Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4)
gp: K = bnfinit(y^24 - 3*y^23 + 6*y^22 - 19*y^21 + 39*y^20 - 30*y^19 - 9*y^18 - 45*y^17 + 93*y^16 + 204*y^15 - 201*y^14 - 129*y^13 - 793*y^12 + 738*y^11 + 141*y^10 + 1177*y^9 - 666*y^8 - 597*y^7 - 913*y^6 + 168*y^5 + 474*y^4 + 464*y^3 + 168*y^2 + 24*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4)
\( x^{24} - 3 x^{23} + 6 x^{22} - 19 x^{21} + 39 x^{20} - 30 x^{19} - 9 x^{18} - 45 x^{17} + 93 x^{16} + \cdots - 4 \)
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: |
\(7654476884289954721505578121232384\)
\(\medspace = 2^{16}\cdot 3^{26}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(25.81\) | 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: | $2^{2/3}3^{7/6}11^{4/5}\approx 38.94415416349148$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{12}a^{18}+\frac{1}{12}a^{15}-\frac{1}{4}a^{14}+\frac{1}{12}a^{12}+\frac{1}{4}a^{11}-\frac{1}{2}a^{10}+\frac{1}{12}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}$, $\frac{1}{24}a^{19}+\frac{1}{24}a^{16}+\frac{1}{8}a^{15}-\frac{1}{4}a^{14}+\frac{1}{24}a^{13}+\frac{1}{8}a^{12}-\frac{1}{2}a^{11}-\frac{11}{24}a^{10}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{3}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{24}a^{20}+\frac{1}{24}a^{17}-\frac{1}{8}a^{16}-\frac{1}{4}a^{15}-\frac{5}{24}a^{14}-\frac{1}{8}a^{13}+\frac{7}{24}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{3}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{48}a^{21}-\frac{1}{48}a^{19}+\frac{1}{48}a^{18}-\frac{1}{16}a^{17}-\frac{1}{48}a^{16}-\frac{1}{6}a^{15}+\frac{3}{16}a^{14}-\frac{7}{48}a^{13}+\frac{1}{12}a^{12}+\frac{5}{16}a^{11}+\frac{23}{48}a^{10}+\frac{1}{16}a^{9}-\frac{3}{16}a^{8}+\frac{1}{8}a^{7}-\frac{7}{16}a^{6}-\frac{3}{8}a^{5}-\frac{3}{8}a^{4}+\frac{1}{6}a^{3}+\frac{1}{12}a+\frac{1}{4}$, $\frac{1}{48}a^{22}-\frac{1}{48}a^{20}-\frac{1}{48}a^{19}+\frac{1}{48}a^{18}-\frac{1}{48}a^{17}+\frac{1}{24}a^{16}+\frac{7}{48}a^{15}+\frac{5}{48}a^{14}-\frac{5}{24}a^{13}-\frac{11}{48}a^{12}+\frac{23}{48}a^{11}-\frac{11}{48}a^{10}-\frac{11}{48}a^{9}+\frac{3}{8}a^{8}+\frac{5}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{12}a^{4}+\frac{1}{12}a^{2}+\frac{5}{12}a+\frac{1}{6}$, $\frac{1}{18\!\cdots\!56}a^{23}-\frac{14\!\cdots\!93}{15\!\cdots\!88}a^{22}+\frac{56\!\cdots\!25}{22\!\cdots\!32}a^{21}+\frac{18\!\cdots\!99}{60\!\cdots\!52}a^{20}+\frac{17\!\cdots\!19}{45\!\cdots\!64}a^{19}+\frac{13\!\cdots\!29}{15\!\cdots\!88}a^{18}-\frac{66\!\cdots\!77}{60\!\cdots\!52}a^{17}-\frac{90\!\cdots\!98}{57\!\cdots\!83}a^{16}-\frac{29\!\cdots\!33}{60\!\cdots\!52}a^{15}+\frac{13\!\cdots\!79}{60\!\cdots\!52}a^{14}-\frac{68\!\cdots\!35}{57\!\cdots\!83}a^{13}+\frac{71\!\cdots\!81}{60\!\cdots\!52}a^{12}+\frac{30\!\cdots\!97}{91\!\cdots\!28}a^{11}+\frac{20\!\cdots\!13}{91\!\cdots\!28}a^{10}-\frac{98\!\cdots\!07}{18\!\cdots\!56}a^{9}+\frac{13\!\cdots\!15}{30\!\cdots\!76}a^{8}-\frac{27\!\cdots\!49}{15\!\cdots\!88}a^{7}+\frac{18\!\cdots\!19}{60\!\cdots\!52}a^{6}-\frac{36\!\cdots\!97}{91\!\cdots\!28}a^{5}-\frac{86\!\cdots\!31}{30\!\cdots\!76}a^{4}-\frac{27\!\cdots\!39}{45\!\cdots\!64}a^{3}+\frac{12\!\cdots\!63}{45\!\cdots\!64}a^{2}+\frac{12\!\cdots\!97}{45\!\cdots\!64}a+\frac{22\!\cdots\!19}{45\!\cdots\!64}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: |
$\frac{82\!\cdots\!21}{76\!\cdots\!44}a^{23}-\frac{10\!\cdots\!85}{30\!\cdots\!76}a^{22}+\frac{61\!\cdots\!05}{91\!\cdots\!28}a^{21}-\frac{62\!\cdots\!21}{30\!\cdots\!76}a^{20}+\frac{24\!\cdots\!09}{57\!\cdots\!83}a^{19}-\frac{14\!\cdots\!79}{45\!\cdots\!64}a^{18}-\frac{30\!\cdots\!73}{15\!\cdots\!88}a^{17}-\frac{28\!\cdots\!27}{91\!\cdots\!28}a^{16}+\frac{96\!\cdots\!93}{91\!\cdots\!28}a^{15}+\frac{41\!\cdots\!45}{19\!\cdots\!61}a^{14}-\frac{25\!\cdots\!13}{91\!\cdots\!28}a^{13}-\frac{13\!\cdots\!37}{91\!\cdots\!28}a^{12}-\frac{14\!\cdots\!31}{19\!\cdots\!61}a^{11}+\frac{20\!\cdots\!41}{22\!\cdots\!32}a^{10}+\frac{35\!\cdots\!21}{15\!\cdots\!88}a^{9}+\frac{27\!\cdots\!03}{30\!\cdots\!76}a^{8}-\frac{23\!\cdots\!71}{30\!\cdots\!76}a^{7}-\frac{24\!\cdots\!61}{30\!\cdots\!76}a^{6}-\frac{21\!\cdots\!99}{38\!\cdots\!22}a^{5}+\frac{55\!\cdots\!77}{15\!\cdots\!88}a^{4}+\frac{60\!\cdots\!97}{11\!\cdots\!66}a^{3}+\frac{22\!\cdots\!49}{76\!\cdots\!44}a^{2}+\frac{64\!\cdots\!69}{11\!\cdots\!66}a-\frac{32\!\cdots\!11}{76\!\cdots\!44}$, $\frac{87\!\cdots\!57}{45\!\cdots\!64}a^{23}-\frac{59\!\cdots\!73}{91\!\cdots\!28}a^{22}+\frac{39\!\cdots\!75}{30\!\cdots\!76}a^{21}-\frac{34\!\cdots\!23}{91\!\cdots\!28}a^{20}+\frac{37\!\cdots\!47}{45\!\cdots\!64}a^{19}-\frac{15\!\cdots\!43}{22\!\cdots\!32}a^{18}-\frac{20\!\cdots\!79}{45\!\cdots\!64}a^{17}-\frac{93\!\cdots\!91}{91\!\cdots\!28}a^{16}+\frac{16\!\cdots\!73}{91\!\cdots\!28}a^{15}+\frac{77\!\cdots\!15}{22\!\cdots\!32}a^{14}-\frac{57\!\cdots\!41}{91\!\cdots\!28}a^{13}-\frac{12\!\cdots\!01}{91\!\cdots\!28}a^{12}-\frac{16\!\cdots\!03}{15\!\cdots\!88}a^{11}+\frac{82\!\cdots\!15}{45\!\cdots\!64}a^{10}+\frac{95\!\cdots\!37}{45\!\cdots\!64}a^{9}+\frac{28\!\cdots\!35}{30\!\cdots\!76}a^{8}-\frac{38\!\cdots\!49}{30\!\cdots\!76}a^{7}-\frac{45\!\cdots\!71}{30\!\cdots\!76}a^{6}-\frac{18\!\cdots\!43}{11\!\cdots\!66}a^{5}+\frac{27\!\cdots\!01}{45\!\cdots\!64}a^{4}+\frac{30\!\cdots\!11}{38\!\cdots\!22}a^{3}+\frac{83\!\cdots\!41}{76\!\cdots\!44}a^{2}+\frac{13\!\cdots\!79}{11\!\cdots\!66}a-\frac{13\!\cdots\!55}{22\!\cdots\!32}$, $\frac{21\!\cdots\!55}{60\!\cdots\!52}a^{23}-\frac{16\!\cdots\!23}{15\!\cdots\!88}a^{22}+\frac{66\!\cdots\!29}{30\!\cdots\!76}a^{21}-\frac{41\!\cdots\!33}{60\!\cdots\!52}a^{20}+\frac{12\!\cdots\!89}{91\!\cdots\!28}a^{19}-\frac{95\!\cdots\!67}{91\!\cdots\!28}a^{18}-\frac{31\!\cdots\!99}{60\!\cdots\!52}a^{17}-\frac{10\!\cdots\!35}{91\!\cdots\!28}a^{16}+\frac{54\!\cdots\!13}{18\!\cdots\!56}a^{15}+\frac{45\!\cdots\!89}{60\!\cdots\!52}a^{14}-\frac{74\!\cdots\!85}{91\!\cdots\!28}a^{13}-\frac{78\!\cdots\!69}{18\!\cdots\!56}a^{12}-\frac{50\!\cdots\!15}{19\!\cdots\!61}a^{11}+\frac{12\!\cdots\!43}{45\!\cdots\!64}a^{10}+\frac{12\!\cdots\!99}{18\!\cdots\!56}a^{9}+\frac{52\!\cdots\!83}{15\!\cdots\!88}a^{8}-\frac{29\!\cdots\!29}{15\!\cdots\!88}a^{7}-\frac{17\!\cdots\!11}{60\!\cdots\!52}a^{6}-\frac{62\!\cdots\!49}{30\!\cdots\!76}a^{5}+\frac{87\!\cdots\!87}{30\!\cdots\!76}a^{4}+\frac{29\!\cdots\!59}{15\!\cdots\!88}a^{3}+\frac{17\!\cdots\!69}{15\!\cdots\!88}a^{2}+\frac{24\!\cdots\!51}{45\!\cdots\!64}a+\frac{93\!\cdots\!89}{45\!\cdots\!64}$, $\frac{11\!\cdots\!09}{22\!\cdots\!32}a^{23}-\frac{11\!\cdots\!83}{30\!\cdots\!76}a^{22}+\frac{33\!\cdots\!89}{30\!\cdots\!76}a^{21}-\frac{23\!\cdots\!17}{91\!\cdots\!28}a^{20}+\frac{76\!\cdots\!35}{11\!\cdots\!66}a^{19}-\frac{53\!\cdots\!99}{45\!\cdots\!64}a^{18}+\frac{38\!\cdots\!35}{45\!\cdots\!64}a^{17}+\frac{95\!\cdots\!19}{91\!\cdots\!28}a^{16}+\frac{80\!\cdots\!35}{91\!\cdots\!28}a^{15}-\frac{76\!\cdots\!26}{57\!\cdots\!83}a^{14}-\frac{43\!\cdots\!23}{91\!\cdots\!28}a^{13}+\frac{59\!\cdots\!33}{91\!\cdots\!28}a^{12}-\frac{31\!\cdots\!73}{11\!\cdots\!66}a^{11}+\frac{40\!\cdots\!47}{22\!\cdots\!32}a^{10}-\frac{98\!\cdots\!51}{45\!\cdots\!64}a^{9}+\frac{19\!\cdots\!37}{30\!\cdots\!76}a^{8}-\frac{60\!\cdots\!13}{30\!\cdots\!76}a^{7}+\frac{49\!\cdots\!25}{30\!\cdots\!76}a^{6}+\frac{40\!\cdots\!84}{57\!\cdots\!83}a^{5}+\frac{12\!\cdots\!75}{15\!\cdots\!88}a^{4}-\frac{16\!\cdots\!93}{19\!\cdots\!61}a^{3}-\frac{12\!\cdots\!31}{22\!\cdots\!32}a^{2}-\frac{32\!\cdots\!67}{11\!\cdots\!66}a+\frac{22\!\cdots\!69}{22\!\cdots\!32}$, $\frac{20\!\cdots\!13}{60\!\cdots\!52}a^{23}+\frac{47\!\cdots\!63}{91\!\cdots\!28}a^{22}-\frac{11\!\cdots\!83}{45\!\cdots\!64}a^{21}+\frac{38\!\cdots\!87}{60\!\cdots\!52}a^{20}-\frac{16\!\cdots\!79}{91\!\cdots\!28}a^{19}+\frac{39\!\cdots\!93}{91\!\cdots\!28}a^{18}-\frac{38\!\cdots\!29}{60\!\cdots\!52}a^{17}+\frac{18\!\cdots\!85}{45\!\cdots\!64}a^{16}-\frac{49\!\cdots\!35}{18\!\cdots\!56}a^{15}+\frac{52\!\cdots\!39}{60\!\cdots\!52}a^{14}+\frac{12\!\cdots\!05}{45\!\cdots\!64}a^{13}-\frac{48\!\cdots\!29}{18\!\cdots\!56}a^{12}+\frac{14\!\cdots\!47}{76\!\cdots\!44}a^{11}-\frac{11\!\cdots\!29}{22\!\cdots\!32}a^{10}+\frac{19\!\cdots\!45}{18\!\cdots\!56}a^{9}-\frac{26\!\cdots\!71}{30\!\cdots\!76}a^{8}+\frac{29\!\cdots\!47}{30\!\cdots\!76}a^{7}-\frac{75\!\cdots\!19}{60\!\cdots\!52}a^{6}+\frac{15\!\cdots\!37}{30\!\cdots\!76}a^{5}-\frac{24\!\cdots\!31}{91\!\cdots\!28}a^{4}+\frac{17\!\cdots\!23}{45\!\cdots\!64}a^{3}-\frac{21\!\cdots\!71}{15\!\cdots\!88}a^{2}+\frac{43\!\cdots\!77}{45\!\cdots\!64}a+\frac{13\!\cdots\!65}{45\!\cdots\!64}$, $\frac{32\!\cdots\!21}{91\!\cdots\!28}a^{23}-\frac{51\!\cdots\!39}{45\!\cdots\!64}a^{22}+\frac{20\!\cdots\!81}{91\!\cdots\!28}a^{21}-\frac{61\!\cdots\!31}{91\!\cdots\!28}a^{20}+\frac{43\!\cdots\!53}{30\!\cdots\!76}a^{19}-\frac{33\!\cdots\!63}{30\!\cdots\!76}a^{18}-\frac{28\!\cdots\!25}{45\!\cdots\!64}a^{17}-\frac{26\!\cdots\!59}{30\!\cdots\!76}a^{16}+\frac{10\!\cdots\!51}{30\!\cdots\!76}a^{15}+\frac{30\!\cdots\!91}{45\!\cdots\!64}a^{14}-\frac{28\!\cdots\!65}{30\!\cdots\!76}a^{13}-\frac{12\!\cdots\!15}{30\!\cdots\!76}a^{12}-\frac{21\!\cdots\!91}{91\!\cdots\!28}a^{11}+\frac{27\!\cdots\!07}{91\!\cdots\!28}a^{10}+\frac{13\!\cdots\!83}{22\!\cdots\!32}a^{9}+\frac{85\!\cdots\!75}{30\!\cdots\!76}a^{8}-\frac{38\!\cdots\!07}{15\!\cdots\!88}a^{7}-\frac{37\!\cdots\!89}{15\!\cdots\!88}a^{6}-\frac{18\!\cdots\!31}{11\!\cdots\!66}a^{5}+\frac{68\!\cdots\!53}{57\!\cdots\!83}a^{4}+\frac{37\!\cdots\!05}{22\!\cdots\!32}a^{3}+\frac{20\!\cdots\!77}{22\!\cdots\!32}a^{2}+\frac{76\!\cdots\!60}{57\!\cdots\!83}a-\frac{94\!\cdots\!05}{57\!\cdots\!83}$, $\frac{35\!\cdots\!17}{22\!\cdots\!32}a^{23}+\frac{20\!\cdots\!31}{45\!\cdots\!64}a^{22}-\frac{56\!\cdots\!39}{22\!\cdots\!32}a^{21}+\frac{34\!\cdots\!23}{76\!\cdots\!44}a^{20}-\frac{86\!\cdots\!05}{57\!\cdots\!83}a^{19}+\frac{65\!\cdots\!39}{15\!\cdots\!88}a^{18}-\frac{40\!\cdots\!29}{76\!\cdots\!44}a^{17}+\frac{15\!\cdots\!17}{11\!\cdots\!66}a^{16}-\frac{81\!\cdots\!55}{76\!\cdots\!44}a^{15}+\frac{50\!\cdots\!45}{38\!\cdots\!22}a^{14}+\frac{10\!\cdots\!49}{11\!\cdots\!66}a^{13}-\frac{25\!\cdots\!25}{76\!\cdots\!44}a^{12}-\frac{16\!\cdots\!33}{22\!\cdots\!32}a^{11}-\frac{77\!\cdots\!33}{15\!\cdots\!88}a^{10}+\frac{25\!\cdots\!55}{22\!\cdots\!32}a^{9}-\frac{15\!\cdots\!69}{76\!\cdots\!44}a^{8}+\frac{59\!\cdots\!29}{76\!\cdots\!44}a^{7}-\frac{17\!\cdots\!95}{15\!\cdots\!88}a^{6}-\frac{59\!\cdots\!41}{22\!\cdots\!32}a^{5}-\frac{65\!\cdots\!35}{22\!\cdots\!32}a^{4}+\frac{59\!\cdots\!11}{11\!\cdots\!66}a^{3}+\frac{29\!\cdots\!39}{11\!\cdots\!66}a^{2}+\frac{46\!\cdots\!15}{38\!\cdots\!22}a-\frac{75\!\cdots\!11}{11\!\cdots\!66}$, $\frac{13\!\cdots\!01}{30\!\cdots\!76}a^{23}-\frac{56\!\cdots\!37}{45\!\cdots\!64}a^{22}+\frac{21\!\cdots\!49}{91\!\cdots\!28}a^{21}-\frac{69\!\cdots\!99}{91\!\cdots\!28}a^{20}+\frac{46\!\cdots\!61}{30\!\cdots\!76}a^{19}-\frac{27\!\cdots\!71}{30\!\cdots\!76}a^{18}-\frac{39\!\cdots\!71}{45\!\cdots\!64}a^{17}-\frac{62\!\cdots\!45}{30\!\cdots\!76}a^{16}+\frac{13\!\cdots\!49}{30\!\cdots\!76}a^{15}+\frac{42\!\cdots\!61}{45\!\cdots\!64}a^{14}-\frac{25\!\cdots\!27}{30\!\cdots\!76}a^{13}-\frac{28\!\cdots\!65}{30\!\cdots\!76}a^{12}-\frac{29\!\cdots\!29}{91\!\cdots\!28}a^{11}+\frac{27\!\cdots\!33}{91\!\cdots\!28}a^{10}+\frac{67\!\cdots\!17}{45\!\cdots\!64}a^{9}+\frac{14\!\cdots\!87}{30\!\cdots\!76}a^{8}-\frac{23\!\cdots\!87}{76\!\cdots\!44}a^{7}-\frac{54\!\cdots\!28}{19\!\cdots\!61}a^{6}-\frac{16\!\cdots\!07}{38\!\cdots\!22}a^{5}+\frac{41\!\cdots\!55}{22\!\cdots\!32}a^{4}+\frac{46\!\cdots\!77}{22\!\cdots\!32}a^{3}+\frac{44\!\cdots\!25}{22\!\cdots\!32}a^{2}+\frac{22\!\cdots\!90}{57\!\cdots\!83}a+\frac{34\!\cdots\!45}{11\!\cdots\!66}$, $\frac{33\!\cdots\!27}{18\!\cdots\!56}a^{23}-\frac{32\!\cdots\!33}{30\!\cdots\!76}a^{22}+\frac{73\!\cdots\!81}{22\!\cdots\!32}a^{21}-\frac{15\!\cdots\!95}{18\!\cdots\!56}a^{20}+\frac{19\!\cdots\!39}{91\!\cdots\!28}a^{19}-\frac{33\!\cdots\!23}{91\!\cdots\!28}a^{18}+\frac{70\!\cdots\!69}{18\!\cdots\!56}a^{17}-\frac{14\!\cdots\!06}{57\!\cdots\!83}a^{16}+\frac{57\!\cdots\!73}{18\!\cdots\!56}a^{15}-\frac{40\!\cdots\!11}{18\!\cdots\!56}a^{14}-\frac{19\!\cdots\!07}{22\!\cdots\!32}a^{13}+\frac{33\!\cdots\!87}{18\!\cdots\!56}a^{12}-\frac{38\!\cdots\!73}{15\!\cdots\!88}a^{11}+\frac{11\!\cdots\!25}{22\!\cdots\!32}a^{10}-\frac{13\!\cdots\!55}{18\!\cdots\!56}a^{9}+\frac{20\!\cdots\!99}{30\!\cdots\!76}a^{8}-\frac{21\!\cdots\!89}{30\!\cdots\!76}a^{7}+\frac{40\!\cdots\!33}{60\!\cdots\!52}a^{6}-\frac{27\!\cdots\!45}{91\!\cdots\!28}a^{5}+\frac{39\!\cdots\!07}{30\!\cdots\!76}a^{4}-\frac{80\!\cdots\!57}{45\!\cdots\!64}a^{3}+\frac{60\!\cdots\!45}{15\!\cdots\!88}a^{2}+\frac{12\!\cdots\!25}{45\!\cdots\!64}a+\frac{92\!\cdots\!21}{45\!\cdots\!64}$, $\frac{32\!\cdots\!21}{91\!\cdots\!28}a^{23}-\frac{44\!\cdots\!87}{22\!\cdots\!32}a^{22}+\frac{23\!\cdots\!99}{45\!\cdots\!64}a^{21}-\frac{37\!\cdots\!87}{30\!\cdots\!76}a^{20}+\frac{14\!\cdots\!41}{45\!\cdots\!64}a^{19}-\frac{21\!\cdots\!39}{45\!\cdots\!64}a^{18}+\frac{78\!\cdots\!19}{30\!\cdots\!76}a^{17}+\frac{17\!\cdots\!48}{57\!\cdots\!83}a^{16}+\frac{42\!\cdots\!25}{91\!\cdots\!28}a^{15}-\frac{49\!\cdots\!61}{30\!\cdots\!76}a^{14}-\frac{53\!\cdots\!95}{22\!\cdots\!32}a^{13}+\frac{20\!\cdots\!63}{91\!\cdots\!28}a^{12}-\frac{84\!\cdots\!39}{45\!\cdots\!64}a^{11}+\frac{12\!\cdots\!47}{15\!\cdots\!88}a^{10}-\frac{23\!\cdots\!15}{30\!\cdots\!76}a^{9}+\frac{20\!\cdots\!49}{76\!\cdots\!44}a^{8}-\frac{11\!\cdots\!39}{15\!\cdots\!88}a^{7}+\frac{13\!\cdots\!05}{30\!\cdots\!76}a^{6}+\frac{15\!\cdots\!35}{45\!\cdots\!64}a^{5}+\frac{11\!\cdots\!01}{45\!\cdots\!64}a^{4}-\frac{41\!\cdots\!81}{22\!\cdots\!32}a^{3}-\frac{56\!\cdots\!91}{22\!\cdots\!32}a^{2}-\frac{35\!\cdots\!85}{76\!\cdots\!44}a+\frac{43\!\cdots\!91}{76\!\cdots\!44}$, $\frac{15\!\cdots\!79}{11\!\cdots\!66}a^{23}-\frac{53\!\cdots\!31}{91\!\cdots\!28}a^{22}+\frac{83\!\cdots\!55}{57\!\cdots\!83}a^{21}-\frac{11\!\cdots\!49}{30\!\cdots\!76}a^{20}+\frac{84\!\cdots\!97}{91\!\cdots\!28}a^{19}-\frac{11\!\cdots\!27}{91\!\cdots\!28}a^{18}+\frac{21\!\cdots\!51}{30\!\cdots\!76}a^{17}-\frac{18\!\cdots\!83}{45\!\cdots\!64}a^{16}+\frac{14\!\cdots\!01}{91\!\cdots\!28}a^{15}+\frac{25\!\cdots\!33}{30\!\cdots\!76}a^{14}-\frac{26\!\cdots\!57}{45\!\cdots\!64}a^{13}+\frac{41\!\cdots\!99}{91\!\cdots\!28}a^{12}-\frac{90\!\cdots\!51}{91\!\cdots\!28}a^{11}+\frac{67\!\cdots\!15}{30\!\cdots\!76}a^{10}-\frac{53\!\cdots\!15}{30\!\cdots\!76}a^{9}+\frac{25\!\cdots\!11}{15\!\cdots\!88}a^{8}-\frac{69\!\cdots\!69}{30\!\cdots\!76}a^{7}+\frac{11\!\cdots\!49}{15\!\cdots\!88}a^{6}-\frac{50\!\cdots\!03}{45\!\cdots\!64}a^{5}+\frac{66\!\cdots\!07}{11\!\cdots\!66}a^{4}-\frac{66\!\cdots\!27}{11\!\cdots\!66}a^{3}-\frac{27\!\cdots\!89}{22\!\cdots\!32}a^{2}-\frac{13\!\cdots\!37}{76\!\cdots\!44}a+\frac{26\!\cdots\!68}{19\!\cdots\!61}$, $\frac{10\!\cdots\!69}{30\!\cdots\!76}a^{23}-\frac{18\!\cdots\!17}{15\!\cdots\!88}a^{22}+\frac{25\!\cdots\!17}{91\!\cdots\!28}a^{21}-\frac{73\!\cdots\!27}{91\!\cdots\!28}a^{20}+\frac{16\!\cdots\!77}{91\!\cdots\!28}a^{19}-\frac{19\!\cdots\!89}{91\!\cdots\!28}a^{18}+\frac{38\!\cdots\!41}{45\!\cdots\!64}a^{17}-\frac{14\!\cdots\!25}{91\!\cdots\!28}a^{16}+\frac{34\!\cdots\!17}{91\!\cdots\!28}a^{15}+\frac{20\!\cdots\!73}{45\!\cdots\!64}a^{14}-\frac{93\!\cdots\!59}{91\!\cdots\!28}a^{13}+\frac{24\!\cdots\!83}{91\!\cdots\!28}a^{12}-\frac{24\!\cdots\!69}{91\!\cdots\!28}a^{11}+\frac{37\!\cdots\!43}{91\!\cdots\!28}a^{10}-\frac{22\!\cdots\!97}{11\!\cdots\!66}a^{9}+\frac{13\!\cdots\!95}{30\!\cdots\!76}a^{8}-\frac{34\!\cdots\!57}{76\!\cdots\!44}a^{7}+\frac{52\!\cdots\!55}{15\!\cdots\!88}a^{6}-\frac{17\!\cdots\!11}{76\!\cdots\!44}a^{5}+\frac{30\!\cdots\!00}{19\!\cdots\!61}a^{4}+\frac{17\!\cdots\!09}{22\!\cdots\!32}a^{3}+\frac{17\!\cdots\!15}{22\!\cdots\!32}a^{2}+\frac{14\!\cdots\!15}{11\!\cdots\!66}a-\frac{17\!\cdots\!65}{57\!\cdots\!83}$, $\frac{25\!\cdots\!75}{91\!\cdots\!28}a^{23}-\frac{93\!\cdots\!63}{91\!\cdots\!28}a^{22}+\frac{20\!\cdots\!69}{91\!\cdots\!28}a^{21}-\frac{28\!\cdots\!07}{45\!\cdots\!64}a^{20}+\frac{64\!\cdots\!35}{45\!\cdots\!64}a^{19}-\frac{67\!\cdots\!79}{45\!\cdots\!64}a^{18}+\frac{73\!\cdots\!15}{91\!\cdots\!28}a^{17}-\frac{49\!\cdots\!95}{91\!\cdots\!28}a^{16}+\frac{13\!\cdots\!67}{45\!\cdots\!64}a^{15}+\frac{36\!\cdots\!61}{91\!\cdots\!28}a^{14}-\frac{88\!\cdots\!01}{91\!\cdots\!28}a^{13}+\frac{58\!\cdots\!39}{45\!\cdots\!64}a^{12}-\frac{41\!\cdots\!95}{22\!\cdots\!32}a^{11}+\frac{48\!\cdots\!81}{15\!\cdots\!88}a^{10}-\frac{85\!\cdots\!01}{91\!\cdots\!28}a^{9}+\frac{69\!\cdots\!97}{30\!\cdots\!76}a^{8}-\frac{88\!\cdots\!85}{30\!\cdots\!76}a^{7}-\frac{12\!\cdots\!35}{15\!\cdots\!88}a^{6}-\frac{21\!\cdots\!91}{45\!\cdots\!64}a^{5}+\frac{25\!\cdots\!25}{22\!\cdots\!32}a^{4}+\frac{17\!\cdots\!41}{22\!\cdots\!32}a^{3}+\frac{11\!\cdots\!72}{57\!\cdots\!83}a^{2}-\frac{12\!\cdots\!45}{76\!\cdots\!44}a-\frac{40\!\cdots\!69}{11\!\cdots\!66}$
| 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: | \( 494355842.3116425 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 494355842.3116425 \cdot 1}{2\cdot\sqrt{7654476884289954721505578121232384}}\cr\approx \mathstrut & 4.33481516125310 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4)
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 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4, 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 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4);
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 - 3*x^23 + 6*x^22 - 19*x^21 + 39*x^20 - 30*x^19 - 9*x^18 - 45*x^17 + 93*x^16 + 204*x^15 - 201*x^14 - 129*x^13 - 793*x^12 + 738*x^11 + 141*x^10 + 1177*x^9 - 666*x^8 - 597*x^7 - 913*x^6 + 168*x^5 + 474*x^4 + 464*x^3 + 168*x^2 + 24*x - 4);
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
| 6.2.170772624.2, 12.4.29163289107845376.1 |
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: | 24.4.7654476884289954721505578121232384.1 |
| 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 | R | $20{,}\,{\href{/padicField/5.4.0.1}{4} }$ | ${\href{/padicField/7.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | $20{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.12.0.1}{12} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.12.12.5 | $x^{12} - 12 x^{11} + 132 x^{10} - 204 x^{9} + 3024 x^{8} - 432 x^{7} + 3510 x^{6} + 2268 x^{5} - 972 x^{4} + 756 x^{3} + 1620 x^{2} + 648 x + 81$ | $3$ | $4$ | $12$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ |
| 3.12.14.9 | $x^{12} - 6 x^{9} + 12 x^{6} + 18 x^{5} + 9 x^{4} + 36 x^{3} + 18 x^{2} + 18$ | $6$ | $2$ | $14$ | $S_3 \times C_4$ | $[3/2]_{2}^{4}$ | |
|
\(11\)
| 11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.20.16.2 | $x^{20} + 968 x^{10} - 13310 x^{5} + 29282$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.13068.120.a.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.13068.120.a.b | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.13068.120.a.c | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 2.13068.120.a.d | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| * | 3.13068.12t76.a.a | $3$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| * | 3.13068.12t76.a.b | $3$ | $ 2^{2} \cdot 3^{3} \cdot 11^{2}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $1$ |
| 3.39204.12t33.a.a | $3$ | $ 2^{2} \cdot 3^{4} \cdot 11^{2}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.39204.12t33.a.b | $3$ | $ 2^{2} \cdot 3^{4} \cdot 11^{2}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 4.4743684.10t11.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $0$ | |
| 4.4743684.5t4.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $0$ | |
| 4.4743684.40t188.a.a | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 4.4743684.40t188.a.b | $4$ | $ 2^{2} \cdot 3^{4} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ | |
| 5.512317872.12t75.a.a | $5$ | $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ | 10.0.67507613675568.1 | $A_5\times C_2$ (as 10T11) | $1$ | $-1$ | |
| * | 5.170772624.6t12.a.a | $5$ | $ 2^{4} \cdot 3^{6} \cdot 11^{4}$ | 5.1.4743684.1 | $A_5$ (as 5T4) | $1$ | $1$ |
| * | 6.512317872.24t576.a.a | $6$ | $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (as 24T576) | $0$ | $0$ |
| * | 6.512317872.24t576.a.b | $6$ | $ 2^{4} \cdot 3^{7} \cdot 11^{4}$ | 24.4.7654476884289954721505578121232384.2 | $\SL(2,5):C_2$ (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 *.