Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888)
gp: K = bnfinit(y^20 - 5*y^19 + 76*y^18 - 247*y^17 + 1197*y^16 - 8474*y^15 + 15561*y^14 - 112347*y^13 + 325793*y^12 - 787322*y^11 + 3851661*y^10 - 5756183*y^9 + 20865344*y^8 - 48001353*y^7 + 45895165*y^6 - 245996344*y^5 + 8889264*y^4 - 588303992*y^3 - 54940704*y^2 - 538817408*y + 31141888, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888)
\( x^{20} - 5 x^{19} + 76 x^{18} - 247 x^{17} + 1197 x^{16} - 8474 x^{15} + 15561 x^{14} - 112347 x^{13} + \cdots + 31141888 \)
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: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-206007596521214410095208558252435839890349094339\)
\(\medspace = -\,19^{37}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(232.11\) | 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: | $19^{683/342}\approx 357.90532196302917$ | ||
| Ramified primes: |
\(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{7}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{8}a^{5}-\frac{1}{16}a^{4}$, $\frac{1}{32}a^{12}-\frac{1}{16}a^{9}-\frac{1}{32}a^{8}-\frac{1}{16}a^{7}-\frac{1}{16}a^{6}+\frac{3}{16}a^{5}-\frac{1}{32}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a$, $\frac{1}{32}a^{13}+\frac{1}{32}a^{9}+\frac{1}{16}a^{8}-\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{3}{32}a^{5}+\frac{1}{16}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{32}a^{14}-\frac{1}{32}a^{10}-\frac{1}{8}a^{7}+\frac{1}{32}a^{6}-\frac{1}{8}a^{5}+\frac{1}{16}a^{4}-\frac{1}{8}a^{3}$, $\frac{1}{64}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{11}+\frac{1}{64}a^{10}-\frac{3}{64}a^{9}-\frac{1}{8}a^{8}+\frac{7}{64}a^{7}-\frac{1}{64}a^{6}-\frac{11}{64}a^{5}-\frac{1}{32}a^{4}-\frac{1}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{64}a^{16}-\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{11}+\frac{1}{64}a^{9}-\frac{3}{64}a^{8}+\frac{1}{32}a^{7}+\frac{3}{32}a^{6}+\frac{3}{64}a^{5}+\frac{1}{16}a^{4}+\frac{1}{8}a^{2}$, $\frac{1}{128}a^{17}-\frac{1}{128}a^{15}-\frac{1}{64}a^{13}+\frac{3}{128}a^{11}+\frac{3}{64}a^{9}-\frac{1}{128}a^{7}+\frac{1}{16}a^{6}+\frac{25}{128}a^{5}+\frac{1}{16}a^{4}+\frac{3}{16}a^{3}+\frac{1}{16}a^{2}+\frac{1}{4}a$, $\frac{1}{512}a^{18}-\frac{1}{256}a^{17}+\frac{1}{512}a^{16}-\frac{1}{256}a^{15}+\frac{1}{256}a^{14}-\frac{1}{256}a^{13}-\frac{7}{512}a^{12}+\frac{5}{256}a^{11}-\frac{7}{256}a^{10}-\frac{7}{256}a^{9}+\frac{9}{512}a^{8}+\frac{29}{256}a^{7}+\frac{49}{512}a^{6}-\frac{5}{32}a^{5}+\frac{7}{64}a^{4}+\frac{9}{64}a^{3}+\frac{5}{16}a^{2}$, $\frac{1}{10\!\cdots\!32}a^{19}-\frac{32\!\cdots\!85}{10\!\cdots\!32}a^{18}-\frac{96\!\cdots\!71}{35\!\cdots\!44}a^{17}-\frac{45\!\cdots\!89}{10\!\cdots\!32}a^{16}-\frac{18\!\cdots\!37}{26\!\cdots\!08}a^{15}-\frac{91\!\cdots\!93}{44\!\cdots\!68}a^{14}-\frac{72\!\cdots\!67}{35\!\cdots\!44}a^{13}-\frac{32\!\cdots\!35}{35\!\cdots\!44}a^{12}+\frac{33\!\cdots\!11}{13\!\cdots\!04}a^{11}+\frac{93\!\cdots\!33}{88\!\cdots\!36}a^{10}-\frac{20\!\cdots\!51}{35\!\cdots\!44}a^{9}+\frac{61\!\cdots\!63}{10\!\cdots\!32}a^{8}-\frac{42\!\cdots\!79}{35\!\cdots\!44}a^{7}+\frac{24\!\cdots\!27}{35\!\cdots\!44}a^{6}-\frac{32\!\cdots\!15}{26\!\cdots\!08}a^{5}-\frac{22\!\cdots\!99}{11\!\cdots\!92}a^{4}+\frac{16\!\cdots\!19}{44\!\cdots\!68}a^{3}+\frac{95\!\cdots\!05}{41\!\cdots\!22}a^{2}+\frac{17\!\cdots\!95}{83\!\cdots\!44}a-\frac{18\!\cdots\!77}{20\!\cdots\!61}$
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: | $10$ | 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: |
$\frac{52\!\cdots\!05}{17\!\cdots\!72}a^{19}-\frac{22\!\cdots\!89}{17\!\cdots\!72}a^{18}+\frac{37\!\cdots\!45}{17\!\cdots\!72}a^{17}-\frac{99\!\cdots\!01}{17\!\cdots\!72}a^{16}+\frac{24\!\cdots\!45}{88\!\cdots\!36}a^{15}-\frac{98\!\cdots\!95}{44\!\cdots\!68}a^{14}+\frac{47\!\cdots\!35}{17\!\cdots\!72}a^{13}-\frac{48\!\cdots\!21}{17\!\cdots\!72}a^{12}+\frac{65\!\cdots\!71}{88\!\cdots\!36}a^{11}-\frac{15\!\cdots\!51}{11\!\cdots\!92}a^{10}+\frac{16\!\cdots\!71}{17\!\cdots\!72}a^{9}-\frac{16\!\cdots\!89}{17\!\cdots\!72}a^{8}+\frac{74\!\cdots\!93}{17\!\cdots\!72}a^{7}-\frac{18\!\cdots\!75}{17\!\cdots\!72}a^{6}+\frac{20\!\cdots\!17}{88\!\cdots\!36}a^{5}-\frac{12\!\cdots\!85}{22\!\cdots\!84}a^{4}-\frac{67\!\cdots\!13}{22\!\cdots\!84}a^{3}-\frac{13\!\cdots\!65}{11\!\cdots\!92}a^{2}-\frac{72\!\cdots\!81}{13\!\cdots\!74}a-\frac{59\!\cdots\!93}{69\!\cdots\!87}$, $\frac{13\!\cdots\!99}{17\!\cdots\!72}a^{19}-\frac{39\!\cdots\!07}{17\!\cdots\!72}a^{18}+\frac{77\!\cdots\!41}{17\!\cdots\!72}a^{17}-\frac{91\!\cdots\!87}{17\!\cdots\!72}a^{16}+\frac{38\!\cdots\!63}{22\!\cdots\!84}a^{15}-\frac{22\!\cdots\!59}{55\!\cdots\!96}a^{14}-\frac{49\!\cdots\!31}{17\!\cdots\!72}a^{13}-\frac{49\!\cdots\!63}{17\!\cdots\!72}a^{12}+\frac{37\!\cdots\!05}{44\!\cdots\!68}a^{11}+\frac{61\!\cdots\!99}{44\!\cdots\!68}a^{10}+\frac{16\!\cdots\!69}{17\!\cdots\!72}a^{9}+\frac{15\!\cdots\!49}{17\!\cdots\!72}a^{8}-\frac{16\!\cdots\!35}{17\!\cdots\!72}a^{7}-\frac{12\!\cdots\!93}{17\!\cdots\!72}a^{6}-\frac{16\!\cdots\!69}{44\!\cdots\!68}a^{5}-\frac{99\!\cdots\!47}{22\!\cdots\!84}a^{4}-\frac{26\!\cdots\!29}{22\!\cdots\!84}a^{3}-\frac{17\!\cdots\!41}{27\!\cdots\!48}a^{2}-\frac{32\!\cdots\!11}{27\!\cdots\!48}a+\frac{48\!\cdots\!25}{69\!\cdots\!87}$, $\frac{31\!\cdots\!47}{35\!\cdots\!44}a^{19}-\frac{14\!\cdots\!03}{35\!\cdots\!44}a^{18}+\frac{20\!\cdots\!57}{35\!\cdots\!44}a^{17}-\frac{53\!\cdots\!83}{35\!\cdots\!44}a^{16}+\frac{85\!\cdots\!49}{22\!\cdots\!84}a^{15}-\frac{52\!\cdots\!67}{11\!\cdots\!92}a^{14}+\frac{13\!\cdots\!41}{35\!\cdots\!44}a^{13}-\frac{13\!\cdots\!71}{35\!\cdots\!44}a^{12}+\frac{10\!\cdots\!05}{88\!\cdots\!36}a^{11}-\frac{90\!\cdots\!53}{88\!\cdots\!36}a^{10}+\frac{33\!\cdots\!09}{35\!\cdots\!44}a^{9}+\frac{48\!\cdots\!33}{35\!\cdots\!44}a^{8}-\frac{20\!\cdots\!95}{35\!\cdots\!44}a^{7}-\frac{18\!\cdots\!49}{35\!\cdots\!44}a^{6}-\frac{14\!\cdots\!87}{44\!\cdots\!68}a^{5}-\frac{63\!\cdots\!69}{22\!\cdots\!84}a^{4}-\frac{46\!\cdots\!53}{44\!\cdots\!68}a^{3}-\frac{44\!\cdots\!97}{11\!\cdots\!92}a^{2}-\frac{29\!\cdots\!61}{27\!\cdots\!48}a+\frac{43\!\cdots\!83}{69\!\cdots\!87}$, $\frac{11\!\cdots\!35}{88\!\cdots\!36}a^{19}-\frac{31\!\cdots\!53}{35\!\cdots\!44}a^{18}+\frac{18\!\cdots\!61}{17\!\cdots\!72}a^{17}-\frac{15\!\cdots\!73}{35\!\cdots\!44}a^{16}+\frac{26\!\cdots\!67}{17\!\cdots\!72}a^{15}-\frac{17\!\cdots\!09}{17\!\cdots\!72}a^{14}+\frac{49\!\cdots\!11}{17\!\cdots\!72}a^{13}-\frac{39\!\cdots\!81}{35\!\cdots\!44}a^{12}+\frac{73\!\cdots\!29}{17\!\cdots\!72}a^{11}-\frac{15\!\cdots\!33}{17\!\cdots\!72}a^{10}+\frac{56\!\cdots\!77}{17\!\cdots\!72}a^{9}-\frac{23\!\cdots\!89}{35\!\cdots\!44}a^{8}+\frac{22\!\cdots\!95}{17\!\cdots\!72}a^{7}-\frac{11\!\cdots\!05}{35\!\cdots\!44}a^{6}+\frac{22\!\cdots\!29}{88\!\cdots\!36}a^{5}-\frac{35\!\cdots\!87}{44\!\cdots\!68}a^{4}+\frac{15\!\cdots\!55}{44\!\cdots\!68}a^{3}-\frac{19\!\cdots\!25}{27\!\cdots\!48}a^{2}+\frac{94\!\cdots\!25}{27\!\cdots\!48}a+\frac{92\!\cdots\!59}{69\!\cdots\!87}$, $\frac{12\!\cdots\!57}{10\!\cdots\!32}a^{19}-\frac{31\!\cdots\!27}{53\!\cdots\!16}a^{18}+\frac{31\!\cdots\!19}{35\!\cdots\!44}a^{17}-\frac{14\!\cdots\!19}{53\!\cdots\!16}a^{16}+\frac{69\!\cdots\!85}{53\!\cdots\!16}a^{15}-\frac{17\!\cdots\!27}{17\!\cdots\!72}a^{14}+\frac{58\!\cdots\!23}{35\!\cdots\!44}a^{13}-\frac{22\!\cdots\!75}{17\!\cdots\!72}a^{12}+\frac{19\!\cdots\!37}{53\!\cdots\!16}a^{11}-\frac{14\!\cdots\!33}{17\!\cdots\!72}a^{10}+\frac{15\!\cdots\!15}{35\!\cdots\!44}a^{9}-\frac{32\!\cdots\!21}{53\!\cdots\!16}a^{8}+\frac{77\!\cdots\!07}{35\!\cdots\!44}a^{7}-\frac{46\!\cdots\!13}{88\!\cdots\!36}a^{6}+\frac{17\!\cdots\!19}{41\!\cdots\!22}a^{5}-\frac{11\!\cdots\!05}{44\!\cdots\!68}a^{4}-\frac{91\!\cdots\!35}{55\!\cdots\!96}a^{3}-\frac{97\!\cdots\!69}{16\!\cdots\!88}a^{2}-\frac{22\!\cdots\!77}{20\!\cdots\!61}a-\frac{96\!\cdots\!53}{20\!\cdots\!61}$, $\frac{20\!\cdots\!17}{17\!\cdots\!72}a^{19}+\frac{68\!\cdots\!09}{88\!\cdots\!36}a^{18}-\frac{21\!\cdots\!93}{17\!\cdots\!72}a^{17}+\frac{75\!\cdots\!63}{88\!\cdots\!36}a^{16}-\frac{84\!\cdots\!85}{22\!\cdots\!84}a^{15}+\frac{72\!\cdots\!83}{88\!\cdots\!36}a^{14}-\frac{16\!\cdots\!79}{17\!\cdots\!72}a^{13}+\frac{18\!\cdots\!21}{88\!\cdots\!36}a^{12}-\frac{20\!\cdots\!29}{22\!\cdots\!84}a^{11}+\frac{35\!\cdots\!61}{88\!\cdots\!36}a^{10}-\frac{12\!\cdots\!39}{17\!\cdots\!72}a^{9}+\frac{27\!\cdots\!21}{88\!\cdots\!36}a^{8}-\frac{10\!\cdots\!97}{17\!\cdots\!72}a^{7}+\frac{48\!\cdots\!19}{44\!\cdots\!68}a^{6}-\frac{32\!\cdots\!13}{88\!\cdots\!36}a^{5}+\frac{15\!\cdots\!01}{11\!\cdots\!92}a^{4}-\frac{10\!\cdots\!73}{11\!\cdots\!92}a^{3}+\frac{35\!\cdots\!87}{11\!\cdots\!92}a^{2}-\frac{70\!\cdots\!88}{69\!\cdots\!87}a+\frac{40\!\cdots\!07}{69\!\cdots\!87}$, $\frac{19\!\cdots\!59}{10\!\cdots\!32}a^{19}-\frac{13\!\cdots\!09}{26\!\cdots\!08}a^{18}+\frac{33\!\cdots\!41}{35\!\cdots\!44}a^{17}-\frac{10\!\cdots\!91}{26\!\cdots\!08}a^{16}-\frac{30\!\cdots\!83}{53\!\cdots\!16}a^{15}-\frac{69\!\cdots\!59}{17\!\cdots\!72}a^{14}-\frac{99\!\cdots\!27}{35\!\cdots\!44}a^{13}+\frac{10\!\cdots\!87}{44\!\cdots\!68}a^{12}-\frac{99\!\cdots\!15}{53\!\cdots\!16}a^{11}+\frac{27\!\cdots\!35}{17\!\cdots\!72}a^{10}-\frac{66\!\cdots\!71}{35\!\cdots\!44}a^{9}+\frac{19\!\cdots\!17}{13\!\cdots\!04}a^{8}-\frac{10\!\cdots\!11}{35\!\cdots\!44}a^{7}+\frac{91\!\cdots\!67}{17\!\cdots\!72}a^{6}-\frac{69\!\cdots\!73}{33\!\cdots\!76}a^{5}+\frac{25\!\cdots\!05}{44\!\cdots\!68}a^{4}-\frac{12\!\cdots\!63}{22\!\cdots\!84}a^{3}-\frac{14\!\cdots\!63}{83\!\cdots\!44}a^{2}-\frac{50\!\cdots\!97}{83\!\cdots\!44}a+\frac{71\!\cdots\!79}{20\!\cdots\!61}$, $\frac{10\!\cdots\!75}{26\!\cdots\!08}a^{19}-\frac{88\!\cdots\!59}{53\!\cdots\!16}a^{18}+\frac{12\!\cdots\!21}{44\!\cdots\!68}a^{17}-\frac{39\!\cdots\!79}{53\!\cdots\!16}a^{16}+\frac{10\!\cdots\!53}{26\!\cdots\!08}a^{15}-\frac{25\!\cdots\!75}{88\!\cdots\!36}a^{14}+\frac{83\!\cdots\!11}{22\!\cdots\!84}a^{13}-\frac{66\!\cdots\!25}{17\!\cdots\!72}a^{12}+\frac{26\!\cdots\!51}{26\!\cdots\!08}a^{11}-\frac{18\!\cdots\!47}{88\!\cdots\!36}a^{10}+\frac{55\!\cdots\!71}{44\!\cdots\!68}a^{9}-\frac{69\!\cdots\!59}{53\!\cdots\!16}a^{8}+\frac{67\!\cdots\!63}{11\!\cdots\!92}a^{7}-\frac{25\!\cdots\!33}{17\!\cdots\!72}a^{6}+\frac{66\!\cdots\!27}{13\!\cdots\!04}a^{5}-\frac{89\!\cdots\!87}{11\!\cdots\!92}a^{4}-\frac{79\!\cdots\!93}{22\!\cdots\!84}a^{3}-\frac{15\!\cdots\!15}{83\!\cdots\!44}a^{2}-\frac{47\!\cdots\!61}{83\!\cdots\!44}a-\frac{27\!\cdots\!79}{20\!\cdots\!61}$, $\frac{13\!\cdots\!71}{35\!\cdots\!44}a^{19}-\frac{27\!\cdots\!55}{44\!\cdots\!68}a^{18}+\frac{40\!\cdots\!63}{35\!\cdots\!44}a^{17}+\frac{40\!\cdots\!47}{44\!\cdots\!68}a^{16}-\frac{15\!\cdots\!11}{17\!\cdots\!72}a^{15}+\frac{64\!\cdots\!55}{17\!\cdots\!72}a^{14}-\frac{93\!\cdots\!49}{35\!\cdots\!44}a^{13}+\frac{77\!\cdots\!75}{88\!\cdots\!36}a^{12}-\frac{55\!\cdots\!55}{17\!\cdots\!72}a^{11}+\frac{19\!\cdots\!97}{17\!\cdots\!72}a^{10}-\frac{87\!\cdots\!53}{35\!\cdots\!44}a^{9}+\frac{66\!\cdots\!63}{88\!\cdots\!36}a^{8}-\frac{53\!\cdots\!97}{35\!\cdots\!44}a^{7}+\frac{44\!\cdots\!19}{17\!\cdots\!72}a^{6}-\frac{17\!\cdots\!13}{27\!\cdots\!48}a^{5}+\frac{16\!\cdots\!93}{44\!\cdots\!68}a^{4}-\frac{31\!\cdots\!95}{22\!\cdots\!84}a^{3}+\frac{59\!\cdots\!89}{27\!\cdots\!48}a^{2}-\frac{92\!\cdots\!61}{69\!\cdots\!87}a+\frac{52\!\cdots\!67}{69\!\cdots\!87}$, $\frac{21\!\cdots\!47}{53\!\cdots\!16}a^{19}-\frac{47\!\cdots\!83}{13\!\cdots\!04}a^{18}+\frac{55\!\cdots\!11}{17\!\cdots\!72}a^{17}-\frac{21\!\cdots\!05}{13\!\cdots\!04}a^{16}+\frac{21\!\cdots\!81}{83\!\cdots\!44}a^{15}-\frac{21\!\cdots\!83}{88\!\cdots\!36}a^{14}+\frac{18\!\cdots\!49}{17\!\cdots\!72}a^{13}-\frac{16\!\cdots\!07}{22\!\cdots\!84}a^{12}+\frac{36\!\cdots\!45}{33\!\cdots\!76}a^{11}-\frac{15\!\cdots\!89}{88\!\cdots\!36}a^{10}+\frac{49\!\cdots\!89}{17\!\cdots\!72}a^{9}-\frac{12\!\cdots\!39}{66\!\cdots\!52}a^{8}-\frac{42\!\cdots\!21}{17\!\cdots\!72}a^{7}-\frac{18\!\cdots\!57}{88\!\cdots\!36}a^{6}+\frac{14\!\cdots\!81}{26\!\cdots\!08}a^{5}+\frac{43\!\cdots\!40}{69\!\cdots\!87}a^{4}+\frac{18\!\cdots\!11}{27\!\cdots\!48}a^{3}+\frac{73\!\cdots\!97}{33\!\cdots\!76}a^{2}+\frac{48\!\cdots\!31}{41\!\cdots\!22}a+\frac{47\!\cdots\!83}{20\!\cdots\!61}$
| 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: | \( 496592090559000000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 496592090559000000 \cdot 1}{2\cdot\sqrt{206007596521214410095208558252435839890349094339}}\cr\approx \mathstrut & 33.3969642233715 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888)
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 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888, 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 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888);
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 - 5*x^19 + 76*x^18 - 247*x^17 + 1197*x^16 - 8474*x^15 + 15561*x^14 - 112347*x^13 + 325793*x^12 - 787322*x^11 + 3851661*x^10 - 5756183*x^9 + 20865344*x^8 - 48001353*x^7 + 45895165*x^6 - 245996344*x^5 + 8889264*x^4 - 588303992*x^3 - 54940704*x^2 - 538817408*x + 31141888);
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
$\PGL(2,19)$ (as 20T362):
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 6840 |
| The 21 conjugacy class representatives for $\PGL(2,19)$ |
| Character table for $\PGL(2,19)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 sibling: | 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 | ${\href{/padicField/2.2.0.1}{2} }^{9}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/17.9.0.1}{9} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | $20$ | $18{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20$ | $18{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | $18{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 19.19.37.13 | $x^{19} + 4332 x^{2} + 4332 x + 19$ | $19$ | $1$ | $37$ | $F_{19}$ | $[37/18]_{18}$ |
Additional information
This field is associated with the 19-torsion points on any elliptic curve in the isogeny class 19.a.