Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464)
gp: K = bnfinit(y^20 - 4*y^19 + 76*y^17 - 608*y^15 + 57*y^14 + 3116*y^13 - 10792*y^11 + 304*y^10 + 26600*y^9 - 1349*y^8 - 46436*y^7 + 8512*y^6 + 52060*y^5 - 17024*y^4 - 34504*y^3 + 26315*y^2 - 5892*y + 464, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 4*x^19 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464)
\( x^{20} - 4 x^{19} + 76 x^{17} - 608 x^{15} + 57 x^{14} + 3116 x^{13} - 10792 x^{11} + 304 x^{10} + \cdots + 464 \)
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: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(14595833152648338774900083206236247150592\)
\(\medspace = 2^{12}\cdot 19^{19}\cdot 23^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(101.91\) | 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: | $2^{2/3}19^{359/342}23^{1/2}\approx 167.44337081125943$ | ||
| Ramified primes: |
\(2\), \(19\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{437}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{3}{8}a^{4}+\frac{1}{8}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{3}{8}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{7}+\frac{3}{8}a^{4}-\frac{1}{4}a$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{8}+\frac{3}{8}a^{5}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{18}-\frac{1}{16}a^{16}-\frac{1}{16}a^{14}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}+\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{16}a^{6}-\frac{1}{2}a^{5}-\frac{5}{16}a^{4}+\frac{7}{16}a^{2}-\frac{1}{4}a$, $\frac{1}{89\!\cdots\!92}a^{19}+\frac{16\!\cdots\!91}{89\!\cdots\!92}a^{18}-\frac{14\!\cdots\!23}{89\!\cdots\!92}a^{17}+\frac{33\!\cdots\!63}{89\!\cdots\!92}a^{16}-\frac{38\!\cdots\!59}{89\!\cdots\!92}a^{15}-\frac{83\!\cdots\!37}{89\!\cdots\!92}a^{14}-\frac{20\!\cdots\!09}{44\!\cdots\!96}a^{13}+\frac{51\!\cdots\!11}{44\!\cdots\!96}a^{12}+\frac{96\!\cdots\!65}{44\!\cdots\!96}a^{11}+\frac{25\!\cdots\!23}{44\!\cdots\!96}a^{10}-\frac{76\!\cdots\!19}{44\!\cdots\!96}a^{9}+\frac{21\!\cdots\!15}{44\!\cdots\!96}a^{8}-\frac{17\!\cdots\!35}{89\!\cdots\!92}a^{7}+\frac{54\!\cdots\!67}{89\!\cdots\!92}a^{6}-\frac{31\!\cdots\!35}{89\!\cdots\!92}a^{5}+\frac{17\!\cdots\!11}{89\!\cdots\!92}a^{4}-\frac{12\!\cdots\!07}{89\!\cdots\!92}a^{3}+\frac{44\!\cdots\!03}{89\!\cdots\!92}a^{2}+\frac{71\!\cdots\!23}{22\!\cdots\!48}a-\frac{27\!\cdots\!57}{56\!\cdots\!62}$
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: | $9$ | 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: |
$\frac{26\!\cdots\!05}{11\!\cdots\!24}a^{19}-\frac{19\!\cdots\!03}{22\!\cdots\!48}a^{18}-\frac{72\!\cdots\!61}{22\!\cdots\!48}a^{17}+\frac{20\!\cdots\!11}{11\!\cdots\!24}a^{16}+\frac{17\!\cdots\!60}{28\!\cdots\!81}a^{15}-\frac{32\!\cdots\!57}{22\!\cdots\!48}a^{14}-\frac{37\!\cdots\!73}{11\!\cdots\!24}a^{13}+\frac{20\!\cdots\!39}{28\!\cdots\!81}a^{12}+\frac{25\!\cdots\!69}{11\!\cdots\!24}a^{11}-\frac{14\!\cdots\!01}{56\!\cdots\!62}a^{10}-\frac{72\!\cdots\!15}{11\!\cdots\!24}a^{9}+\frac{69\!\cdots\!33}{11\!\cdots\!24}a^{8}+\frac{14\!\cdots\!43}{11\!\cdots\!24}a^{7}-\frac{23\!\cdots\!77}{22\!\cdots\!48}a^{6}-\frac{11\!\cdots\!61}{22\!\cdots\!48}a^{5}+\frac{13\!\cdots\!09}{11\!\cdots\!24}a^{4}-\frac{17\!\cdots\!89}{11\!\cdots\!24}a^{3}-\frac{17\!\cdots\!89}{22\!\cdots\!48}a^{2}+\frac{51\!\cdots\!39}{11\!\cdots\!24}a-\frac{17\!\cdots\!33}{28\!\cdots\!81}$, $\frac{16\!\cdots\!27}{44\!\cdots\!96}a^{19}-\frac{14\!\cdots\!59}{11\!\cdots\!24}a^{18}-\frac{26\!\cdots\!45}{44\!\cdots\!96}a^{17}+\frac{62\!\cdots\!99}{22\!\cdots\!48}a^{16}+\frac{55\!\cdots\!73}{44\!\cdots\!96}a^{15}-\frac{49\!\cdots\!29}{22\!\cdots\!48}a^{14}-\frac{23\!\cdots\!05}{28\!\cdots\!81}a^{13}+\frac{25\!\cdots\!71}{22\!\cdots\!48}a^{12}+\frac{60\!\cdots\!97}{11\!\cdots\!24}a^{11}-\frac{83\!\cdots\!17}{22\!\cdots\!48}a^{10}-\frac{48\!\cdots\!85}{28\!\cdots\!81}a^{9}+\frac{19\!\cdots\!13}{22\!\cdots\!48}a^{8}+\frac{16\!\cdots\!69}{44\!\cdots\!96}a^{7}-\frac{32\!\cdots\!01}{22\!\cdots\!48}a^{6}-\frac{17\!\cdots\!83}{44\!\cdots\!96}a^{5}+\frac{17\!\cdots\!61}{11\!\cdots\!24}a^{4}+\frac{26\!\cdots\!75}{44\!\cdots\!96}a^{3}-\frac{60\!\cdots\!15}{56\!\cdots\!62}a^{2}+\frac{29\!\cdots\!13}{56\!\cdots\!62}a-\frac{17\!\cdots\!03}{28\!\cdots\!81}$, $\frac{28\!\cdots\!67}{89\!\cdots\!92}a^{19}-\frac{10\!\cdots\!03}{89\!\cdots\!92}a^{18}-\frac{10\!\cdots\!45}{89\!\cdots\!92}a^{17}+\frac{27\!\cdots\!01}{89\!\cdots\!92}a^{16}-\frac{92\!\cdots\!81}{89\!\cdots\!92}a^{15}-\frac{22\!\cdots\!15}{89\!\cdots\!92}a^{14}+\frac{70\!\cdots\!27}{44\!\cdots\!96}a^{13}+\frac{56\!\cdots\!55}{44\!\cdots\!96}a^{12}-\frac{45\!\cdots\!87}{44\!\cdots\!96}a^{11}-\frac{18\!\cdots\!89}{44\!\cdots\!96}a^{10}+\frac{17\!\cdots\!57}{44\!\cdots\!96}a^{9}+\frac{39\!\cdots\!91}{44\!\cdots\!96}a^{8}-\frac{96\!\cdots\!45}{89\!\cdots\!92}a^{7}-\frac{94\!\cdots\!19}{89\!\cdots\!92}a^{6}+\frac{16\!\cdots\!83}{89\!\cdots\!92}a^{5}+\frac{35\!\cdots\!77}{89\!\cdots\!92}a^{4}-\frac{15\!\cdots\!85}{89\!\cdots\!92}a^{3}+\frac{89\!\cdots\!01}{89\!\cdots\!92}a^{2}-\frac{46\!\cdots\!09}{22\!\cdots\!48}a+\frac{86\!\cdots\!41}{56\!\cdots\!62}$, $\frac{66\!\cdots\!23}{28\!\cdots\!81}a^{19}-\frac{44\!\cdots\!53}{44\!\cdots\!96}a^{18}-\frac{76\!\cdots\!39}{28\!\cdots\!81}a^{17}+\frac{85\!\cdots\!03}{44\!\cdots\!96}a^{16}-\frac{86\!\cdots\!49}{22\!\cdots\!48}a^{15}-\frac{69\!\cdots\!17}{44\!\cdots\!96}a^{14}+\frac{59\!\cdots\!41}{11\!\cdots\!24}a^{13}+\frac{17\!\cdots\!87}{22\!\cdots\!48}a^{12}-\frac{27\!\cdots\!39}{11\!\cdots\!24}a^{11}-\frac{61\!\cdots\!87}{22\!\cdots\!48}a^{10}+\frac{11\!\cdots\!03}{11\!\cdots\!24}a^{9}+\frac{15\!\cdots\!85}{22\!\cdots\!48}a^{8}-\frac{30\!\cdots\!07}{11\!\cdots\!24}a^{7}-\frac{50\!\cdots\!17}{44\!\cdots\!96}a^{6}+\frac{67\!\cdots\!85}{11\!\cdots\!24}a^{5}+\frac{52\!\cdots\!35}{44\!\cdots\!96}a^{4}-\frac{18\!\cdots\!33}{22\!\cdots\!48}a^{3}-\frac{27\!\cdots\!33}{44\!\cdots\!96}a^{2}+\frac{18\!\cdots\!83}{28\!\cdots\!81}a-\frac{45\!\cdots\!79}{28\!\cdots\!81}$, $\frac{23\!\cdots\!65}{11\!\cdots\!24}a^{19}+\frac{15\!\cdots\!37}{44\!\cdots\!96}a^{18}-\frac{12\!\cdots\!73}{28\!\cdots\!81}a^{17}+\frac{80\!\cdots\!83}{44\!\cdots\!96}a^{16}+\frac{17\!\cdots\!05}{22\!\cdots\!48}a^{15}-\frac{51\!\cdots\!91}{44\!\cdots\!96}a^{14}-\frac{51\!\cdots\!13}{11\!\cdots\!24}a^{13}+\frac{13\!\cdots\!85}{22\!\cdots\!48}a^{12}+\frac{20\!\cdots\!61}{11\!\cdots\!24}a^{11}-\frac{35\!\cdots\!25}{22\!\cdots\!48}a^{10}-\frac{11\!\cdots\!56}{28\!\cdots\!81}a^{9}+\frac{87\!\cdots\!25}{22\!\cdots\!48}a^{8}+\frac{16\!\cdots\!03}{28\!\cdots\!81}a^{7}-\frac{22\!\cdots\!51}{44\!\cdots\!96}a^{6}-\frac{27\!\cdots\!21}{11\!\cdots\!24}a^{5}+\frac{31\!\cdots\!71}{44\!\cdots\!96}a^{4}-\frac{68\!\cdots\!61}{22\!\cdots\!48}a^{3}-\frac{10\!\cdots\!19}{44\!\cdots\!96}a^{2}+\frac{64\!\cdots\!97}{56\!\cdots\!62}a-\frac{42\!\cdots\!37}{28\!\cdots\!81}$, $\frac{14\!\cdots\!83}{89\!\cdots\!92}a^{19}+\frac{38\!\cdots\!81}{89\!\cdots\!92}a^{18}-\frac{34\!\cdots\!49}{89\!\cdots\!92}a^{17}+\frac{95\!\cdots\!45}{89\!\cdots\!92}a^{16}+\frac{72\!\cdots\!75}{89\!\cdots\!92}a^{15}-\frac{61\!\cdots\!63}{89\!\cdots\!92}a^{14}-\frac{28\!\cdots\!81}{44\!\cdots\!96}a^{13}+\frac{14\!\cdots\!55}{44\!\cdots\!96}a^{12}+\frac{14\!\cdots\!97}{44\!\cdots\!96}a^{11}-\frac{26\!\cdots\!17}{44\!\cdots\!96}a^{10}-\frac{49\!\cdots\!75}{44\!\cdots\!96}a^{9}+\frac{34\!\cdots\!51}{44\!\cdots\!96}a^{8}+\frac{24\!\cdots\!63}{89\!\cdots\!92}a^{7}+\frac{31\!\cdots\!81}{89\!\cdots\!92}a^{6}-\frac{41\!\cdots\!09}{89\!\cdots\!92}a^{5}+\frac{13\!\cdots\!85}{89\!\cdots\!92}a^{4}+\frac{46\!\cdots\!87}{89\!\cdots\!92}a^{3}-\frac{50\!\cdots\!39}{89\!\cdots\!92}a^{2}-\frac{76\!\cdots\!45}{22\!\cdots\!48}a+\frac{81\!\cdots\!07}{56\!\cdots\!62}$, $\frac{90\!\cdots\!97}{89\!\cdots\!92}a^{19}-\frac{35\!\cdots\!11}{89\!\cdots\!92}a^{18}-\frac{37\!\cdots\!55}{89\!\cdots\!92}a^{17}+\frac{68\!\cdots\!53}{89\!\cdots\!92}a^{16}+\frac{71\!\cdots\!13}{89\!\cdots\!92}a^{15}-\frac{55\!\cdots\!39}{89\!\cdots\!92}a^{14}-\frac{26\!\cdots\!59}{44\!\cdots\!96}a^{13}+\frac{14\!\cdots\!35}{44\!\cdots\!96}a^{12}+\frac{14\!\cdots\!39}{44\!\cdots\!96}a^{11}-\frac{48\!\cdots\!85}{44\!\cdots\!96}a^{10}-\frac{37\!\cdots\!13}{44\!\cdots\!96}a^{9}+\frac{12\!\cdots\!43}{44\!\cdots\!96}a^{8}+\frac{12\!\cdots\!69}{89\!\cdots\!92}a^{7}-\frac{42\!\cdots\!19}{89\!\cdots\!92}a^{6}+\frac{32\!\cdots\!17}{89\!\cdots\!92}a^{5}+\frac{47\!\cdots\!61}{89\!\cdots\!92}a^{4}-\frac{10\!\cdots\!75}{89\!\cdots\!92}a^{3}-\frac{32\!\cdots\!71}{89\!\cdots\!92}a^{2}+\frac{51\!\cdots\!43}{22\!\cdots\!48}a-\frac{18\!\cdots\!69}{56\!\cdots\!62}$, $\frac{21\!\cdots\!73}{22\!\cdots\!48}a^{19}-\frac{13\!\cdots\!69}{44\!\cdots\!96}a^{18}-\frac{58\!\cdots\!47}{22\!\cdots\!48}a^{17}+\frac{31\!\cdots\!95}{44\!\cdots\!96}a^{16}+\frac{67\!\cdots\!17}{11\!\cdots\!24}a^{15}-\frac{23\!\cdots\!97}{44\!\cdots\!96}a^{14}-\frac{94\!\cdots\!93}{22\!\cdots\!48}a^{13}+\frac{69\!\cdots\!32}{28\!\cdots\!81}a^{12}+\frac{52\!\cdots\!19}{22\!\cdots\!48}a^{11}-\frac{21\!\cdots\!49}{28\!\cdots\!81}a^{10}-\frac{16\!\cdots\!01}{22\!\cdots\!48}a^{9}+\frac{17\!\cdots\!63}{11\!\cdots\!24}a^{8}+\frac{17\!\cdots\!97}{11\!\cdots\!24}a^{7}-\frac{10\!\cdots\!79}{44\!\cdots\!96}a^{6}-\frac{10\!\cdots\!15}{56\!\cdots\!62}a^{5}+\frac{89\!\cdots\!33}{44\!\cdots\!96}a^{4}+\frac{25\!\cdots\!87}{22\!\cdots\!48}a^{3}-\frac{60\!\cdots\!19}{44\!\cdots\!96}a^{2}+\frac{37\!\cdots\!21}{11\!\cdots\!24}a-\frac{77\!\cdots\!24}{28\!\cdots\!81}$, $\frac{12\!\cdots\!09}{89\!\cdots\!92}a^{19}+\frac{19\!\cdots\!75}{89\!\cdots\!92}a^{18}-\frac{49\!\cdots\!67}{89\!\cdots\!92}a^{17}+\frac{22\!\cdots\!51}{89\!\cdots\!92}a^{16}+\frac{33\!\cdots\!89}{89\!\cdots\!92}a^{15}-\frac{22\!\cdots\!25}{89\!\cdots\!92}a^{14}-\frac{65\!\cdots\!97}{44\!\cdots\!96}a^{13}+\frac{68\!\cdots\!11}{44\!\cdots\!96}a^{12}+\frac{45\!\cdots\!97}{44\!\cdots\!96}a^{11}-\frac{24\!\cdots\!69}{44\!\cdots\!96}a^{10}+\frac{78\!\cdots\!01}{44\!\cdots\!96}a^{9}+\frac{60\!\cdots\!43}{44\!\cdots\!96}a^{8}-\frac{82\!\cdots\!63}{89\!\cdots\!92}a^{7}-\frac{17\!\cdots\!33}{89\!\cdots\!92}a^{6}+\frac{18\!\cdots\!17}{89\!\cdots\!92}a^{5}+\frac{11\!\cdots\!99}{89\!\cdots\!92}a^{4}-\frac{24\!\cdots\!19}{89\!\cdots\!92}a^{3}+\frac{12\!\cdots\!07}{89\!\cdots\!92}a^{2}-\frac{64\!\cdots\!63}{22\!\cdots\!48}a+\frac{12\!\cdots\!19}{56\!\cdots\!62}$
| 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: | \( 7144225924900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 7144225924900 \cdot 1}{2\cdot\sqrt{14595833152648338774900083206236247150592}}\cr\approx \mathstrut & 2.83536792936576 \end{aligned}\] (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 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464)
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 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464, 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 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464);
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 + 76*x^17 - 608*x^15 + 57*x^14 + 3116*x^13 - 10792*x^11 + 304*x^10 + 26600*x^9 - 1349*x^8 - 46436*x^7 + 8512*x^6 + 52060*x^5 - 17024*x^4 - 34504*x^3 + 26315*x^2 - 5892*x + 464);
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 | R | $20$ | $20$ | $18{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $20$ | $18{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $20$ | R | R | $18{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/37.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/43.2.0.1}{2} }^{9}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 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 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 19.19.19.1 | $x^{19} + 38 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ | |
|
\(23\)
| 23.2.1.1 | $x^{2} + 115$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |