Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241)
gp: K = bnfinit(y^24 - 2*y^23 - 8*y^22 + 22*y^21 + 51*y^20 - 228*y^19 - 3*y^18 + 1321*y^17 - 2220*y^16 - 1748*y^15 + 9722*y^14 - 7588*y^13 - 14572*y^12 + 34524*y^11 - 16026*y^10 - 34263*y^9 + 60097*y^8 - 26110*y^7 - 33112*y^6 + 60245*y^5 - 45651*y^4 + 20521*y^3 - 6288*y^2 + 1547*y - 241, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241)
\( x^{24} - 2 x^{23} - 8 x^{22} + 22 x^{21} + 51 x^{20} - 228 x^{19} - 3 x^{18} + 1321 x^{17} - 2220 x^{16} + \cdots - 241 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(19756778413055716819205133752664064\)
\(\medspace = 2^{16}\cdot 887^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(26.85\) | 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: | $2^{2/3}887^{1/2}\approx 47.2768436183456$ | ||
| Ramified primes: |
\(2\), \(887\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{13}a^{22}-\frac{6}{13}a^{21}-\frac{1}{13}a^{20}-\frac{2}{13}a^{19}-\frac{2}{13}a^{18}-\frac{4}{13}a^{17}-\frac{5}{13}a^{16}+\frac{5}{13}a^{15}+\frac{3}{13}a^{14}+\frac{1}{13}a^{13}-\frac{5}{13}a^{12}-\frac{6}{13}a^{11}+\frac{6}{13}a^{10}-\frac{4}{13}a^{9}-\frac{5}{13}a^{8}+\frac{2}{13}a^{7}-\frac{3}{13}a^{6}-\frac{2}{13}a^{5}+\frac{6}{13}a^{4}-\frac{6}{13}a^{2}+\frac{5}{13}a-\frac{5}{13}$, $\frac{1}{16\!\cdots\!11}a^{23}+\frac{43\!\cdots\!42}{16\!\cdots\!11}a^{22}+\frac{11\!\cdots\!56}{41\!\cdots\!49}a^{21}+\frac{47\!\cdots\!12}{16\!\cdots\!11}a^{20}+\frac{60\!\cdots\!02}{16\!\cdots\!11}a^{19}+\frac{91\!\cdots\!84}{16\!\cdots\!11}a^{18}-\frac{19\!\cdots\!19}{12\!\cdots\!47}a^{17}+\frac{15\!\cdots\!81}{16\!\cdots\!11}a^{16}-\frac{69\!\cdots\!45}{12\!\cdots\!47}a^{15}+\frac{30\!\cdots\!08}{53\!\cdots\!37}a^{14}-\frac{61\!\cdots\!40}{16\!\cdots\!11}a^{13}+\frac{48\!\cdots\!75}{16\!\cdots\!11}a^{12}-\frac{18\!\cdots\!97}{53\!\cdots\!37}a^{11}+\frac{21\!\cdots\!08}{53\!\cdots\!37}a^{10}-\frac{68\!\cdots\!00}{41\!\cdots\!49}a^{9}-\frac{18\!\cdots\!35}{53\!\cdots\!37}a^{8}+\frac{17\!\cdots\!46}{16\!\cdots\!11}a^{7}-\frac{36\!\cdots\!32}{53\!\cdots\!37}a^{6}+\frac{79\!\cdots\!08}{16\!\cdots\!11}a^{5}+\frac{61\!\cdots\!47}{16\!\cdots\!11}a^{4}+\frac{67\!\cdots\!76}{16\!\cdots\!11}a^{3}-\frac{33\!\cdots\!36}{16\!\cdots\!11}a^{2}-\frac{79\!\cdots\!23}{16\!\cdots\!11}a-\frac{78\!\cdots\!20}{16\!\cdots\!11}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| 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: |
$\frac{14\!\cdots\!57}{53\!\cdots\!37}a^{23}-\frac{19\!\cdots\!26}{53\!\cdots\!37}a^{22}-\frac{77\!\cdots\!98}{31\!\cdots\!73}a^{21}+\frac{23\!\cdots\!48}{53\!\cdots\!37}a^{20}+\frac{91\!\cdots\!96}{53\!\cdots\!37}a^{19}-\frac{27\!\cdots\!54}{53\!\cdots\!37}a^{18}-\frac{15\!\cdots\!54}{41\!\cdots\!49}a^{17}+\frac{18\!\cdots\!51}{53\!\cdots\!37}a^{16}-\frac{15\!\cdots\!78}{41\!\cdots\!49}a^{15}-\frac{40\!\cdots\!41}{53\!\cdots\!37}a^{14}+\frac{11\!\cdots\!70}{53\!\cdots\!37}a^{13}-\frac{30\!\cdots\!67}{53\!\cdots\!37}a^{12}-\frac{23\!\cdots\!90}{53\!\cdots\!37}a^{11}+\frac{34\!\cdots\!99}{53\!\cdots\!37}a^{10}+\frac{53\!\cdots\!27}{41\!\cdots\!49}a^{9}-\frac{50\!\cdots\!33}{53\!\cdots\!37}a^{8}+\frac{53\!\cdots\!31}{53\!\cdots\!37}a^{7}-\frac{54\!\cdots\!91}{53\!\cdots\!37}a^{6}-\frac{50\!\cdots\!03}{53\!\cdots\!37}a^{5}+\frac{54\!\cdots\!31}{53\!\cdots\!37}a^{4}-\frac{28\!\cdots\!34}{53\!\cdots\!37}a^{3}+\frac{93\!\cdots\!20}{53\!\cdots\!37}a^{2}-\frac{24\!\cdots\!45}{53\!\cdots\!37}a+\frac{50\!\cdots\!88}{53\!\cdots\!37}$, $\frac{17\!\cdots\!89}{53\!\cdots\!37}a^{23}-\frac{20\!\cdots\!59}{53\!\cdots\!37}a^{22}-\frac{12\!\cdots\!00}{41\!\cdots\!49}a^{21}+\frac{25\!\cdots\!20}{53\!\cdots\!37}a^{20}+\frac{11\!\cdots\!79}{53\!\cdots\!37}a^{19}-\frac{31\!\cdots\!51}{53\!\cdots\!37}a^{18}-\frac{19\!\cdots\!36}{41\!\cdots\!49}a^{17}+\frac{21\!\cdots\!90}{53\!\cdots\!37}a^{16}-\frac{17\!\cdots\!97}{41\!\cdots\!49}a^{15}-\frac{47\!\cdots\!04}{53\!\cdots\!37}a^{14}+\frac{13\!\cdots\!41}{53\!\cdots\!37}a^{13}-\frac{29\!\cdots\!75}{53\!\cdots\!37}a^{12}-\frac{27\!\cdots\!05}{53\!\cdots\!37}a^{11}+\frac{38\!\cdots\!47}{53\!\cdots\!37}a^{10}+\frac{10\!\cdots\!03}{41\!\cdots\!49}a^{9}-\frac{57\!\cdots\!77}{53\!\cdots\!37}a^{8}+\frac{60\!\cdots\!67}{53\!\cdots\!37}a^{7}-\frac{94\!\cdots\!29}{53\!\cdots\!37}a^{6}-\frac{56\!\cdots\!47}{53\!\cdots\!37}a^{5}+\frac{61\!\cdots\!84}{53\!\cdots\!37}a^{4}-\frac{34\!\cdots\!21}{53\!\cdots\!37}a^{3}+\frac{11\!\cdots\!94}{53\!\cdots\!37}a^{2}-\frac{29\!\cdots\!80}{53\!\cdots\!37}a+\frac{59\!\cdots\!00}{53\!\cdots\!37}$, $\frac{11\!\cdots\!60}{53\!\cdots\!37}a^{23}-\frac{14\!\cdots\!03}{53\!\cdots\!37}a^{22}-\frac{76\!\cdots\!47}{41\!\cdots\!49}a^{21}+\frac{17\!\cdots\!52}{53\!\cdots\!37}a^{20}+\frac{69\!\cdots\!22}{53\!\cdots\!37}a^{19}-\frac{20\!\cdots\!86}{53\!\cdots\!37}a^{18}-\frac{12\!\cdots\!10}{41\!\cdots\!49}a^{17}+\frac{13\!\cdots\!42}{53\!\cdots\!37}a^{16}-\frac{11\!\cdots\!36}{41\!\cdots\!49}a^{15}-\frac{31\!\cdots\!68}{53\!\cdots\!37}a^{14}+\frac{85\!\cdots\!25}{53\!\cdots\!37}a^{13}-\frac{16\!\cdots\!19}{53\!\cdots\!37}a^{12}-\frac{18\!\cdots\!45}{53\!\cdots\!37}a^{11}+\frac{24\!\cdots\!22}{53\!\cdots\!37}a^{10}+\frac{20\!\cdots\!85}{41\!\cdots\!49}a^{9}-\frac{38\!\cdots\!78}{53\!\cdots\!37}a^{8}+\frac{37\!\cdots\!95}{53\!\cdots\!37}a^{7}+\frac{31\!\cdots\!97}{53\!\cdots\!37}a^{6}-\frac{38\!\cdots\!13}{53\!\cdots\!37}a^{5}+\frac{38\!\cdots\!08}{53\!\cdots\!37}a^{4}-\frac{18\!\cdots\!87}{53\!\cdots\!37}a^{3}+\frac{49\!\cdots\!35}{53\!\cdots\!37}a^{2}-\frac{10\!\cdots\!16}{53\!\cdots\!37}a+\frac{22\!\cdots\!23}{53\!\cdots\!37}$, $\frac{23\!\cdots\!71}{53\!\cdots\!37}a^{23}-\frac{25\!\cdots\!70}{53\!\cdots\!37}a^{22}-\frac{16\!\cdots\!85}{41\!\cdots\!49}a^{21}+\frac{32\!\cdots\!35}{53\!\cdots\!37}a^{20}+\frac{14\!\cdots\!17}{53\!\cdots\!37}a^{19}-\frac{39\!\cdots\!47}{53\!\cdots\!37}a^{18}-\frac{27\!\cdots\!17}{41\!\cdots\!49}a^{17}+\frac{27\!\cdots\!58}{53\!\cdots\!37}a^{16}-\frac{20\!\cdots\!40}{41\!\cdots\!49}a^{15}-\frac{64\!\cdots\!74}{53\!\cdots\!37}a^{14}+\frac{16\!\cdots\!57}{53\!\cdots\!37}a^{13}-\frac{27\!\cdots\!38}{53\!\cdots\!37}a^{12}-\frac{36\!\cdots\!56}{53\!\cdots\!37}a^{11}+\frac{48\!\cdots\!66}{53\!\cdots\!37}a^{10}+\frac{38\!\cdots\!35}{41\!\cdots\!49}a^{9}-\frac{74\!\cdots\!28}{53\!\cdots\!37}a^{8}+\frac{73\!\cdots\!63}{53\!\cdots\!37}a^{7}+\frac{36\!\cdots\!76}{53\!\cdots\!37}a^{6}-\frac{73\!\cdots\!94}{53\!\cdots\!37}a^{5}+\frac{75\!\cdots\!37}{53\!\cdots\!37}a^{4}-\frac{39\!\cdots\!47}{53\!\cdots\!37}a^{3}+\frac{12\!\cdots\!00}{53\!\cdots\!37}a^{2}-\frac{34\!\cdots\!10}{53\!\cdots\!37}a+\frac{67\!\cdots\!20}{53\!\cdots\!37}$, $\frac{11\!\cdots\!36}{16\!\cdots\!11}a^{23}-\frac{12\!\cdots\!37}{16\!\cdots\!11}a^{22}-\frac{26\!\cdots\!11}{41\!\cdots\!49}a^{21}+\frac{15\!\cdots\!76}{16\!\cdots\!11}a^{20}+\frac{73\!\cdots\!75}{16\!\cdots\!11}a^{19}-\frac{19\!\cdots\!84}{16\!\cdots\!11}a^{18}-\frac{14\!\cdots\!61}{12\!\cdots\!47}a^{17}+\frac{13\!\cdots\!58}{16\!\cdots\!11}a^{16}-\frac{10\!\cdots\!17}{12\!\cdots\!47}a^{15}-\frac{10\!\cdots\!49}{53\!\cdots\!37}a^{14}+\frac{83\!\cdots\!23}{16\!\cdots\!11}a^{13}-\frac{12\!\cdots\!70}{16\!\cdots\!11}a^{12}-\frac{59\!\cdots\!95}{53\!\cdots\!37}a^{11}+\frac{78\!\cdots\!23}{53\!\cdots\!37}a^{10}+\frac{71\!\cdots\!57}{41\!\cdots\!49}a^{9}-\frac{12\!\cdots\!07}{53\!\cdots\!37}a^{8}+\frac{36\!\cdots\!79}{16\!\cdots\!11}a^{7}+\frac{73\!\cdots\!87}{53\!\cdots\!37}a^{6}-\frac{35\!\cdots\!90}{16\!\cdots\!11}a^{5}+\frac{37\!\cdots\!18}{16\!\cdots\!11}a^{4}-\frac{19\!\cdots\!17}{16\!\cdots\!11}a^{3}+\frac{63\!\cdots\!39}{16\!\cdots\!11}a^{2}-\frac{16\!\cdots\!41}{16\!\cdots\!11}a+\frac{32\!\cdots\!73}{16\!\cdots\!11}$, $\frac{14\!\cdots\!08}{16\!\cdots\!11}a^{23}-\frac{10\!\cdots\!56}{16\!\cdots\!11}a^{22}-\frac{33\!\cdots\!53}{41\!\cdots\!49}a^{21}+\frac{14\!\cdots\!76}{16\!\cdots\!11}a^{20}+\frac{95\!\cdots\!36}{16\!\cdots\!11}a^{19}-\frac{20\!\cdots\!83}{16\!\cdots\!11}a^{18}-\frac{22\!\cdots\!17}{12\!\cdots\!47}a^{17}+\frac{15\!\cdots\!91}{16\!\cdots\!11}a^{16}-\frac{84\!\cdots\!32}{12\!\cdots\!47}a^{15}-\frac{14\!\cdots\!31}{53\!\cdots\!37}a^{14}+\frac{85\!\cdots\!92}{16\!\cdots\!11}a^{13}+\frac{11\!\cdots\!25}{16\!\cdots\!11}a^{12}-\frac{69\!\cdots\!33}{53\!\cdots\!37}a^{11}+\frac{72\!\cdots\!83}{53\!\cdots\!37}a^{10}+\frac{23\!\cdots\!17}{41\!\cdots\!49}a^{9}-\frac{13\!\cdots\!53}{53\!\cdots\!37}a^{8}+\frac{31\!\cdots\!80}{16\!\cdots\!11}a^{7}+\frac{32\!\cdots\!43}{53\!\cdots\!37}a^{6}-\frac{37\!\cdots\!61}{16\!\cdots\!11}a^{5}+\frac{33\!\cdots\!15}{16\!\cdots\!11}a^{4}-\frac{15\!\cdots\!03}{16\!\cdots\!11}a^{3}+\frac{51\!\cdots\!49}{16\!\cdots\!11}a^{2}-\frac{14\!\cdots\!00}{16\!\cdots\!11}a+\frac{28\!\cdots\!60}{16\!\cdots\!11}$, $\frac{32\!\cdots\!03}{16\!\cdots\!11}a^{23}-\frac{40\!\cdots\!64}{16\!\cdots\!11}a^{22}-\frac{74\!\cdots\!81}{41\!\cdots\!49}a^{21}+\frac{49\!\cdots\!39}{16\!\cdots\!11}a^{20}+\frac{20\!\cdots\!34}{16\!\cdots\!11}a^{19}-\frac{58\!\cdots\!21}{16\!\cdots\!11}a^{18}-\frac{35\!\cdots\!49}{12\!\cdots\!47}a^{17}+\frac{39\!\cdots\!05}{16\!\cdots\!11}a^{16}-\frac{32\!\cdots\!32}{12\!\cdots\!47}a^{15}-\frac{29\!\cdots\!45}{53\!\cdots\!37}a^{14}+\frac{24\!\cdots\!03}{16\!\cdots\!11}a^{13}-\frac{53\!\cdots\!69}{16\!\cdots\!11}a^{12}-\frac{17\!\cdots\!60}{53\!\cdots\!37}a^{11}+\frac{23\!\cdots\!65}{53\!\cdots\!37}a^{10}+\frac{98\!\cdots\!01}{31\!\cdots\!73}a^{9}-\frac{36\!\cdots\!86}{53\!\cdots\!37}a^{8}+\frac{10\!\cdots\!27}{16\!\cdots\!11}a^{7}+\frac{11\!\cdots\!35}{53\!\cdots\!37}a^{6}-\frac{10\!\cdots\!65}{16\!\cdots\!11}a^{5}+\frac{11\!\cdots\!37}{16\!\cdots\!11}a^{4}-\frac{58\!\cdots\!47}{16\!\cdots\!11}a^{3}+\frac{18\!\cdots\!30}{16\!\cdots\!11}a^{2}-\frac{48\!\cdots\!11}{16\!\cdots\!11}a+\frac{10\!\cdots\!19}{16\!\cdots\!11}$, $\frac{14\!\cdots\!85}{53\!\cdots\!37}a^{23}-\frac{21\!\cdots\!10}{53\!\cdots\!37}a^{22}-\frac{97\!\cdots\!24}{31\!\cdots\!73}a^{21}+\frac{18\!\cdots\!65}{53\!\cdots\!37}a^{20}+\frac{16\!\cdots\!59}{53\!\cdots\!37}a^{19}-\frac{14\!\cdots\!37}{53\!\cdots\!37}a^{18}+\frac{12\!\cdots\!18}{41\!\cdots\!49}a^{17}+\frac{60\!\cdots\!60}{53\!\cdots\!37}a^{16}-\frac{14\!\cdots\!74}{41\!\cdots\!49}a^{15}+\frac{27\!\cdots\!31}{53\!\cdots\!37}a^{14}+\frac{61\!\cdots\!70}{53\!\cdots\!37}a^{13}-\frac{84\!\cdots\!50}{53\!\cdots\!37}a^{12}-\frac{52\!\cdots\!30}{53\!\cdots\!37}a^{11}+\frac{25\!\cdots\!68}{53\!\cdots\!37}a^{10}-\frac{16\!\cdots\!68}{41\!\cdots\!49}a^{9}-\frac{18\!\cdots\!96}{53\!\cdots\!37}a^{8}+\frac{50\!\cdots\!63}{53\!\cdots\!37}a^{7}-\frac{30\!\cdots\!22}{53\!\cdots\!37}a^{6}-\frac{22\!\cdots\!23}{53\!\cdots\!37}a^{5}+\frac{51\!\cdots\!05}{53\!\cdots\!37}a^{4}-\frac{37\!\cdots\!78}{53\!\cdots\!37}a^{3}+\frac{98\!\cdots\!67}{53\!\cdots\!37}a^{2}+\frac{10\!\cdots\!49}{53\!\cdots\!37}a-\frac{30\!\cdots\!08}{53\!\cdots\!37}$, $\frac{72\!\cdots\!18}{16\!\cdots\!11}a^{23}-\frac{67\!\cdots\!53}{16\!\cdots\!11}a^{22}-\frac{17\!\cdots\!71}{41\!\cdots\!49}a^{21}+\frac{89\!\cdots\!86}{16\!\cdots\!11}a^{20}+\frac{47\!\cdots\!31}{16\!\cdots\!11}a^{19}-\frac{11\!\cdots\!56}{16\!\cdots\!11}a^{18}-\frac{10\!\cdots\!56}{12\!\cdots\!47}a^{17}+\frac{83\!\cdots\!32}{16\!\cdots\!11}a^{16}-\frac{53\!\cdots\!23}{12\!\cdots\!47}a^{15}-\frac{70\!\cdots\!98}{53\!\cdots\!37}a^{14}+\frac{48\!\cdots\!15}{16\!\cdots\!11}a^{13}-\frac{11\!\cdots\!06}{16\!\cdots\!11}a^{12}-\frac{37\!\cdots\!30}{53\!\cdots\!37}a^{11}+\frac{43\!\cdots\!26}{53\!\cdots\!37}a^{10}+\frac{92\!\cdots\!30}{41\!\cdots\!49}a^{9}-\frac{73\!\cdots\!81}{53\!\cdots\!37}a^{8}+\frac{19\!\cdots\!15}{16\!\cdots\!11}a^{7}+\frac{13\!\cdots\!34}{53\!\cdots\!37}a^{6}-\frac{21\!\cdots\!65}{16\!\cdots\!11}a^{5}+\frac{19\!\cdots\!59}{16\!\cdots\!11}a^{4}-\frac{95\!\cdots\!41}{16\!\cdots\!11}a^{3}+\frac{28\!\cdots\!04}{16\!\cdots\!11}a^{2}-\frac{78\!\cdots\!80}{16\!\cdots\!11}a+\frac{16\!\cdots\!39}{16\!\cdots\!11}$, $\frac{18\!\cdots\!91}{16\!\cdots\!11}a^{23}-\frac{31\!\cdots\!88}{16\!\cdots\!11}a^{22}-\frac{48\!\cdots\!66}{41\!\cdots\!49}a^{21}+\frac{95\!\cdots\!87}{16\!\cdots\!11}a^{20}+\frac{14\!\cdots\!62}{16\!\cdots\!11}a^{19}-\frac{20\!\cdots\!26}{16\!\cdots\!11}a^{18}-\frac{46\!\cdots\!59}{12\!\cdots\!47}a^{17}+\frac{18\!\cdots\!80}{16\!\cdots\!11}a^{16}+\frac{17\!\cdots\!05}{12\!\cdots\!47}a^{15}-\frac{23\!\cdots\!69}{53\!\cdots\!37}a^{14}+\frac{78\!\cdots\!77}{16\!\cdots\!11}a^{13}+\frac{11\!\cdots\!49}{16\!\cdots\!11}a^{12}-\frac{96\!\cdots\!58}{53\!\cdots\!37}a^{11}+\frac{24\!\cdots\!98}{53\!\cdots\!37}a^{10}+\frac{98\!\cdots\!06}{41\!\cdots\!49}a^{9}-\frac{15\!\cdots\!26}{53\!\cdots\!37}a^{8}-\frac{36\!\cdots\!29}{16\!\cdots\!11}a^{7}+\frac{16\!\cdots\!82}{53\!\cdots\!37}a^{6}-\frac{39\!\cdots\!27}{16\!\cdots\!11}a^{5}+\frac{30\!\cdots\!04}{16\!\cdots\!11}a^{4}+\frac{14\!\cdots\!73}{16\!\cdots\!11}a^{3}-\frac{66\!\cdots\!69}{16\!\cdots\!11}a^{2}+\frac{13\!\cdots\!00}{16\!\cdots\!11}a-\frac{84\!\cdots\!78}{16\!\cdots\!11}$, $\frac{25\!\cdots\!23}{16\!\cdots\!11}a^{23}-\frac{39\!\cdots\!32}{16\!\cdots\!11}a^{22}-\frac{55\!\cdots\!01}{41\!\cdots\!49}a^{21}+\frac{46\!\cdots\!18}{16\!\cdots\!11}a^{20}+\frac{14\!\cdots\!40}{16\!\cdots\!11}a^{19}-\frac{51\!\cdots\!47}{16\!\cdots\!11}a^{18}-\frac{15\!\cdots\!71}{12\!\cdots\!47}a^{17}+\frac{32\!\cdots\!52}{16\!\cdots\!11}a^{16}-\frac{33\!\cdots\!73}{12\!\cdots\!47}a^{15}-\frac{19\!\cdots\!86}{53\!\cdots\!37}a^{14}+\frac{21\!\cdots\!05}{16\!\cdots\!11}a^{13}-\frac{10\!\cdots\!02}{16\!\cdots\!11}a^{12}-\frac{13\!\cdots\!97}{53\!\cdots\!37}a^{11}+\frac{23\!\cdots\!80}{53\!\cdots\!37}a^{10}-\frac{36\!\cdots\!76}{41\!\cdots\!49}a^{9}-\frac{29\!\cdots\!49}{53\!\cdots\!37}a^{8}+\frac{11\!\cdots\!97}{16\!\cdots\!11}a^{7}-\frac{81\!\cdots\!60}{53\!\cdots\!37}a^{6}-\frac{87\!\cdots\!52}{16\!\cdots\!11}a^{5}+\frac{11\!\cdots\!27}{16\!\cdots\!11}a^{4}-\frac{73\!\cdots\!37}{16\!\cdots\!11}a^{3}+\frac{28\!\cdots\!24}{16\!\cdots\!11}a^{2}-\frac{79\!\cdots\!77}{16\!\cdots\!11}a+\frac{16\!\cdots\!61}{16\!\cdots\!11}$, $\frac{14\!\cdots\!42}{53\!\cdots\!37}a^{23}-\frac{12\!\cdots\!61}{53\!\cdots\!37}a^{22}-\frac{95\!\cdots\!46}{41\!\cdots\!49}a^{21}+\frac{17\!\cdots\!86}{53\!\cdots\!37}a^{20}+\frac{88\!\cdots\!41}{53\!\cdots\!37}a^{19}-\frac{22\!\cdots\!81}{53\!\cdots\!37}a^{18}-\frac{17\!\cdots\!93}{41\!\cdots\!49}a^{17}+\frac{15\!\cdots\!02}{53\!\cdots\!37}a^{16}-\frac{11\!\cdots\!02}{41\!\cdots\!49}a^{15}-\frac{37\!\cdots\!78}{53\!\cdots\!37}a^{14}+\frac{93\!\cdots\!85}{53\!\cdots\!37}a^{13}-\frac{12\!\cdots\!84}{53\!\cdots\!37}a^{12}-\frac{20\!\cdots\!96}{53\!\cdots\!37}a^{11}+\frac{26\!\cdots\!54}{53\!\cdots\!37}a^{10}+\frac{17\!\cdots\!04}{41\!\cdots\!49}a^{9}-\frac{40\!\cdots\!85}{53\!\cdots\!37}a^{8}+\frac{41\!\cdots\!10}{53\!\cdots\!37}a^{7}+\frac{18\!\cdots\!84}{53\!\cdots\!37}a^{6}-\frac{39\!\cdots\!92}{53\!\cdots\!37}a^{5}+\frac{43\!\cdots\!58}{53\!\cdots\!37}a^{4}-\frac{24\!\cdots\!08}{53\!\cdots\!37}a^{3}+\frac{84\!\cdots\!05}{53\!\cdots\!37}a^{2}-\frac{21\!\cdots\!33}{53\!\cdots\!37}a+\frac{48\!\cdots\!78}{53\!\cdots\!37}$, $\frac{18\!\cdots\!88}{16\!\cdots\!11}a^{23}+\frac{21\!\cdots\!73}{16\!\cdots\!11}a^{22}-\frac{91\!\cdots\!33}{31\!\cdots\!73}a^{21}-\frac{20\!\cdots\!48}{16\!\cdots\!11}a^{20}+\frac{47\!\cdots\!11}{16\!\cdots\!11}a^{19}+\frac{13\!\cdots\!13}{16\!\cdots\!11}a^{18}-\frac{35\!\cdots\!18}{12\!\cdots\!47}a^{17}-\frac{24\!\cdots\!30}{16\!\cdots\!11}a^{16}+\frac{21\!\cdots\!51}{12\!\cdots\!47}a^{15}-\frac{10\!\cdots\!63}{53\!\cdots\!37}a^{14}-\frac{66\!\cdots\!03}{16\!\cdots\!11}a^{13}+\frac{17\!\cdots\!74}{16\!\cdots\!11}a^{12}-\frac{42\!\cdots\!35}{53\!\cdots\!37}a^{11}-\frac{13\!\cdots\!33}{53\!\cdots\!37}a^{10}+\frac{11\!\cdots\!15}{41\!\cdots\!49}a^{9}+\frac{54\!\cdots\!00}{53\!\cdots\!37}a^{8}-\frac{78\!\cdots\!09}{16\!\cdots\!11}a^{7}+\frac{19\!\cdots\!01}{53\!\cdots\!37}a^{6}+\frac{23\!\cdots\!76}{16\!\cdots\!11}a^{5}-\frac{73\!\cdots\!78}{16\!\cdots\!11}a^{4}+\frac{55\!\cdots\!99}{16\!\cdots\!11}a^{3}-\frac{20\!\cdots\!91}{16\!\cdots\!11}a^{2}+\frac{56\!\cdots\!24}{16\!\cdots\!11}a-\frac{14\!\cdots\!38}{16\!\cdots\!11}$
| 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: | \( 33313220.242250312 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 33313220.242250312 \cdot 1}{2\cdot\sqrt{19756778413055716819205133752664064}}\cr\approx \mathstrut & 0.181822331396691 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$\SL(2,5):C_2$ (as 24T576):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| 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
| 6.2.12588304.1, 12.4.158465397596416.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 40 siblings: | data not computed |
| Arithmetically equvalently sibling: | 24.4.19756778413055716819205133752664064.2 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.3.0.1}{3} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{8}$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.12.0.1}{12} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | $20{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
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 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(887\)
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $8$ | $2$ | $4$ | $4$ | ||||
| Deg $8$ | $2$ | $4$ | $4$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.887.2t1.a.a | $1$ | $ 887 $ | \(\Q(\sqrt{-887}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.3548.120.a.a | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3548.120.a.b | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3548.120.a.c | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.3548.120.a.d | $2$ | $ 2^{2} \cdot 887 $ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 3.3147076.12t33.a.a | $3$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.3147076.12t33.a.b | $3$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| * | 3.3548.12t76.a.a | $3$ | $ 2^{2} \cdot 887 $ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| * | 3.3548.12t76.a.b | $3$ | $ 2^{2} \cdot 887 $ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| 4.3147076.10t11.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
| 4.3147076.5t4.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
| 4.3147076.40t188.a.a | $4$ | $ 2^{2} \cdot 887^{2}$ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 4.3147076.40t188.a.b | $4$ | $ 2^{2} \cdot 887^{2}$ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 5.11165825648.12t75.a.a | $5$ | $ 2^{4} \cdot 887^{3}$ | 10.0.8784925479251312.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
| * | 5.12588304.6t12.a.a | $5$ | $ 2^{4} \cdot 887^{2}$ | 5.1.3147076.1 | $A_5$ (as 5T4) | $1$ | $1$ |
| * | 6.11165825648.24t576.a.a | $6$ | $ 2^{4} \cdot 887^{3}$ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
| * | 6.11165825648.24t576.a.b | $6$ | $ 2^{4} \cdot 887^{3}$ | 24.4.19756778413055716819205133752664064.1 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.