Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144)
gp: K = bnfinit(y^20 - 4*y^19 - 192*y^18 - 368*y^17 + 5900*y^16 + 11612*y^15 - 82376*y^14 - 147412*y^13 + 496249*y^12 + 5837552*y^11 + 28472784*y^10 + 9581508*y^9 - 257955946*y^8 - 848288280*y^7 - 488954264*y^6 + 5231676656*y^5 + 10164474572*y^4 - 6966146992*y^3 - 22153914848*y^2 - 5610833232*y + 1443605144, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144)
\( x^{20} - 4 x^{19} - 192 x^{18} - 368 x^{17} + 5900 x^{16} + 11612 x^{15} - 82376 x^{14} + \cdots + 1443605144 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[10, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4478964844204240999374053727702242866567597682327552\)
\(\medspace = -\,2^{38}\cdot 17^{8}\cdot 23^{5}\cdot 881^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(382.44\) | 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: | $2^{125/48}17^{1/2}23^{1/2}881^{2/3}\approx 11049.38055118573$ | ||
| Ramified primes: |
\(2\), \(17\), \(23\), \(881\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{11}+\frac{1}{8}a^{10}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{17}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{24}a^{18}-\frac{1}{24}a^{16}-\frac{1}{8}a^{14}+\frac{1}{12}a^{13}-\frac{1}{8}a^{12}-\frac{1}{12}a^{10}-\frac{1}{4}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{6}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{44\!\cdots\!28}a^{19}-\frac{43\!\cdots\!39}{44\!\cdots\!28}a^{18}+\frac{14\!\cdots\!47}{44\!\cdots\!28}a^{17}+\frac{17\!\cdots\!71}{44\!\cdots\!28}a^{16}+\frac{57\!\cdots\!91}{14\!\cdots\!76}a^{15}-\frac{49\!\cdots\!47}{44\!\cdots\!28}a^{14}+\frac{62\!\cdots\!23}{44\!\cdots\!28}a^{13}+\frac{28\!\cdots\!61}{86\!\cdots\!28}a^{12}+\frac{18\!\cdots\!85}{22\!\cdots\!64}a^{11}+\frac{41\!\cdots\!77}{22\!\cdots\!64}a^{10}-\frac{24\!\cdots\!07}{22\!\cdots\!64}a^{9}-\frac{28\!\cdots\!79}{22\!\cdots\!64}a^{8}+\frac{25\!\cdots\!32}{27\!\cdots\!83}a^{7}+\frac{11\!\cdots\!01}{11\!\cdots\!32}a^{6}-\frac{32\!\cdots\!99}{11\!\cdots\!32}a^{5}+\frac{29\!\cdots\!99}{12\!\cdots\!48}a^{4}-\frac{10\!\cdots\!65}{16\!\cdots\!99}a^{3}+\frac{17\!\cdots\!83}{61\!\cdots\!74}a^{2}-\frac{19\!\cdots\!21}{55\!\cdots\!66}a-\frac{26\!\cdots\!51}{55\!\cdots\!66}$
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: | $14$ | 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: |
$\frac{10\!\cdots\!09}{74\!\cdots\!14}a^{19}-\frac{48\!\cdots\!09}{74\!\cdots\!14}a^{18}-\frac{39\!\cdots\!61}{14\!\cdots\!28}a^{17}-\frac{12\!\cdots\!31}{37\!\cdots\!07}a^{16}+\frac{41\!\cdots\!13}{49\!\cdots\!76}a^{15}+\frac{15\!\cdots\!31}{14\!\cdots\!28}a^{14}-\frac{17\!\cdots\!71}{14\!\cdots\!28}a^{13}-\frac{15\!\cdots\!84}{12\!\cdots\!69}a^{12}+\frac{11\!\cdots\!91}{14\!\cdots\!28}a^{11}+\frac{11\!\cdots\!51}{14\!\cdots\!28}a^{10}+\frac{12\!\cdots\!86}{37\!\cdots\!07}a^{9}-\frac{35\!\cdots\!85}{37\!\cdots\!07}a^{8}-\frac{12\!\cdots\!30}{37\!\cdots\!07}a^{7}-\frac{70\!\cdots\!97}{74\!\cdots\!14}a^{6}-\frac{19\!\cdots\!19}{37\!\cdots\!07}a^{5}+\frac{59\!\cdots\!53}{82\!\cdots\!46}a^{4}+\frac{34\!\cdots\!05}{37\!\cdots\!07}a^{3}-\frac{21\!\cdots\!26}{13\!\cdots\!41}a^{2}-\frac{80\!\cdots\!74}{37\!\cdots\!07}a+\frac{13\!\cdots\!06}{37\!\cdots\!07}$, $\frac{49\!\cdots\!13}{24\!\cdots\!38}a^{19}-\frac{16\!\cdots\!59}{12\!\cdots\!69}a^{18}-\frac{16\!\cdots\!75}{49\!\cdots\!76}a^{17}+\frac{30\!\cdots\!01}{49\!\cdots\!76}a^{16}+\frac{15\!\cdots\!35}{16\!\cdots\!92}a^{15}+\frac{69\!\cdots\!61}{49\!\cdots\!76}a^{14}-\frac{48\!\cdots\!65}{49\!\cdots\!76}a^{13}-\frac{82\!\cdots\!53}{82\!\cdots\!46}a^{12}+\frac{21\!\cdots\!25}{49\!\cdots\!76}a^{11}+\frac{55\!\cdots\!55}{49\!\cdots\!76}a^{10}+\frac{38\!\cdots\!87}{12\!\cdots\!69}a^{9}-\frac{13\!\cdots\!59}{49\!\cdots\!76}a^{8}-\frac{42\!\cdots\!84}{12\!\cdots\!69}a^{7}-\frac{30\!\cdots\!99}{24\!\cdots\!38}a^{6}+\frac{26\!\cdots\!77}{12\!\cdots\!69}a^{5}+\frac{49\!\cdots\!91}{82\!\cdots\!46}a^{4}+\frac{16\!\cdots\!83}{12\!\cdots\!69}a^{3}-\frac{23\!\cdots\!40}{41\!\cdots\!23}a^{2}-\frac{51\!\cdots\!42}{12\!\cdots\!69}a-\frac{23\!\cdots\!25}{12\!\cdots\!69}$, $\frac{45\!\cdots\!83}{49\!\cdots\!76}a^{19}-\frac{21\!\cdots\!23}{49\!\cdots\!76}a^{18}-\frac{85\!\cdots\!31}{49\!\cdots\!76}a^{17}-\frac{10\!\cdots\!09}{49\!\cdots\!76}a^{16}+\frac{91\!\cdots\!63}{16\!\cdots\!92}a^{15}+\frac{32\!\cdots\!03}{49\!\cdots\!76}a^{14}-\frac{39\!\cdots\!43}{49\!\cdots\!76}a^{13}-\frac{29\!\cdots\!67}{41\!\cdots\!23}a^{12}+\frac{62\!\cdots\!23}{12\!\cdots\!69}a^{11}+\frac{12\!\cdots\!19}{24\!\cdots\!38}a^{10}+\frac{56\!\cdots\!91}{24\!\cdots\!38}a^{9}-\frac{40\!\cdots\!01}{49\!\cdots\!76}a^{8}-\frac{28\!\cdots\!71}{12\!\cdots\!69}a^{7}-\frac{74\!\cdots\!88}{12\!\cdots\!69}a^{6}-\frac{36\!\cdots\!39}{12\!\cdots\!69}a^{5}+\frac{65\!\cdots\!00}{13\!\cdots\!41}a^{4}+\frac{68\!\cdots\!72}{12\!\cdots\!69}a^{3}-\frac{13\!\cdots\!24}{13\!\cdots\!41}a^{2}-\frac{14\!\cdots\!54}{12\!\cdots\!69}a+\frac{14\!\cdots\!44}{12\!\cdots\!69}$, $\frac{78\!\cdots\!43}{49\!\cdots\!76}a^{19}-\frac{60\!\cdots\!13}{49\!\cdots\!76}a^{18}-\frac{33\!\cdots\!39}{12\!\cdots\!69}a^{17}+\frac{56\!\cdots\!92}{12\!\cdots\!69}a^{16}+\frac{38\!\cdots\!65}{41\!\cdots\!23}a^{15}-\frac{38\!\cdots\!29}{24\!\cdots\!38}a^{14}-\frac{32\!\cdots\!77}{24\!\cdots\!38}a^{13}+\frac{19\!\cdots\!03}{82\!\cdots\!46}a^{12}+\frac{34\!\cdots\!81}{49\!\cdots\!76}a^{11}+\frac{32\!\cdots\!33}{49\!\cdots\!76}a^{10}+\frac{42\!\cdots\!47}{24\!\cdots\!38}a^{9}-\frac{11\!\cdots\!97}{12\!\cdots\!69}a^{8}-\frac{31\!\cdots\!01}{12\!\cdots\!69}a^{7}-\frac{44\!\cdots\!31}{24\!\cdots\!38}a^{6}+\frac{22\!\cdots\!08}{12\!\cdots\!69}a^{5}+\frac{16\!\cdots\!33}{27\!\cdots\!82}a^{4}-\frac{11\!\cdots\!41}{12\!\cdots\!69}a^{3}-\frac{26\!\cdots\!24}{13\!\cdots\!41}a^{2}+\frac{22\!\cdots\!56}{12\!\cdots\!69}a+\frac{17\!\cdots\!84}{12\!\cdots\!69}$, $\frac{69\!\cdots\!73}{27\!\cdots\!83}a^{19}-\frac{21\!\cdots\!61}{22\!\cdots\!64}a^{18}-\frac{10\!\cdots\!01}{22\!\cdots\!64}a^{17}-\frac{28\!\cdots\!13}{27\!\cdots\!83}a^{16}+\frac{10\!\cdots\!95}{73\!\cdots\!88}a^{15}+\frac{88\!\cdots\!09}{27\!\cdots\!83}a^{14}-\frac{45\!\cdots\!25}{22\!\cdots\!64}a^{13}-\frac{22\!\cdots\!08}{54\!\cdots\!33}a^{12}+\frac{26\!\cdots\!57}{22\!\cdots\!64}a^{11}+\frac{33\!\cdots\!33}{22\!\cdots\!64}a^{10}+\frac{82\!\cdots\!19}{11\!\cdots\!32}a^{9}+\frac{20\!\cdots\!61}{55\!\cdots\!66}a^{8}-\frac{71\!\cdots\!03}{11\!\cdots\!32}a^{7}-\frac{12\!\cdots\!75}{55\!\cdots\!66}a^{6}-\frac{88\!\cdots\!37}{55\!\cdots\!66}a^{5}+\frac{26\!\cdots\!65}{20\!\cdots\!58}a^{4}+\frac{45\!\cdots\!89}{16\!\cdots\!99}a^{3}-\frac{81\!\cdots\!27}{61\!\cdots\!74}a^{2}-\frac{16\!\cdots\!13}{27\!\cdots\!83}a-\frac{64\!\cdots\!50}{27\!\cdots\!83}$, $\frac{18\!\cdots\!11}{22\!\cdots\!64}a^{19}-\frac{12\!\cdots\!13}{22\!\cdots\!64}a^{18}-\frac{15\!\cdots\!59}{11\!\cdots\!32}a^{17}+\frac{28\!\cdots\!84}{27\!\cdots\!83}a^{16}+\frac{33\!\cdots\!05}{73\!\cdots\!88}a^{15}-\frac{73\!\cdots\!27}{22\!\cdots\!64}a^{14}-\frac{31\!\cdots\!75}{55\!\cdots\!66}a^{13}+\frac{39\!\cdots\!47}{10\!\cdots\!66}a^{12}+\frac{75\!\cdots\!40}{27\!\cdots\!83}a^{11}+\frac{23\!\cdots\!29}{55\!\cdots\!66}a^{10}+\frac{13\!\cdots\!03}{11\!\cdots\!32}a^{9}-\frac{13\!\cdots\!47}{55\!\cdots\!66}a^{8}-\frac{15\!\cdots\!69}{11\!\cdots\!32}a^{7}-\frac{18\!\cdots\!73}{55\!\cdots\!66}a^{6}+\frac{12\!\cdots\!78}{27\!\cdots\!83}a^{5}+\frac{89\!\cdots\!40}{30\!\cdots\!87}a^{4}+\frac{18\!\cdots\!11}{32\!\cdots\!98}a^{3}-\frac{16\!\cdots\!70}{30\!\cdots\!87}a^{2}-\frac{81\!\cdots\!59}{27\!\cdots\!83}a-\frac{57\!\cdots\!17}{27\!\cdots\!83}$, $\frac{42\!\cdots\!81}{11\!\cdots\!32}a^{19}-\frac{19\!\cdots\!59}{11\!\cdots\!32}a^{18}-\frac{20\!\cdots\!32}{27\!\cdots\!83}a^{17}-\frac{12\!\cdots\!23}{11\!\cdots\!32}a^{16}+\frac{85\!\cdots\!61}{36\!\cdots\!44}a^{15}+\frac{37\!\cdots\!33}{11\!\cdots\!32}a^{14}-\frac{36\!\cdots\!85}{11\!\cdots\!32}a^{13}-\frac{22\!\cdots\!54}{54\!\cdots\!33}a^{12}+\frac{23\!\cdots\!03}{11\!\cdots\!32}a^{11}+\frac{59\!\cdots\!23}{27\!\cdots\!83}a^{10}+\frac{11\!\cdots\!93}{11\!\cdots\!32}a^{9}-\frac{70\!\cdots\!43}{11\!\cdots\!32}a^{8}-\frac{10\!\cdots\!51}{11\!\cdots\!32}a^{7}-\frac{15\!\cdots\!23}{55\!\cdots\!66}a^{6}-\frac{35\!\cdots\!59}{55\!\cdots\!66}a^{5}+\frac{41\!\cdots\!25}{20\!\cdots\!58}a^{4}+\frac{94\!\cdots\!35}{32\!\cdots\!98}a^{3}-\frac{12\!\cdots\!22}{30\!\cdots\!87}a^{2}-\frac{17\!\cdots\!69}{27\!\cdots\!83}a+\frac{30\!\cdots\!79}{27\!\cdots\!83}$, $\frac{44\!\cdots\!05}{22\!\cdots\!64}a^{19}-\frac{18\!\cdots\!15}{22\!\cdots\!64}a^{18}-\frac{86\!\cdots\!93}{22\!\cdots\!64}a^{17}-\frac{76\!\cdots\!57}{11\!\cdots\!32}a^{16}+\frac{91\!\cdots\!79}{73\!\cdots\!88}a^{15}+\frac{11\!\cdots\!71}{55\!\cdots\!66}a^{14}-\frac{40\!\cdots\!83}{22\!\cdots\!64}a^{13}-\frac{14\!\cdots\!91}{54\!\cdots\!33}a^{12}+\frac{13\!\cdots\!51}{11\!\cdots\!32}a^{11}+\frac{25\!\cdots\!75}{22\!\cdots\!64}a^{10}+\frac{30\!\cdots\!77}{55\!\cdots\!66}a^{9}+\frac{21\!\cdots\!63}{11\!\cdots\!32}a^{8}-\frac{15\!\cdots\!50}{27\!\cdots\!83}a^{7}-\frac{87\!\cdots\!85}{55\!\cdots\!66}a^{6}-\frac{26\!\cdots\!39}{55\!\cdots\!66}a^{5}+\frac{34\!\cdots\!08}{30\!\cdots\!87}a^{4}+\frac{28\!\cdots\!20}{16\!\cdots\!99}a^{3}-\frac{47\!\cdots\!79}{20\!\cdots\!58}a^{2}-\frac{10\!\cdots\!68}{27\!\cdots\!83}a+\frac{18\!\cdots\!02}{27\!\cdots\!83}$, $\frac{28\!\cdots\!47}{44\!\cdots\!28}a^{19}-\frac{20\!\cdots\!35}{44\!\cdots\!28}a^{18}-\frac{48\!\cdots\!65}{44\!\cdots\!28}a^{17}+\frac{32\!\cdots\!37}{44\!\cdots\!28}a^{16}+\frac{47\!\cdots\!35}{14\!\cdots\!76}a^{15}-\frac{91\!\cdots\!17}{44\!\cdots\!28}a^{14}-\frac{13\!\cdots\!41}{44\!\cdots\!28}a^{13}+\frac{31\!\cdots\!31}{86\!\cdots\!28}a^{12}-\frac{53\!\cdots\!33}{11\!\cdots\!32}a^{11}+\frac{10\!\cdots\!55}{27\!\cdots\!83}a^{10}+\frac{28\!\cdots\!01}{22\!\cdots\!64}a^{9}-\frac{42\!\cdots\!75}{22\!\cdots\!64}a^{8}-\frac{15\!\cdots\!51}{11\!\cdots\!32}a^{7}-\frac{35\!\cdots\!69}{11\!\cdots\!32}a^{6}+\frac{56\!\cdots\!77}{11\!\cdots\!32}a^{5}+\frac{32\!\cdots\!45}{12\!\cdots\!48}a^{4}-\frac{24\!\cdots\!54}{16\!\cdots\!99}a^{3}-\frac{15\!\cdots\!40}{30\!\cdots\!87}a^{2}-\frac{84\!\cdots\!95}{55\!\cdots\!66}a+\frac{20\!\cdots\!07}{55\!\cdots\!66}$, $\frac{27\!\cdots\!71}{81\!\cdots\!32}a^{19}-\frac{40\!\cdots\!99}{24\!\cdots\!96}a^{18}-\frac{50\!\cdots\!13}{81\!\cdots\!32}a^{17}-\frac{22\!\cdots\!42}{30\!\cdots\!87}a^{16}+\frac{15\!\cdots\!27}{81\!\cdots\!32}a^{15}+\frac{39\!\cdots\!21}{20\!\cdots\!58}a^{14}-\frac{62\!\cdots\!29}{24\!\cdots\!96}a^{13}-\frac{11\!\cdots\!46}{60\!\cdots\!37}a^{12}+\frac{13\!\cdots\!12}{10\!\cdots\!29}a^{11}+\frac{42\!\cdots\!51}{24\!\cdots\!96}a^{10}+\frac{16\!\cdots\!93}{20\!\cdots\!58}a^{9}-\frac{75\!\cdots\!61}{61\!\cdots\!74}a^{8}-\frac{79\!\cdots\!07}{12\!\cdots\!48}a^{7}-\frac{12\!\cdots\!61}{61\!\cdots\!74}a^{6}-\frac{21\!\cdots\!23}{20\!\cdots\!58}a^{5}+\frac{39\!\cdots\!85}{30\!\cdots\!87}a^{4}+\frac{54\!\cdots\!73}{36\!\cdots\!22}a^{3}-\frac{98\!\cdots\!81}{61\!\cdots\!74}a^{2}-\frac{22\!\cdots\!60}{30\!\cdots\!87}a+\frac{44\!\cdots\!95}{30\!\cdots\!87}$, $\frac{93\!\cdots\!09}{22\!\cdots\!64}a^{19}-\frac{21\!\cdots\!21}{22\!\cdots\!64}a^{18}-\frac{18\!\cdots\!53}{22\!\cdots\!64}a^{17}-\frac{82\!\cdots\!33}{27\!\cdots\!83}a^{16}+\frac{14\!\cdots\!53}{73\!\cdots\!88}a^{15}+\frac{91\!\cdots\!55}{11\!\cdots\!32}a^{14}-\frac{44\!\cdots\!37}{22\!\cdots\!64}a^{13}-\frac{21\!\cdots\!79}{21\!\cdots\!32}a^{12}+\frac{44\!\cdots\!85}{11\!\cdots\!32}a^{11}+\frac{56\!\cdots\!79}{22\!\cdots\!64}a^{10}+\frac{18\!\cdots\!41}{11\!\cdots\!32}a^{9}+\frac{35\!\cdots\!49}{11\!\cdots\!32}a^{8}-\frac{57\!\cdots\!35}{11\!\cdots\!32}a^{7}-\frac{12\!\cdots\!05}{27\!\cdots\!83}a^{6}-\frac{54\!\cdots\!25}{55\!\cdots\!66}a^{5}+\frac{15\!\cdots\!52}{30\!\cdots\!87}a^{4}+\frac{16\!\cdots\!31}{32\!\cdots\!98}a^{3}+\frac{36\!\cdots\!99}{61\!\cdots\!74}a^{2}+\frac{30\!\cdots\!53}{27\!\cdots\!83}a-\frac{93\!\cdots\!02}{27\!\cdots\!83}$, $\frac{94\!\cdots\!61}{11\!\cdots\!32}a^{19}-\frac{11\!\cdots\!97}{22\!\cdots\!64}a^{18}-\frac{17\!\cdots\!21}{11\!\cdots\!32}a^{17}-\frac{62\!\cdots\!89}{22\!\cdots\!64}a^{16}+\frac{38\!\cdots\!35}{73\!\cdots\!88}a^{15}+\frac{17\!\cdots\!42}{27\!\cdots\!83}a^{14}-\frac{20\!\cdots\!94}{27\!\cdots\!83}a^{13}+\frac{16\!\cdots\!65}{43\!\cdots\!64}a^{12}+\frac{95\!\cdots\!99}{22\!\cdots\!64}a^{11}+\frac{93\!\cdots\!73}{22\!\cdots\!64}a^{10}+\frac{18\!\cdots\!49}{11\!\cdots\!32}a^{9}-\frac{26\!\cdots\!77}{11\!\cdots\!32}a^{8}-\frac{20\!\cdots\!59}{11\!\cdots\!32}a^{7}-\frac{22\!\cdots\!99}{55\!\cdots\!66}a^{6}+\frac{92\!\cdots\!16}{27\!\cdots\!83}a^{5}+\frac{12\!\cdots\!40}{30\!\cdots\!87}a^{4}+\frac{27\!\cdots\!94}{16\!\cdots\!99}a^{3}-\frac{19\!\cdots\!47}{20\!\cdots\!58}a^{2}-\frac{89\!\cdots\!86}{27\!\cdots\!83}a+\frac{19\!\cdots\!03}{27\!\cdots\!83}$, $\frac{29\!\cdots\!25}{44\!\cdots\!28}a^{19}-\frac{52\!\cdots\!45}{44\!\cdots\!28}a^{18}-\frac{58\!\cdots\!01}{44\!\cdots\!28}a^{17}-\frac{24\!\cdots\!09}{44\!\cdots\!28}a^{16}+\frac{41\!\cdots\!11}{14\!\cdots\!76}a^{15}+\frac{63\!\cdots\!97}{44\!\cdots\!28}a^{14}-\frac{10\!\cdots\!13}{44\!\cdots\!28}a^{13}-\frac{13\!\cdots\!71}{86\!\cdots\!28}a^{12}-\frac{41\!\cdots\!33}{22\!\cdots\!64}a^{11}+\frac{21\!\cdots\!01}{55\!\cdots\!66}a^{10}+\frac{61\!\cdots\!33}{22\!\cdots\!64}a^{9}+\frac{15\!\cdots\!89}{22\!\cdots\!64}a^{8}-\frac{15\!\cdots\!77}{55\!\cdots\!66}a^{7}-\frac{74\!\cdots\!87}{11\!\cdots\!32}a^{6}-\frac{21\!\cdots\!51}{11\!\cdots\!32}a^{5}-\frac{38\!\cdots\!13}{40\!\cdots\!16}a^{4}+\frac{79\!\cdots\!01}{16\!\cdots\!99}a^{3}+\frac{21\!\cdots\!68}{30\!\cdots\!87}a^{2}+\frac{79\!\cdots\!97}{55\!\cdots\!66}a-\frac{23\!\cdots\!75}{55\!\cdots\!66}$, $\frac{71\!\cdots\!17}{22\!\cdots\!64}a^{19}-\frac{27\!\cdots\!93}{22\!\cdots\!64}a^{18}-\frac{13\!\cdots\!21}{22\!\cdots\!64}a^{17}-\frac{71\!\cdots\!05}{55\!\cdots\!66}a^{16}+\frac{13\!\cdots\!63}{73\!\cdots\!88}a^{15}+\frac{22\!\cdots\!89}{55\!\cdots\!66}a^{14}-\frac{57\!\cdots\!13}{22\!\cdots\!64}a^{13}-\frac{56\!\cdots\!21}{10\!\cdots\!66}a^{12}+\frac{42\!\cdots\!03}{27\!\cdots\!83}a^{11}+\frac{42\!\cdots\!85}{22\!\cdots\!64}a^{10}+\frac{10\!\cdots\!73}{11\!\cdots\!32}a^{9}+\frac{12\!\cdots\!60}{27\!\cdots\!83}a^{8}-\frac{91\!\cdots\!23}{11\!\cdots\!32}a^{7}-\frac{15\!\cdots\!41}{55\!\cdots\!66}a^{6}-\frac{11\!\cdots\!23}{55\!\cdots\!66}a^{5}+\frac{51\!\cdots\!89}{30\!\cdots\!87}a^{4}+\frac{11\!\cdots\!87}{32\!\cdots\!98}a^{3}-\frac{10\!\cdots\!33}{61\!\cdots\!74}a^{2}-\frac{20\!\cdots\!05}{27\!\cdots\!83}a-\frac{82\!\cdots\!77}{27\!\cdots\!83}$
| 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: | \( 12096615465700000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 12096615465700000000 \cdot 1}{2\cdot\sqrt{4478964844204240999374053727702242866567597682327552}}\cr\approx \mathstrut & 0.906242345172334 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144)
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^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144, 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^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144);
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^20 - 4*x^19 - 192*x^18 - 368*x^17 + 5900*x^16 + 11612*x^15 - 82376*x^14 - 147412*x^13 + 496249*x^12 + 5837552*x^11 + 28472784*x^10 + 9581508*x^9 - 257955946*x^8 - 848288280*x^7 - 488954264*x^6 + 5231676656*x^5 + 10164474572*x^4 - 6966146992*x^3 - 22153914848*x^2 - 5610833232*x + 1443605144);
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
$C_2^8.(S_3\times A_5)$ (as 20T754):
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 92160 |
| The 39 conjugacy class representatives for $C_2^8.(S_3\times A_5)$ |
| Character table for $C_2^8.(S_3\times A_5)$ |
Intermediate fields
| 5.5.3104644.1 |
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 30 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| 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 | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | $15{,}\,{\href{/padicField/13.5.0.1}{5} }$ | R | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.12.16 | $x^{8} + 8 x^{7} + 32 x^{6} + 78 x^{5} + 137 x^{4} + 186 x^{3} + 128 x^{2} - 10 x + 7$ | $4$ | $2$ | $12$ | $A_4\times C_2$ | $[2, 2]^{6}$ |
| 2.12.26.29 | $x^{12} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 2$ | $12$ | $1$ | $26$ | 12T128 | $[8/3, 8/3, 8/3, 8/3]_{3}^{6}$ | |
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.0.1 | $x^{4} + 7 x^{2} + 10 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.12.6.2 | $x^{12} + 578 x^{8} + 835210 x^{4} - 4259571 x^{2} + 72412707$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
|
\(23\)
| 23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 23.5.0.1 | $x^{5} + 3 x + 18$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 23.10.5.2 | $x^{10} + 115 x^{8} + 5296 x^{6} + 36 x^{5} + 120980 x^{4} - 8280 x^{3} + 1383344 x^{2} + 95328 x + 6509876$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(881\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $3$ | $3$ | $1$ | $2$ | ||||
| Deg $6$ | $3$ | $2$ | $4$ |