!85}a^{16}-\frac{64\!\cdots\!03}{10\!\cdots\!25}a^{15}+\frac{87\!\cdots\!83}{10\!\cdots\!25}a^{14}+\frac{71\!\cdots\!67}{10\!\cdots\!25}a^{13}-\frac{15\!\cdots\!72}{10\!\cdots\!25}a^{12}+\frac{15\!\cdots\!77}{10\!\cdots\!25}a^{11}+\frac{14\!\cdots\!03}{10\!\cdots\!25}a^{10}-\frac{57\!\cdots\!31}{10\!\cdots\!25}a^{9}+\frac{23\!\cdots\!89}{20\!\cdots\!85}a^{8}+\frac{73\!\cdots\!33}{10\!\cdots\!25}a^{7}-\frac{46\!\cdots\!47}{10\!\cdots\!25}a^{6}-\frac{54\!\cdots\!76}{10\!\cdots\!25}a^{5}+\frac{49\!\cdots\!69}{10\!\cdots\!25}a^{4}+\frac{22\!\cdots\!12}{10\!\cdots\!25}a^{3}-\frac{45\!\cdots\!12}{20\!\cdots\!85}a^{2}-\frac{36\!\cdots\!31}{10\!\cdots\!25}a+\frac{42\!\cdots\!99}{10\!\cdots\!25}$, $\frac{46\!\cdots\!37}{10\!\cdots\!25}a^{23}-\frac{25\!\cdots\!73}{10\!\cdots\!25}a^{22}+\frac{36\!\cdots\!76}{10\!\cdots\!25}a^{21}+\frac{32\!\cdots\!87}{20\!\cdots\!85}a^{20}-\frac{18\!\cdots\!69}{10\!\cdots\!25}a^{19}+\frac{50\!\cdots\!01}{10\!\cdots\!25}a^{18}-\frac{43\!\cdots\!62}{10\!\cdots\!25}a^{17}-\frac{13\!\cdots\!37}{20\!\cdots\!85}a^{16}+\frac{17\!\cdots\!21}{10\!\cdots\!25}a^{15}-\frac{13\!\cdots\!86}{10\!\cdots\!25}a^{14}-\frac{31\!\cdots\!84}{10\!\cdots\!25}a^{13}+\frac{32\!\cdots\!34}{10\!\cdots\!25}a^{12}-\frac{55\!\cdots\!84}{10\!\cdots\!25}a^{11}+\frac{67\!\cdots\!99}{10\!\cdots\!25}a^{10}+\frac{12\!\cdots\!12}{10\!\cdots\!25}a^{9}-\frac{17\!\cdots\!66}{20\!\cdots\!85}a^{8}-\frac{11\!\cdots\!71}{10\!\cdots\!25}a^{7}+\frac{14\!\cdots\!19}{10\!\cdots\!25}a^{6}+\frac{52\!\cdots\!72}{10\!\cdots\!25}a^{5}-\frac{10\!\cdots\!53}{10\!\cdots\!25}a^{4}-\frac{11\!\cdots\!54}{10\!\cdots\!25}a^{3}+\frac{78\!\cdots\!14}{20\!\cdots\!85}a^{2}+\frac{11\!\cdots\!87}{10\!\cdots\!25}a-\frac{55\!\cdots\!68}{10\!\cdots\!25}$, $\frac{32\!\cdots\!83}{10\!\cdots\!25}a^{23}+\frac{17\!\cdots\!47}{10\!\cdots\!25}a^{22}-\frac{23\!\cdots\!64}{10\!\cdots\!25}a^{21}-\frac{29\!\cdots\!09}{20\!\cdots\!85}a^{20}+\frac{13\!\cdots\!91}{10\!\cdots\!25}a^{19}-\frac{34\!\cdots\!74}{10\!\cdots\!25}a^{18}+\frac{27\!\cdots\!73}{10\!\cdots\!25}a^{17}+\frac{10\!\cdots\!26}{20\!\cdots\!85}a^{16}-\frac{12\!\cdots\!84}{10\!\cdots\!25}a^{15}+\frac{82\!\cdots\!44}{10\!\cdots\!25}a^{14}+\frac{36\!\cdots\!31}{10\!\cdots\!25}a^{13}-\frac{23\!\cdots\!56}{10\!\cdots\!25}a^{12}+\frac{38\!\cdots\!56}{10\!\cdots\!25}a^{11}-\frac{58\!\cdots\!81}{10\!\cdots\!25}a^{10}-\frac{88\!\cdots\!28}{10\!\cdots\!25}a^{9}+\frac{22\!\cdots\!87}{40\!\cdots\!37}a^{8}+\frac{86\!\cdots\!04}{10\!\cdots\!25}a^{7}-\frac{99\!\cdots\!16}{10\!\cdots\!25}a^{6}-\frac{45\!\cdots\!38}{10\!\cdots\!25}a^{5}+\frac{79\!\cdots\!52}{10\!\cdots\!25}a^{4}+\frac{12\!\cdots\!46}{10\!\cdots\!25}a^{3}-\frac{59\!\cdots\!54}{20\!\cdots\!85}a^{2}-\frac{17\!\cdots\!33}{10\!\cdots\!25}a+\frac{44\!\cdots\!42}{10\!\cdots\!25}$, $\frac{10\!\cdots\!42}{10\!\cdots\!25}a^{23}+\frac{56\!\cdots\!74}{10\!\cdots\!25}a^{22}-\frac{80\!\cdots\!08}{10\!\cdots\!25}a^{21}-\frac{40\!\cdots\!81}{10\!\cdots\!25}a^{20}+\frac{41\!\cdots\!12}{10\!\cdots\!25}a^{19}-\frac{11\!\cdots\!02}{10\!\cdots\!25}a^{18}+\frac{94\!\cdots\!98}{10\!\cdots\!25}a^{17}+\frac{15\!\cdots\!96}{10\!\cdots\!25}a^{16}-\frac{38\!\cdots\!94}{10\!\cdots\!25}a^{15}+\frac{28\!\cdots\!53}{10\!\cdots\!25}a^{14}+\frac{17\!\cdots\!07}{20\!\cdots\!85}a^{13}-\frac{73\!\cdots\!39}{10\!\cdots\!25}a^{12}+\frac{12\!\cdots\!06}{10\!\cdots\!25}a^{11}-\frac{95\!\cdots\!47}{10\!\cdots\!25}a^{10}-\frac{27\!\cdots\!76}{10\!\cdots\!25}a^{9}+\frac{18\!\cdots\!58}{10\!\cdots\!25}a^{8}+\frac{26\!\cdots\!62}{10\!\cdots\!25}a^{7}-\frac{63\!\cdots\!23}{20\!\cdots\!85}a^{6}-\frac{13\!\cdots\!72}{10\!\cdots\!25}a^{5}+\frac{24\!\cdots\!69}{10\!\cdots\!25}a^{4}+\frac{34\!\cdots\!57}{10\!\cdots\!25}a^{3}-\frac{91\!\cdots\!36}{10\!\cdots\!25}a^{2}-\frac{45\!\cdots\!84}{10\!\cdots\!25}a+\frac{26\!\cdots\!26}{20\!\cdots\!85}$, $\frac{21\!\cdots\!89}{40\!\cdots\!37}a^{23}-\frac{29\!\cdots\!06}{10\!\cdots\!25}a^{22}+\frac{42\!\cdots\!87}{10\!\cdots\!25}a^{21}+\frac{19\!\cdots\!01}{10\!\cdots\!25}a^{20}-\frac{21\!\cdots\!98}{10\!\cdots\!25}a^{19}+\frac{58\!\cdots\!16}{10\!\cdots\!25}a^{18}-\frac{49\!\cdots\!21}{10\!\cdots\!25}a^{17}-\frac{80\!\cdots\!06}{10\!\cdots\!25}a^{16}+\frac{20\!\cdots\!28}{10\!\cdots\!25}a^{15}-\frac{15\!\cdots\!72}{10\!\cdots\!25}a^{14}-\frac{44\!\cdots\!36}{10\!\cdots\!25}a^{13}+\frac{15\!\cdots\!76}{40\!\cdots\!37}a^{12}-\frac{65\!\cdots\!47}{10\!\cdots\!25}a^{11}+\frac{67\!\cdots\!28}{10\!\cdots\!25}a^{10}+\frac{14\!\cdots\!44}{10\!\cdots\!25}a^{9}-\frac{10\!\cdots\!83}{10\!\cdots\!25}a^{8}-\frac{13\!\cdots\!41}{10\!\cdots\!25}a^{7}+\frac{17\!\cdots\!51}{10\!\cdots\!25}a^{6}+\frac{13\!\cdots\!62}{20\!\cdots\!85}a^{5}-\frac{13\!\cdots\!66}{10\!\cdots\!25}a^{4}-\frac{16\!\cdots\!93}{10\!\cdots\!25}a^{3}+\frac{50\!\cdots\!71}{10\!\cdots\!25}a^{2}+\frac{25\!\cdots\!92}{10\!\cdots\!25}a-\frac{79\!\cdots\!47}{10\!\cdots\!25}$, $\frac{59\!\cdots\!47}{10\!\cdots\!25}a^{23}-\frac{32\!\cdots\!61}{10\!\cdots\!25}a^{22}+\frac{45\!\cdots\!02}{10\!\cdots\!25}a^{21}+\frac{23\!\cdots\!83}{10\!\cdots\!25}a^{20}-\frac{23\!\cdots\!08}{10\!\cdots\!25}a^{19}+\frac{64\!\cdots\!49}{10\!\cdots\!25}a^{18}-\frac{21\!\cdots\!10}{40\!\cdots\!37}a^{17}-\frac{90\!\cdots\!18}{10\!\cdots\!25}a^{16}+\frac{44\!\cdots\!54}{20\!\cdots\!85}a^{15}-\frac{16\!\cdots\!07}{10\!\cdots\!25}a^{14}-\frac{53\!\cdots\!47}{10\!\cdots\!25}a^{13}+\frac{42\!\cdots\!54}{10\!\cdots\!25}a^{12}-\frac{14\!\cdots\!14}{20\!\cdots\!85}a^{11}+\frac{52\!\cdots\!83}{10\!\cdots\!25}a^{10}+\frac{15\!\cdots\!79}{10\!\cdots\!25}a^{9}-\frac{10\!\cdots\!29}{10\!\cdots\!25}a^{8}-\frac{15\!\cdots\!34}{10\!\cdots\!25}a^{7}+\frac{18\!\cdots\!27}{10\!\cdots\!25}a^{6}+\frac{76\!\cdots\!57}{10\!\cdots\!25}a^{5}-\frac{14\!\cdots\!91}{10\!\cdots\!25}a^{4}-\frac{19\!\cdots\!23}{10\!\cdots\!25}a^{3}+\frac{54\!\cdots\!53}{10\!\cdots\!25}a^{2}+\frac{26\!\cdots\!38}{10\!\cdots\!25}a-\frac{81\!\cdots\!59}{10\!\cdots\!25}$, $\frac{43\!\cdots\!51}{10\!\cdots\!25}a^{23}+\frac{22\!\cdots\!51}{10\!\cdots\!25}a^{22}-\frac{29\!\cdots\!92}{10\!\cdots\!25}a^{21}-\frac{19\!\cdots\!47}{10\!\cdots\!25}a^{20}+\frac{16\!\cdots\!93}{10\!\cdots\!25}a^{19}-\frac{87\!\cdots\!84}{20\!\cdots\!85}a^{18}+\frac{32\!\cdots\!83}{10\!\cdots\!25}a^{17}+\frac{66\!\cdots\!12}{10\!\cdots\!25}a^{16}-\frac{14\!\cdots\!74}{10\!\cdots\!25}a^{15}+\frac{99\!\cdots\!82}{10\!\cdots\!25}a^{14}+\frac{40\!\cdots\!99}{10\!\cdots\!25}a^{13}-\frac{29\!\cdots\!67}{10\!\cdots\!25}a^{12}+\frac{46\!\cdots\!21}{10\!\cdots\!25}a^{11}+\frac{12\!\cdots\!47}{10\!\cdots\!25}a^{10}-\frac{10\!\cdots\!24}{10\!\cdots\!25}a^{9}+\frac{59\!\cdots\!06}{10\!\cdots\!25}a^{8}+\frac{21\!\cdots\!46}{20\!\cdots\!85}a^{7}-\frac{11\!\cdots\!69}{10\!\cdots\!25}a^{6}-\frac{61\!\cdots\!96}{10\!\cdots\!25}a^{5}+\frac{85\!\cdots\!51}{10\!\cdots\!25}a^{4}+\frac{19\!\cdots\!43}{10\!\cdots\!25}a^{3}-\frac{29\!\cdots\!97}{10\!\cdots\!25}a^{2}-\frac{64\!\cdots\!39}{20\!\cdots\!85}a+\frac{40\!\cdots\!98}{10\!\cdots\!25}$, $\frac{10\!\cdots\!92}{20\!\cdots\!85}a^{23}+\frac{29\!\cdots\!48}{10\!\cdots\!25}a^{22}-\frac{44\!\cdots\!36}{10\!\cdots\!25}a^{21}-\frac{18\!\cdots\!33}{10\!\cdots\!25}a^{20}+\frac{21\!\cdots\!64}{10\!\cdots\!25}a^{19}-\frac{60\!\cdots\!48}{10\!\cdots\!25}a^{18}+\frac{53\!\cdots\!33}{10\!\cdots\!25}a^{17}+\frac{79\!\cdots\!73}{10\!\cdots\!25}a^{16}-\frac{21\!\cdots\!14}{10\!\cdots\!25}a^{15}+\frac{16\!\cdots\!71}{10\!\cdots\!25}a^{14}+\frac{39\!\cdots\!13}{10\!\cdots\!25}a^{13}-\frac{78\!\cdots\!41}{20\!\cdots\!85}a^{12}+\frac{67\!\cdots\!81}{10\!\cdots\!25}a^{11}-\frac{91\!\cdots\!09}{10\!\cdots\!25}a^{10}-\frac{14\!\cdots\!92}{10\!\cdots\!25}a^{9}+\frac{10\!\cdots\!24}{10\!\cdots\!25}a^{8}+\frac{13\!\cdots\!18}{10\!\cdots\!25}a^{7}-\frac{17\!\cdots\!98}{10\!\cdots\!25}a^{6}-\frac{12\!\cdots\!72}{20\!\cdots\!85}a^{5}+\frac{13\!\cdots\!88}{10\!\cdots\!25}a^{4}+\frac{13\!\cdots\!69}{10\!\cdots\!25}a^{3}-\frac{51\!\cdots\!98}{10\!\cdots\!25}a^{2}-\frac{14\!\cdots\!61}{10\!\cdots\!25}a+\frac{76\!\cdots\!71}{10\!\cdots\!25}$, $\frac{34\!\cdots\!13}{10\!\cdots\!25}a^{23}+\frac{17\!\cdots\!51}{10\!\cdots\!25}a^{22}-\frac{22\!\cdots\!42}{10\!\cdots\!25}a^{21}-\frac{16\!\cdots\!64}{10\!\cdots\!25}a^{20}+\frac{13\!\cdots\!98}{10\!\cdots\!25}a^{19}-\frac{34\!\cdots\!93}{10\!\cdots\!25}a^{18}+\frac{24\!\cdots\!77}{10\!\cdots\!25}a^{17}+\frac{53\!\cdots\!99}{10\!\cdots\!25}a^{16}-\frac{11\!\cdots\!16}{10\!\cdots\!25}a^{15}+\frac{75\!\cdots\!07}{10\!\cdots\!25}a^{14}+\frac{69\!\cdots\!08}{20\!\cdots\!85}a^{13}-\frac{23\!\cdots\!51}{10\!\cdots\!25}a^{12}+\frac{36\!\cdots\!64}{10\!\cdots\!25}a^{11}+\frac{21\!\cdots\!62}{10\!\cdots\!25}a^{10}-\frac{87\!\cdots\!09}{10\!\cdots\!25}a^{9}+\frac{44\!\cdots\!42}{10\!\cdots\!25}a^{8}+\frac{86\!\cdots\!48}{10\!\cdots\!25}a^{7}-\frac{16\!\cdots\!86}{20\!\cdots\!85}a^{6}-\frac{51\!\cdots\!28}{10\!\cdots\!25}a^{5}+\frac{64\!\cdots\!06}{10\!\cdots\!25}a^{4}+\frac{17\!\cdots\!58}{10\!\cdots\!25}a^{3}-\frac{22\!\cdots\!99}{10\!\cdots\!25}a^{2}-\frac{29\!\cdots\!91}{10\!\cdots\!25}a+\frac{58\!\cdots\!02}{20\!\cdots\!85}$, $\frac{76\!\cdots\!77}{20\!\cdots\!85}a^{23}+\frac{19\!\cdots\!94}{10\!\cdots\!25}a^{22}-\frac{25\!\cdots\!33}{10\!\cdots\!25}a^{21}-\frac{17\!\cdots\!59}{10\!\cdots\!25}a^{20}+\frac{14\!\cdots\!02}{10\!\cdots\!25}a^{19}-\frac{38\!\cdots\!19}{10\!\cdots\!25}a^{18}+\frac{28\!\cdots\!19}{10\!\cdots\!25}a^{17}+\frac{58\!\cdots\!34}{10\!\cdots\!25}a^{16}-\frac{12\!\cdots\!57}{10\!\cdots\!25}a^{15}+\frac{85\!\cdots\!28}{10\!\cdots\!25}a^{14}+\frac{35\!\cdots\!94}{10\!\cdots\!25}a^{13}-\frac{50\!\cdots\!22}{20\!\cdots\!85}a^{12}+\frac{40\!\cdots\!53}{10\!\cdots\!25}a^{11}+\frac{15\!\cdots\!23}{10\!\cdots\!25}a^{10}-\frac{96\!\cdots\!81}{10\!\cdots\!25}a^{9}+\frac{50\!\cdots\!62}{10\!\cdots\!25}a^{8}+\frac{94\!\cdots\!44}{10\!\cdots\!25}a^{7}-\frac{94\!\cdots\!14}{10\!\cdots\!25}a^{6}-\frac{11\!\cdots\!26}{20\!\cdots\!85}a^{5}+\frac{72\!\cdots\!84}{10\!\cdots\!25}a^{4}+\frac{19\!\cdots\!37}{10\!\cdots\!25}a^{3}-\frac{24\!\cdots\!04}{10\!\cdots\!25}a^{2}-\frac{33\!\cdots\!03}{10\!\cdots\!25}a+\frac{34\!\cdots\!93}{10\!\cdots\!25}$, $\frac{22\!\cdots\!16}{10\!\cdots\!25}a^{23}+\frac{11\!\cdots\!71}{10\!\cdots\!25}a^{22}-\frac{15\!\cdots\!22}{10\!\cdots\!25}a^{21}-\frac{10\!\cdots\!27}{10\!\cdots\!25}a^{20}+\frac{85\!\cdots\!13}{10\!\cdots\!25}a^{19}-\frac{90\!\cdots\!06}{40\!\cdots\!37}a^{18}+\frac{16\!\cdots\!93}{10\!\cdots\!25}a^{17}+\frac{34\!\cdots\!07}{10\!\cdots\!25}a^{16}-\frac{75\!\cdots\!24}{10\!\cdots\!25}a^{15}+\frac{50\!\cdots\!77}{10\!\cdots\!25}a^{14}+\frac{21\!\cdots\!54}{10\!\cdots\!25}a^{13}-\frac{15\!\cdots\!07}{10\!\cdots\!25}a^{12}+\frac{24\!\cdots\!96}{10\!\cdots\!25}a^{11}+\frac{86\!\cdots\!77}{10\!\cdots\!25}a^{10}-\frac{56\!\cdots\!69}{10\!\cdots\!25}a^{9}+\frac{29\!\cdots\!76}{10\!\cdots\!25}a^{8}+\frac{10\!\cdots\!16}{20\!\cdots\!85}a^{7}-\frac{55\!\cdots\!74}{10\!\cdots\!25}a^{6}-\frac{31\!\cdots\!56}{10\!\cdots\!25}a^{5}+\frac{42\!\cdots\!51}{10\!\cdots\!25}a^{4}+\frac{96\!\cdots\!38}{10\!\cdots\!25}a^{3}-\frac{13\!\cdots\!12}{10\!\cdots\!25}a^{2}-\frac{26\!\cdots\!23}{20\!\cdots\!85}a+\frac{16\!\cdots\!03}{10\!\cdots\!25}$, $\frac{91\!\cdots\!23}{10\!\cdots\!25}a^{23}-\frac{52\!\cdots\!39}{10\!\cdots\!25}a^{22}+\frac{86\!\cdots\!68}{10\!\cdots\!25}a^{21}+\frac{59\!\cdots\!82}{10\!\cdots\!25}a^{20}-\frac{35\!\cdots\!72}{10\!\cdots\!25}a^{19}+\frac{10\!\cdots\!31}{10\!\cdots\!25}a^{18}-\frac{23\!\cdots\!13}{20\!\cdots\!85}a^{17}-\frac{94\!\cdots\!87}{10\!\cdots\!25}a^{16}+\frac{69\!\cdots\!49}{20\!\cdots\!85}a^{15}-\frac{33\!\cdots\!43}{10\!\cdots\!25}a^{14}+\frac{44\!\cdots\!62}{10\!\cdots\!25}a^{13}+\frac{57\!\cdots\!26}{10\!\cdots\!25}a^{12}-\frac{24\!\cdots\!59}{20\!\cdots\!85}a^{11}+\frac{44\!\cdots\!32}{10\!\cdots\!25}a^{10}+\frac{21\!\cdots\!31}{10\!\cdots\!25}a^{9}-\frac{21\!\cdots\!06}{10\!\cdots\!25}a^{8}-\frac{16\!\cdots\!81}{10\!\cdots\!25}a^{7}+\frac{29\!\cdots\!03}{10\!\cdots\!25}a^{6}+\frac{44\!\cdots\!23}{10\!\cdots\!25}a^{5}-\frac{20\!\cdots\!89}{10\!\cdots\!25}a^{4}+\frac{47\!\cdots\!48}{10\!\cdots\!25}a^{3}+\frac{68\!\cdots\!27}{10\!\cdots\!25}a^{2}-\frac{21\!\cdots\!43}{10\!\cdots\!25}a-\frac{91\!\cdots\!46}{10\!\cdots\!25}$ 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166961596859997956564195837434781/1014050943578398939313708163425*a^7 + 295829633036339059002838028386103/1014050943578398939313708163425*a^6 + 44516900213152863774955845583123/1014050943578398939313708163425*a^5 - 205581429198166541673776185616089/1014050943578398939313708163425*a^4 + 4700009940940634628861249277648/1014050943578398939313708163425*a^3 + 68242372546546505313998922446827/1014050943578398939313708163425*a^2 - 2192719548276577037339318020943/1014050943578398939313708163425*a - 9142791092375949697136954273146/1014050943578398939313708163425 (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:		\( 9344905.652391985 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 9344905.652391985 \cdot 1}{2\cdot\sqrt{641953627807088196277618408203125}}\cr\approx \mathstrut & 0.282951345118995 \end{aligned}\] (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
\(5\) 5	5.4.0.1	$x^{4} + 4 x^{2} + 4 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $20$	$20$	$1$	$31$
\(13\) 13	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.4.0.1	$x^{4} + 3 x^{2} + 12 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	13.8.4.1	$x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	13.8.4.1	$x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$