Normalized defining polynomial
\( x^{24} - 5 x^{23} + x^{22} + 40 x^{21} - 89 x^{20} + 10 x^{19} + 240 x^{18} - 427 x^{17} + 266 x^{16} + \cdots - 1 \)
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: | \(49177850545349555386638457176064\) \(\medspace = 2^{16}\cdot 487^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.92\) | 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: | $2^{2/3}487^{1/2}\approx 35.030887836273955$ | ||
Ramified primes: | \(2\), \(487\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{31\!\cdots\!39}a^{23}+\frac{12\!\cdots\!09}{31\!\cdots\!39}a^{22}+\frac{12\!\cdots\!02}{31\!\cdots\!39}a^{21}-\frac{24\!\cdots\!25}{31\!\cdots\!39}a^{20}+\frac{10\!\cdots\!55}{31\!\cdots\!39}a^{19}-\frac{41\!\cdots\!12}{31\!\cdots\!39}a^{18}-\frac{77\!\cdots\!01}{31\!\cdots\!39}a^{17}-\frac{36\!\cdots\!68}{31\!\cdots\!39}a^{16}-\frac{11\!\cdots\!17}{31\!\cdots\!39}a^{15}+\frac{85\!\cdots\!21}{31\!\cdots\!39}a^{14}-\frac{17\!\cdots\!46}{31\!\cdots\!39}a^{13}+\frac{54\!\cdots\!37}{31\!\cdots\!39}a^{12}+\frac{11\!\cdots\!05}{31\!\cdots\!39}a^{11}+\frac{53\!\cdots\!50}{31\!\cdots\!39}a^{10}-\frac{56\!\cdots\!72}{31\!\cdots\!39}a^{9}+\frac{14\!\cdots\!17}{31\!\cdots\!39}a^{8}+\frac{42\!\cdots\!83}{31\!\cdots\!39}a^{7}+\frac{10\!\cdots\!47}{31\!\cdots\!39}a^{6}+\frac{37\!\cdots\!44}{31\!\cdots\!39}a^{5}-\frac{46\!\cdots\!04}{31\!\cdots\!39}a^{4}+\frac{92\!\cdots\!32}{31\!\cdots\!39}a^{3}-\frac{75\!\cdots\!09}{31\!\cdots\!39}a^{2}+\frac{88\!\cdots\!75}{31\!\cdots\!39}a-\frac{59\!\cdots\!66}{31\!\cdots\!39}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( -1 \) (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: | $\frac{14\!\cdots\!03}{31\!\cdots\!39}a^{23}-\frac{71\!\cdots\!04}{31\!\cdots\!39}a^{22}+\frac{12\!\cdots\!77}{31\!\cdots\!39}a^{21}+\frac{56\!\cdots\!36}{31\!\cdots\!39}a^{20}-\frac{12\!\cdots\!08}{31\!\cdots\!39}a^{19}+\frac{13\!\cdots\!30}{31\!\cdots\!39}a^{18}+\frac{33\!\cdots\!92}{31\!\cdots\!39}a^{17}-\frac{60\!\cdots\!97}{31\!\cdots\!39}a^{16}+\frac{38\!\cdots\!86}{31\!\cdots\!39}a^{15}+\frac{30\!\cdots\!97}{31\!\cdots\!39}a^{14}-\frac{87\!\cdots\!05}{31\!\cdots\!39}a^{13}+\frac{80\!\cdots\!27}{31\!\cdots\!39}a^{12}-\frac{24\!\cdots\!67}{31\!\cdots\!39}a^{11}-\frac{30\!\cdots\!14}{31\!\cdots\!39}a^{10}+\frac{35\!\cdots\!61}{31\!\cdots\!39}a^{9}-\frac{44\!\cdots\!59}{31\!\cdots\!39}a^{8}+\frac{23\!\cdots\!97}{31\!\cdots\!39}a^{7}-\frac{20\!\cdots\!84}{31\!\cdots\!39}a^{6}-\frac{27\!\cdots\!58}{31\!\cdots\!39}a^{5}-\frac{28\!\cdots\!05}{31\!\cdots\!39}a^{4}-\frac{20\!\cdots\!79}{31\!\cdots\!39}a^{3}-\frac{45\!\cdots\!44}{31\!\cdots\!39}a^{2}-\frac{12\!\cdots\!06}{31\!\cdots\!39}a+\frac{26\!\cdots\!26}{31\!\cdots\!39}$, $\frac{11\!\cdots\!41}{31\!\cdots\!39}a^{23}-\frac{56\!\cdots\!66}{31\!\cdots\!39}a^{22}-\frac{37\!\cdots\!00}{31\!\cdots\!39}a^{21}+\frac{48\!\cdots\!13}{31\!\cdots\!39}a^{20}-\frac{94\!\cdots\!64}{31\!\cdots\!39}a^{19}-\frac{17\!\cdots\!00}{31\!\cdots\!39}a^{18}+\frac{29\!\cdots\!57}{31\!\cdots\!39}a^{17}-\frac{44\!\cdots\!72}{31\!\cdots\!39}a^{16}+\frac{17\!\cdots\!16}{31\!\cdots\!39}a^{15}+\frac{37\!\cdots\!45}{31\!\cdots\!39}a^{14}-\frac{68\!\cdots\!33}{31\!\cdots\!39}a^{13}+\frac{44\!\cdots\!43}{31\!\cdots\!39}a^{12}+\frac{41\!\cdots\!27}{31\!\cdots\!39}a^{11}-\frac{34\!\cdots\!91}{31\!\cdots\!39}a^{10}+\frac{19\!\cdots\!48}{31\!\cdots\!39}a^{9}+\frac{96\!\cdots\!53}{31\!\cdots\!39}a^{8}+\frac{15\!\cdots\!66}{31\!\cdots\!39}a^{7}+\frac{21\!\cdots\!24}{31\!\cdots\!39}a^{6}-\frac{24\!\cdots\!12}{31\!\cdots\!39}a^{5}-\frac{29\!\cdots\!73}{31\!\cdots\!39}a^{4}-\frac{18\!\cdots\!80}{31\!\cdots\!39}a^{3}-\frac{42\!\cdots\!30}{31\!\cdots\!39}a^{2}+\frac{24\!\cdots\!67}{31\!\cdots\!39}a+\frac{44\!\cdots\!62}{31\!\cdots\!39}$, $\frac{19\!\cdots\!09}{31\!\cdots\!39}a^{23}-\frac{97\!\cdots\!66}{31\!\cdots\!39}a^{22}+\frac{15\!\cdots\!41}{31\!\cdots\!39}a^{21}+\frac{79\!\cdots\!12}{31\!\cdots\!39}a^{20}-\frac{17\!\cdots\!85}{31\!\cdots\!39}a^{19}+\frac{12\!\cdots\!51}{31\!\cdots\!39}a^{18}+\frac{47\!\cdots\!78}{31\!\cdots\!39}a^{17}-\frac{83\!\cdots\!37}{31\!\cdots\!39}a^{16}+\frac{49\!\cdots\!27}{31\!\cdots\!39}a^{15}+\frac{46\!\cdots\!28}{31\!\cdots\!39}a^{14}-\frac{12\!\cdots\!01}{31\!\cdots\!39}a^{13}+\frac{10\!\cdots\!08}{31\!\cdots\!39}a^{12}-\frac{25\!\cdots\!17}{31\!\cdots\!39}a^{11}-\frac{45\!\cdots\!63}{31\!\cdots\!39}a^{10}+\frac{45\!\cdots\!44}{31\!\cdots\!39}a^{9}-\frac{12\!\cdots\!96}{31\!\cdots\!39}a^{8}+\frac{30\!\cdots\!22}{31\!\cdots\!39}a^{7}-\frac{21\!\cdots\!24}{31\!\cdots\!39}a^{6}-\frac{38\!\cdots\!58}{31\!\cdots\!39}a^{5}-\frac{40\!\cdots\!24}{31\!\cdots\!39}a^{4}-\frac{26\!\cdots\!68}{31\!\cdots\!39}a^{3}-\frac{58\!\cdots\!96}{31\!\cdots\!39}a^{2}-\frac{10\!\cdots\!63}{31\!\cdots\!39}a+\frac{55\!\cdots\!91}{31\!\cdots\!39}$, $\frac{63\!\cdots\!74}{31\!\cdots\!39}a^{23}-\frac{28\!\cdots\!67}{31\!\cdots\!39}a^{22}-\frac{12\!\cdots\!26}{31\!\cdots\!39}a^{21}+\frac{27\!\cdots\!35}{31\!\cdots\!39}a^{20}-\frac{43\!\cdots\!95}{31\!\cdots\!39}a^{19}-\frac{33\!\cdots\!79}{31\!\cdots\!39}a^{18}+\frac{18\!\cdots\!15}{31\!\cdots\!39}a^{17}-\frac{20\!\cdots\!05}{31\!\cdots\!39}a^{16}-\frac{34\!\cdots\!59}{31\!\cdots\!39}a^{15}+\frac{35\!\cdots\!71}{31\!\cdots\!39}a^{14}-\frac{41\!\cdots\!45}{31\!\cdots\!39}a^{13}+\frac{91\!\cdots\!68}{31\!\cdots\!39}a^{12}+\frac{28\!\cdots\!63}{31\!\cdots\!39}a^{11}-\frac{38\!\cdots\!78}{31\!\cdots\!39}a^{10}+\frac{13\!\cdots\!54}{31\!\cdots\!39}a^{9}+\frac{15\!\cdots\!07}{31\!\cdots\!39}a^{8}-\frac{16\!\cdots\!59}{31\!\cdots\!39}a^{7}+\frac{70\!\cdots\!71}{31\!\cdots\!39}a^{6}-\frac{15\!\cdots\!42}{31\!\cdots\!39}a^{5}-\frac{19\!\cdots\!57}{31\!\cdots\!39}a^{4}-\frac{88\!\cdots\!36}{31\!\cdots\!39}a^{3}-\frac{25\!\cdots\!07}{31\!\cdots\!39}a^{2}+\frac{88\!\cdots\!23}{31\!\cdots\!39}a+\frac{27\!\cdots\!11}{31\!\cdots\!39}$, $\frac{49\!\cdots\!76}{31\!\cdots\!39}a^{23}-\frac{26\!\cdots\!42}{31\!\cdots\!39}a^{22}+\frac{13\!\cdots\!90}{31\!\cdots\!39}a^{21}+\frac{19\!\cdots\!41}{31\!\cdots\!39}a^{20}-\frac{51\!\cdots\!96}{31\!\cdots\!39}a^{19}+\frac{21\!\cdots\!15}{31\!\cdots\!39}a^{18}+\frac{11\!\cdots\!50}{31\!\cdots\!39}a^{17}-\frac{25\!\cdots\!40}{31\!\cdots\!39}a^{16}+\frac{21\!\cdots\!18}{31\!\cdots\!39}a^{15}+\frac{46\!\cdots\!65}{31\!\cdots\!39}a^{14}-\frac{33\!\cdots\!19}{31\!\cdots\!39}a^{13}+\frac{38\!\cdots\!36}{31\!\cdots\!39}a^{12}-\frac{19\!\cdots\!07}{31\!\cdots\!39}a^{11}-\frac{54\!\cdots\!32}{31\!\cdots\!39}a^{10}+\frac{14\!\cdots\!99}{31\!\cdots\!39}a^{9}-\frac{60\!\cdots\!55}{31\!\cdots\!39}a^{8}+\frac{95\!\cdots\!76}{31\!\cdots\!39}a^{7}-\frac{34\!\cdots\!96}{31\!\cdots\!39}a^{6}-\frac{98\!\cdots\!19}{31\!\cdots\!39}a^{5}-\frac{60\!\cdots\!15}{31\!\cdots\!39}a^{4}-\frac{37\!\cdots\!96}{31\!\cdots\!39}a^{3}+\frac{75\!\cdots\!10}{31\!\cdots\!39}a^{2}+\frac{10\!\cdots\!41}{31\!\cdots\!39}a+\frac{13\!\cdots\!62}{31\!\cdots\!39}$, $\frac{34\!\cdots\!42}{31\!\cdots\!39}a^{23}-\frac{16\!\cdots\!33}{31\!\cdots\!39}a^{22}+\frac{27\!\cdots\!39}{31\!\cdots\!39}a^{21}+\frac{13\!\cdots\!05}{31\!\cdots\!39}a^{20}-\frac{29\!\cdots\!18}{31\!\cdots\!39}a^{19}+\frac{33\!\cdots\!48}{31\!\cdots\!39}a^{18}+\frac{77\!\cdots\!10}{31\!\cdots\!39}a^{17}-\frac{13\!\cdots\!70}{31\!\cdots\!39}a^{16}+\frac{94\!\cdots\!88}{31\!\cdots\!39}a^{15}+\frac{53\!\cdots\!14}{31\!\cdots\!39}a^{14}-\frac{18\!\cdots\!08}{31\!\cdots\!39}a^{13}+\frac{18\!\cdots\!71}{31\!\cdots\!39}a^{12}-\frac{87\!\cdots\!28}{31\!\cdots\!39}a^{11}-\frac{26\!\cdots\!99}{31\!\cdots\!39}a^{10}+\frac{57\!\cdots\!80}{31\!\cdots\!39}a^{9}-\frac{23\!\cdots\!89}{31\!\cdots\!39}a^{8}+\frac{90\!\cdots\!55}{31\!\cdots\!39}a^{7}-\frac{20\!\cdots\!06}{31\!\cdots\!39}a^{6}-\frac{62\!\cdots\!33}{31\!\cdots\!39}a^{5}-\frac{68\!\cdots\!51}{31\!\cdots\!39}a^{4}-\frac{63\!\cdots\!89}{31\!\cdots\!39}a^{3}-\frac{90\!\cdots\!87}{31\!\cdots\!39}a^{2}+\frac{10\!\cdots\!51}{31\!\cdots\!39}a+\frac{21\!\cdots\!92}{31\!\cdots\!39}$, $\frac{40\!\cdots\!38}{31\!\cdots\!39}a^{23}-\frac{77\!\cdots\!46}{31\!\cdots\!39}a^{22}+\frac{26\!\cdots\!04}{31\!\cdots\!39}a^{21}+\frac{20\!\cdots\!04}{31\!\cdots\!39}a^{20}-\frac{27\!\cdots\!93}{31\!\cdots\!39}a^{19}+\frac{44\!\cdots\!50}{31\!\cdots\!39}a^{18}+\frac{25\!\cdots\!11}{31\!\cdots\!39}a^{17}-\frac{16\!\cdots\!25}{31\!\cdots\!39}a^{16}+\frac{21\!\cdots\!72}{31\!\cdots\!39}a^{15}-\frac{33\!\cdots\!41}{31\!\cdots\!39}a^{14}-\frac{26\!\cdots\!37}{31\!\cdots\!39}a^{13}+\frac{37\!\cdots\!30}{31\!\cdots\!39}a^{12}-\frac{17\!\cdots\!54}{31\!\cdots\!39}a^{11}-\frac{10\!\cdots\!02}{31\!\cdots\!39}a^{10}+\frac{23\!\cdots\!73}{31\!\cdots\!39}a^{9}-\frac{10\!\cdots\!88}{31\!\cdots\!39}a^{8}-\frac{65\!\cdots\!24}{31\!\cdots\!39}a^{7}-\frac{56\!\cdots\!84}{31\!\cdots\!39}a^{6}-\frac{12\!\cdots\!39}{31\!\cdots\!39}a^{5}+\frac{11\!\cdots\!81}{31\!\cdots\!39}a^{4}+\frac{13\!\cdots\!76}{31\!\cdots\!39}a^{3}+\frac{76\!\cdots\!63}{31\!\cdots\!39}a^{2}+\frac{10\!\cdots\!39}{31\!\cdots\!39}a-\frac{57\!\cdots\!39}{31\!\cdots\!39}$, $\frac{13\!\cdots\!62}{31\!\cdots\!39}a^{23}-\frac{64\!\cdots\!34}{31\!\cdots\!39}a^{22}-\frac{12\!\cdots\!80}{31\!\cdots\!39}a^{21}+\frac{56\!\cdots\!70}{31\!\cdots\!39}a^{20}-\frac{10\!\cdots\!77}{31\!\cdots\!39}a^{19}-\frac{37\!\cdots\!76}{31\!\cdots\!39}a^{18}+\frac{35\!\cdots\!95}{31\!\cdots\!39}a^{17}-\frac{47\!\cdots\!24}{31\!\cdots\!39}a^{16}+\frac{11\!\cdots\!52}{31\!\cdots\!39}a^{15}+\frac{51\!\cdots\!48}{31\!\cdots\!39}a^{14}-\frac{78\!\cdots\!73}{31\!\cdots\!39}a^{13}+\frac{40\!\cdots\!27}{31\!\cdots\!39}a^{12}+\frac{17\!\cdots\!50}{31\!\cdots\!39}a^{11}-\frac{46\!\cdots\!49}{31\!\cdots\!39}a^{10}+\frac{21\!\cdots\!58}{31\!\cdots\!39}a^{9}+\frac{15\!\cdots\!57}{31\!\cdots\!39}a^{8}+\frac{16\!\cdots\!65}{31\!\cdots\!39}a^{7}+\frac{44\!\cdots\!82}{31\!\cdots\!39}a^{6}-\frac{29\!\cdots\!28}{31\!\cdots\!39}a^{5}-\frac{37\!\cdots\!43}{31\!\cdots\!39}a^{4}-\frac{23\!\cdots\!41}{31\!\cdots\!39}a^{3}-\frac{50\!\cdots\!16}{31\!\cdots\!39}a^{2}+\frac{89\!\cdots\!72}{31\!\cdots\!39}a+\frac{79\!\cdots\!51}{31\!\cdots\!39}$, $\frac{23\!\cdots\!24}{31\!\cdots\!39}a^{23}-\frac{10\!\cdots\!54}{31\!\cdots\!39}a^{22}-\frac{13\!\cdots\!40}{31\!\cdots\!39}a^{21}+\frac{90\!\cdots\!08}{31\!\cdots\!39}a^{20}-\frac{16\!\cdots\!07}{31\!\cdots\!39}a^{19}-\frac{18\!\cdots\!77}{31\!\cdots\!39}a^{18}+\frac{44\!\cdots\!73}{31\!\cdots\!39}a^{17}-\frac{69\!\cdots\!89}{31\!\cdots\!39}a^{16}+\frac{49\!\cdots\!27}{31\!\cdots\!39}a^{15}+\frac{27\!\cdots\!71}{31\!\cdots\!39}a^{14}-\frac{51\!\cdots\!41}{31\!\cdots\!39}a^{13}+\frac{82\!\cdots\!49}{31\!\cdots\!39}a^{12}-\frac{10\!\cdots\!91}{31\!\cdots\!39}a^{11}+\frac{79\!\cdots\!60}{31\!\cdots\!39}a^{10}-\frac{28\!\cdots\!72}{31\!\cdots\!39}a^{9}-\frac{31\!\cdots\!98}{31\!\cdots\!39}a^{8}+\frac{12\!\cdots\!39}{31\!\cdots\!39}a^{7}-\frac{27\!\cdots\!35}{31\!\cdots\!39}a^{6}-\frac{44\!\cdots\!45}{31\!\cdots\!39}a^{5}-\frac{64\!\cdots\!01}{31\!\cdots\!39}a^{4}-\frac{67\!\cdots\!62}{31\!\cdots\!39}a^{3}-\frac{17\!\cdots\!85}{31\!\cdots\!39}a^{2}-\frac{47\!\cdots\!62}{31\!\cdots\!39}a+\frac{31\!\cdots\!43}{31\!\cdots\!39}$, $\frac{84\!\cdots\!48}{31\!\cdots\!39}a^{23}-\frac{44\!\cdots\!71}{31\!\cdots\!39}a^{22}+\frac{18\!\cdots\!93}{31\!\cdots\!39}a^{21}+\frac{34\!\cdots\!16}{31\!\cdots\!39}a^{20}-\frac{85\!\cdots\!31}{31\!\cdots\!39}a^{19}+\frac{22\!\cdots\!07}{31\!\cdots\!39}a^{18}+\frac{22\!\cdots\!38}{31\!\cdots\!39}a^{17}-\frac{43\!\cdots\!91}{31\!\cdots\!39}a^{16}+\frac{28\!\cdots\!68}{31\!\cdots\!39}a^{15}+\frac{22\!\cdots\!29}{31\!\cdots\!39}a^{14}-\frac{67\!\cdots\!90}{31\!\cdots\!39}a^{13}+\frac{61\!\cdots\!48}{31\!\cdots\!39}a^{12}-\frac{13\!\cdots\!15}{31\!\cdots\!39}a^{11}-\frac{31\!\cdots\!17}{31\!\cdots\!39}a^{10}+\frac{33\!\cdots\!33}{31\!\cdots\!39}a^{9}-\frac{49\!\cdots\!64}{31\!\cdots\!39}a^{8}+\frac{79\!\cdots\!35}{31\!\cdots\!39}a^{7}-\frac{41\!\cdots\!73}{31\!\cdots\!39}a^{6}-\frac{17\!\cdots\!22}{31\!\cdots\!39}a^{5}-\frac{10\!\cdots\!03}{31\!\cdots\!39}a^{4}-\frac{30\!\cdots\!30}{31\!\cdots\!39}a^{3}+\frac{37\!\cdots\!81}{31\!\cdots\!39}a^{2}+\frac{82\!\cdots\!54}{31\!\cdots\!39}a+\frac{18\!\cdots\!61}{31\!\cdots\!39}$, $\frac{16\!\cdots\!27}{31\!\cdots\!39}a^{23}-\frac{77\!\cdots\!22}{31\!\cdots\!39}a^{22}-\frac{66\!\cdots\!65}{31\!\cdots\!39}a^{21}+\frac{66\!\cdots\!19}{31\!\cdots\!39}a^{20}-\frac{13\!\cdots\!90}{31\!\cdots\!39}a^{19}-\frac{15\!\cdots\!02}{31\!\cdots\!39}a^{18}+\frac{40\!\cdots\!40}{31\!\cdots\!39}a^{17}-\frac{65\!\cdots\!19}{31\!\cdots\!39}a^{16}+\frac{36\!\cdots\!16}{31\!\cdots\!39}a^{15}+\frac{46\!\cdots\!55}{31\!\cdots\!39}a^{14}-\frac{11\!\cdots\!85}{31\!\cdots\!39}a^{13}+\frac{11\!\cdots\!77}{31\!\cdots\!39}a^{12}-\frac{47\!\cdots\!01}{31\!\cdots\!39}a^{11}-\frac{46\!\cdots\!65}{31\!\cdots\!39}a^{10}+\frac{85\!\cdots\!23}{31\!\cdots\!39}a^{9}-\frac{64\!\cdots\!88}{31\!\cdots\!39}a^{8}+\frac{70\!\cdots\!28}{31\!\cdots\!39}a^{7}-\frac{37\!\cdots\!21}{31\!\cdots\!39}a^{6}-\frac{31\!\cdots\!56}{31\!\cdots\!39}a^{5}-\frac{38\!\cdots\!37}{31\!\cdots\!39}a^{4}-\frac{43\!\cdots\!68}{31\!\cdots\!39}a^{3}-\frac{10\!\cdots\!09}{31\!\cdots\!39}a^{2}-\frac{29\!\cdots\!64}{31\!\cdots\!39}a+\frac{33\!\cdots\!22}{31\!\cdots\!39}$, $\frac{23\!\cdots\!79}{31\!\cdots\!39}a^{23}-\frac{11\!\cdots\!34}{31\!\cdots\!39}a^{22}+\frac{29\!\cdots\!81}{31\!\cdots\!39}a^{21}+\frac{94\!\cdots\!44}{31\!\cdots\!39}a^{20}-\frac{21\!\cdots\!11}{31\!\cdots\!39}a^{19}+\frac{30\!\cdots\!71}{31\!\cdots\!39}a^{18}+\frac{58\!\cdots\!11}{31\!\cdots\!39}a^{17}-\frac{10\!\cdots\!27}{31\!\cdots\!39}a^{16}+\frac{64\!\cdots\!07}{31\!\cdots\!39}a^{15}+\frac{61\!\cdots\!07}{31\!\cdots\!39}a^{14}-\frac{16\!\cdots\!65}{31\!\cdots\!39}a^{13}+\frac{13\!\cdots\!54}{31\!\cdots\!39}a^{12}-\frac{25\!\cdots\!57}{31\!\cdots\!39}a^{11}-\frac{72\!\cdots\!20}{31\!\cdots\!39}a^{10}+\frac{68\!\cdots\!17}{31\!\cdots\!39}a^{9}+\frac{30\!\cdots\!85}{31\!\cdots\!39}a^{8}+\frac{23\!\cdots\!30}{31\!\cdots\!39}a^{7}-\frac{18\!\cdots\!73}{31\!\cdots\!39}a^{6}-\frac{46\!\cdots\!92}{31\!\cdots\!39}a^{5}-\frac{42\!\cdots\!74}{31\!\cdots\!39}a^{4}-\frac{21\!\cdots\!63}{31\!\cdots\!39}a^{3}-\frac{12\!\cdots\!98}{31\!\cdots\!39}a^{2}-\frac{54\!\cdots\!24}{31\!\cdots\!39}a+\frac{20\!\cdots\!55}{31\!\cdots\!39}$, $\frac{63\!\cdots\!96}{31\!\cdots\!39}a^{23}-\frac{35\!\cdots\!36}{31\!\cdots\!39}a^{22}+\frac{23\!\cdots\!25}{31\!\cdots\!39}a^{21}+\frac{25\!\cdots\!80}{31\!\cdots\!39}a^{20}-\frac{72\!\cdots\!97}{31\!\cdots\!39}a^{19}+\frac{35\!\cdots\!35}{31\!\cdots\!39}a^{18}+\frac{16\!\cdots\!14}{31\!\cdots\!39}a^{17}-\frac{38\!\cdots\!75}{31\!\cdots\!39}a^{16}+\frac{31\!\cdots\!44}{31\!\cdots\!39}a^{15}+\frac{13\!\cdots\!77}{31\!\cdots\!39}a^{14}-\frac{59\!\cdots\!15}{31\!\cdots\!39}a^{13}+\frac{62\!\cdots\!49}{31\!\cdots\!39}a^{12}-\frac{20\!\cdots\!27}{31\!\cdots\!39}a^{11}-\frac{26\!\cdots\!28}{31\!\cdots\!39}a^{10}+\frac{37\!\cdots\!29}{31\!\cdots\!39}a^{9}-\frac{11\!\cdots\!93}{31\!\cdots\!39}a^{8}+\frac{40\!\cdots\!39}{31\!\cdots\!39}a^{7}-\frac{19\!\cdots\!17}{31\!\cdots\!39}a^{6}-\frac{14\!\cdots\!47}{31\!\cdots\!39}a^{5}-\frac{30\!\cdots\!63}{31\!\cdots\!39}a^{4}+\frac{11\!\cdots\!57}{31\!\cdots\!39}a^{3}+\frac{24\!\cdots\!11}{31\!\cdots\!39}a^{2}-\frac{45\!\cdots\!95}{31\!\cdots\!39}a-\frac{16\!\cdots\!54}{31\!\cdots\!39}$ (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: | \( 1432261.5638336123 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1432261.5638336123 \cdot 1}{2\cdot\sqrt{49177850545349555386638457176064}}\cr\approx \mathstrut & 0.156684572333305 \end{aligned}\] (assuming GRH)
Galois group
$\SL(2,5):C_2$ (as 24T576):
A non-solvable group of order 240 |
The 18 conjugacy class representatives for $\SL(2,5):C_2$ |
Character table for $\SL(2,5):C_2$ |
Intermediate fields
6.2.3794704.1, 12.4.14399778447616.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 40 siblings: | data not computed |
Arithmetically equvalently sibling: | 24.4.49177850545349555386638457176064.2 |
Minimal sibling: | 24.4.49177850545349555386638457176064.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $20{,}\,{\href{/padicField/3.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | $20{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}$ | $20{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{6}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ |
2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
\(487\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.487.2t1.a.a | $1$ | $ 487 $ | \(\Q(\sqrt{-487}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.1948.120.b.a | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
2.1948.120.b.b | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
2.1948.120.b.c | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
2.1948.120.b.d | $2$ | $ 2^{2} \cdot 487 $ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
* | 3.1948.12t76.a.a | $3$ | $ 2^{2} \cdot 487 $ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
* | 3.1948.12t76.a.b | $3$ | $ 2^{2} \cdot 487 $ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
3.948676.12t33.a.a | $3$ | $ 2^{2} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
3.948676.12t33.a.b | $3$ | $ 2^{2} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
4.948676.10t11.a.a | $4$ | $ 2^{2} \cdot 487^{2}$ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
4.948676.5t4.a.a | $4$ | $ 2^{2} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
4.948676.40t188.b.a | $4$ | $ 2^{2} \cdot 487^{2}$ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
4.948676.40t188.b.b | $4$ | $ 2^{2} \cdot 487^{2}$ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
5.1848020848.12t75.a.a | $5$ | $ 2^{4} \cdot 487^{3}$ | 10.0.438293256499312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
* | 5.3794704.6t12.a.a | $5$ | $ 2^{4} \cdot 487^{2}$ | 5.1.948676.1 | $A_5$ (as 5T4) | $1$ | $1$ |
* | 6.1848020848.24t576.b.a | $6$ | $ 2^{4} \cdot 487^{3}$ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
* | 6.1848020848.24t576.b.b | $6$ | $ 2^{4} \cdot 487^{3}$ | 24.4.49177850545349555386638457176064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |